Thanh Tuan Nguyen
email: tpnguyen@univ-tln.fr

Thanh Phuong Nguyen

Frédéric Bouchara
email: bouchara@univ-tln.fr

Dynamic texture representation based on oriented magnitudes of Gaussian gradients

Keywords: Dynamic textures, Gaussianfiltered derivatives, Oriented magnitudes, LBP, CLBP, Video representation

Efficiently capturing shape and turbulent motions of dynamic textures (DTs) for video description is a challenge in real applications due to the negative influences of the wellknown problems: environmental elements, illumination, scale, and noise. In this paper, we propose an efficient and simple framework for DT representation based on oriented features of high-order Gaussian gradients. Firstly, 2D/3D Gaussian-based filtering kernels in high-order partial derivatives are taken into account video analysis as a preprocessing to obtain corresponding gradient-filtered images/volumes. After that, oriented features, which are robust against above issues, are extracted by decomposing the Gaussian derivative magnitudes into oriented components. Finally, a shallow local encoding is utilized for structuring spatio-temporal features from these oriented magnitudes. This allows to construct discriminative descriptors with promising performances compared to those based on the non-oriented ones. Experimental results for DT classification task on benchmark datasets have verified the interest of our proposal.

Introduction

Dynamic textures (DTs) are textural characteristics repeated in a temporal domain. Understanding them in effect is one of crucial issues in many applications of computer vision, such as human interaction [START_REF] Ghodsi | Simultaneous joint and object trajectory templates for human activity recognition from 3-d data[END_REF][START_REF] Nguyen | Local derivative pattern for action recognition in depth images[END_REF][START_REF] Kiliboz | A hand gesture recognition technique for human-computer interaction[END_REF], detection and object tracking [START_REF] Nguyen | Spatial motion patterns: Action models from semi-dense trajectories[END_REF][START_REF] Tian | Weighted correlation filters guidance with spatial-temporal attention for online multi-object tracking[END_REF], background subtraction [START_REF] Jeyabharathi | Cut set-based dynamic key frame selection and adaptive layer-based background modeling for background subtraction[END_REF][START_REF] Srivastava | Salient object detection using background subtraction, gabor filters, objectness and minimum directional backgroundness[END_REF], crowded people [START_REF] Dehghan | Binary quadratic programing for online tracking of hundreds of people in extremely crowded scenes[END_REF][START_REF] Al-Zaydi | An adaptive people counting system with dynamic features selection and occlusion handling[END_REF], etc. Due to the turbulent motions along with impacts of environmental changes and illumination, efficiently capturing dynamic characteristics is a major challenge for DT representation. In order to deal with those problems, many techniques have been proposed and they can be categorized into the following groups.

Geometry-based methods: Based on fractal analysis, geometry-based methods attempt to deal with the influence of environmental changes. Dynamic Fractal Spectrum (DFS) [START_REF] Xu | Dynamic texture classification using dynamic fractal analysis[END_REF] and its variant (Multi-Fractal Spectrum (MFS) [START_REF] Xu | Scale-space texture description on sift-like textons[END_REF]) were introduced in order to take advantage of stochastic self-similarities and fractal features for DT representation. After that, Ji et al. [START_REF] Ji | Wavelet domain multifractal analysis for static and dynamic texture classification[END_REF] adapted the MFS model using wavelet coefficients to construct Wavelet-based MFS (WMFS) descriptor with more effectiveness. In addition, Quan et al. [START_REF] Quan | Spatiotemporal lacunarity spectrum for dynamic texture classification[END_REF] proposed a technique of lacunarity analysis, named Spatio-Temporal Lacunarity Spectrum (STLS), to extract lacunarity-based patterns for DT description. Baktashmotlagh et al. [START_REF] Baktashmotlagh | Discriminative non-linear stationary subspace analysis for video classification[END_REF] introduced Stationary Subspace Analysis (SSA) to extract stationary components in videos for DT encoding. In respect of DT recognition, experimental results have validated that the geometry-based methods seem to be adaptive for recognizing DTs with simple motions (e.g., those in UCLA [START_REF] Saisan | Dynamic texture recognition[END_REF] dataset), while being difficult to understand complex DTs (e.g., those in DynTex [START_REF] Péteri | Dyntex: A comprehensive database of dynamic textures[END_REF] and Dyn-Tex++ [START_REF] Ghanem | Maximum margin distance learning for dynamic texture recognition[END_REF] datasets). It may be due to a lack of the temporal information addressed in these analyses.

Optical-flow-based methods: Magnitudes and directions of the normal flow were taken into account video analysis in natural ways [START_REF] Péteri | Dynamic texture recognition using normal flow and texture regularity[END_REF][START_REF] Péteri | Qualitative characterization of dynamic textures for video retrieval[END_REF][START_REF] Peh | Synergizing spatial and temporal texture[END_REF]. However, a supposition of local smooth-ness and brightness in stability can be a limitation for dealing with the chaotic motions of DTs in videos [START_REF] Rivera | Spatiotemporal directional number transitional graph for dynamic texture recognition[END_REF]. Besides, dense trajectories and angles of DT motions were addressed in [START_REF] Nguyen | Directional beams of dense trajectories for dynamic texture recognition[END_REF] to capture directional dynamic features in various directions of motion points. Lu et al. [START_REF] Lu | Dynamic texture recognition by spatiotemporal multiresolution histograms[END_REF] exploited characteristics of velocity and acceleration in multi-resolution analysis to structure probability distributions for DT description. Nevertheless, just motion components were considered in [START_REF] Nguyen | Directional beams of dense trajectories for dynamic texture recognition[END_REF] and [START_REF] Lu | Dynamic texture recognition by spatiotemporal multiresolution histograms[END_REF], lack of textural appearance information for encoding spatial features.

Model-based methods: Most of them are based on Linear Dynamical System (LDS) [START_REF] Saisan | Dynamic texture recognition[END_REF] and its variants to address turbulent dynamic properties of DTs in videos. Kernel-PCA was exploited to adapt the LDS's observation in order to handle DTs with complex motions [START_REF] Chan | Classifying video with kernel dynamic textures[END_REF]. Chan et al. [START_REF] Mumtaz | Clustering dynamic textures with the hierarchical EM algorithm for modeling video[END_REF] introduced a model of DT mixtures (DTM) for addressing characteristics of movable objects in videos. The outcomes were then arranged into k clusters by employing a method of hierarchical expectationmaximization (HEM-DTM). In addition, other efforts also focused on the LDS's concept to be in accordance with analyzing DT features: bag-of-words (BoW) [START_REF] Wang | Chaotic features for dynamic textures recognition[END_REF], bag-of-systems (BoS) [START_REF] Ravichandran | View-invariant dynamic texture recognition using a bag of dynamical systems[END_REF], and BoS Tree [START_REF] Mumtaz | A scalable and accurate descriptor for dynamic textures using bag of system trees[END_REF]. With respect to their ability of classifying DTs, the model-based techniques have achieved moderate results due to without regard to dynamic features, one of important information for DT description [START_REF] Saisan | Dynamic texture recognition[END_REF]. Furthermore, the processes in constructing the models can become more complicated if the dynamic properties are taken into account [START_REF] Ravichandran | View-invariant dynamic texture recognition using a bag of dynamical systems[END_REF].

Learning-based methods: It can group them into two main kinds of approaches as follows. Deep learning techniques often utilize Convolutional Neural Networks (CNNs) to learn DT features in several ways. Such methods are Transferred Con-vNet Features (TCoF) [START_REF] Qi | Dynamic texture and scene classification by transferring deep image features[END_REF] -learning deep structures in still images; DT-CNN [START_REF] Andrearczyk | Convolutional neural network on three orthogonal planes for dynamic texture classification[END_REF] and PCANet-TOP [START_REF]Dynamic texture representation using a deep multi-scale convolutional network[END_REF] -learning DT features based on three orthogonal planes of sequences; D3 [START_REF] Hong | D3: recognizing dynamic scenes with deep dual descriptor based on key frames and key segments[END_REF] -concentrating on "key frames" and "key segments" of videos to extract static and dynamic patterns. Although the deep-learning methods achieved outstanding results in classifying DTs, they addressed tremendous parameters for the learning processes with high complexity of net-computing algorithms. This leads to a strict barrier for mobile applications in practice. The remain group learns dictionaries of DT features based on kernel sparse coding. Quan et al. [START_REF] Quan | Dynamic texture recognition via orthogonal tensor dictionary learning[END_REF] introduced a dictionary learned from atoms of sequences. In the meantime, an equiangular kernel was proposed in [START_REF] Quan | Equiangular kernel dictionary learning with applications to dynamic texture analysis[END_REF] to build a dictionary in reasonable dimension. Like the geometry-based approaches, the dictionary-based methods have arduously faced with "understanding" complex dynamic properties of DTs in DynTex [START_REF] Péteri | Dyntex: A comprehensive database of dynamic textures[END_REF] and DynTex++ [START_REF] Ghanem | Maximum margin distance learning for dynamic texture recognition[END_REF].

Filter-based methods: Thanks to robustness against changes of environmental elements, illumination and noise, filter-based methods have achieved potential results of DT recognition. Arashloo et al. [START_REF] Arashloo | Dynamic texture recognition using multiscale binarized statistical image features[END_REF] proposed to employ filters learned by transformation of ICA (independent component analysis). They then extracted Multi-scale Binarized Statistical Image Features based on three orthogonal planes of sequences (MBSIF-TOP). In the meanwhile, Zhao et al. [START_REF] Zhao | Dynamic texture classification using unsupervised 3d filter learning and local binary encoding[END_REF] utilized CLBP [START_REF] Guo | A completed modeling of local binary pattern operator for texture classification[END_REF] (Completed Local Binary Pattern) to capture spatio-temporal characteristics from 3D filtered volumes. Therein, the 3D filters were learned from various unsupervised techniques: PCA (Principal Component Analysis), ICA, sparse filtering, and k-means clustering. Recently, Nguyen et al. [START_REF] Nguyen | Completed statistical adaptive patterns on three orthogonal planes for recognition of dynamic textures and scenes[END_REF][START_REF] Nguyen | Volumes of blurredinvariant gaussians for dynamic texture classification[END_REF][START_REF] Nguyen | Smooth-invariant gaussian features for dynamic texture recognition[END_REF] addressed filtering methods as a pre-processing to mitigate the negative impacts of environmental changes, illumination noise on DT encoding: moment image model [START_REF] Nguyen | Completed statistical adaptive patterns on three orthogonal planes for recognition of dynamic textures and scenes[END_REF] -a filtering technique based on pre-defined supporting regions; Gaussian-based filtering kernels [START_REF] Nguyen | Volumes of blurredinvariant gaussians for dynamic texture classification[END_REF][START_REF] Nguyen | Smooth-invariant gaussian features for dynamic texture recognition[END_REF]. Experimental results for DT classification have validated that the filter-based methods seem to work well for describing simple motions rather than for complicated ones.

Local-feature-based methods: Most of them are based on Local Binary Pattern (LBP) [START_REF] Ojala | Multiresolution gray-scale and rotation invariant texture classification with local binary patterns[END_REF] and its completed model (CLBP [START_REF] Guo | A completed modeling of local binary pattern operator for texture classification[END_REF]) to encode shape and motion clues for DT representation. Zhao et al. [START_REF] Zhao | Dynamic texture recognition using local binary patterns with an application to facial expressions[END_REF] introduced two approaches taking advantage of LBP to investigate local relationships in spatio-temporal domain of video analysis as follows. For a video, Volume-LBP (VLBP) patterns were formed by using LBP on its three consecutive frames while LBP-TOP patterns were computed by exploiting LBP on its three orthogonal planes. Motivated by VLBP and LBP-TOP, many efforts have addressed LBP's conventional limitations in order to enhance the discrimination power: rotation-invariant problems [START_REF] Zhao | Rotation-invariant image and video description with local binary pattern features[END_REF], sensitivity to noise, and near-uniform regions [START_REF] Tiwari | Dynamic texture recognition based on completed volume local binary pattern[END_REF][START_REF] Tiwari | A novel scheme based on local binary pattern for dynamic texture recognition[END_REF][START_REF] Nguyen | Completed local structure patterns on three orthogonal planes for dynamic texture recognition[END_REF][START_REF] Nguyen | Completed statistical adaptive patterns on three orthogonal planes for recognition of dynamic textures and scenes[END_REF]. Furthermore, Ren et al. [START_REF] Ren | Optimizing LBP structure for visual recognition using binary quadratic programming[END_REF] proposed data-driven LBP (DDLBP) features to deal with problems of grand dimension, while PCA was involved in local encoding to eliminate noise features [START_REF] Ren | Dynamic texture recognition using enhanced LBP features[END_REF].

In spite of having promising performances, the local-featurebased techniques [START_REF] Tiwari | Dynamic texture recognition based on completed volume local binary pattern[END_REF][START_REF] Tiwari | A novel scheme based on local binary pattern for dynamic texture recognition[END_REF][START_REF] Nguyen | Completed local structure patterns on three orthogonal planes for dynamic texture recognition[END_REF][START_REF] Nguyen | Completed statistical adaptive patterns on three orthogonal planes for recognition of dynamic textures and scenes[END_REF]] have yet encountered with the well-known issues of DT representation: environmental elements, illumination, and noise. To mitigate those problems, Nguyen et al. [START_REF] Nguyen | Volumes of blurredinvariant gaussians for dynamic texture classification[END_REF][START_REF] Nguyen | Smooth-invariant gaussian features for dynamic texture recognition[END_REF][START_REF] Nguyen | Rubik gaussian-based patterns for dynamic texture classification[END_REF] used Gaussian-based filtering kernels for denosing before encoding local spatio-temporal features of DTs. In other aspects, Nguyen et al. [START_REF] Nguyen | Completed statistical adaptive patterns on three orthogonal planes for recognition of dynamic textures and scenes[END_REF][START_REF] Nguyen | Momental directional patterns for dynamic texture recognition[END_REF] introduced local DT features structured from moment-filtered outcomes, while other efforts [START_REF] Arashloo | Dynamic texture recognition using multiscale binarized statistical image features[END_REF][START_REF] Zhao | Dynamic texture classification using unsupervised 3d filter learning and local binary encoding[END_REF] addressed the learning-based filtered features. The abilities of the achieved descriptors for DT classification task are encouraging in comparison with other local-feature-based attempts. However, the conventional limitations seem not to be thoroughly dealt with. To this end, we propose in this work a novel approach for exploiting oriented local features containing rich textural information. An efficient and simple framework is then introduced for DT representation based on local patterns of high-order oriented magnitudes. This allows to efficiently reduce the negative impacts of above problems on capturing shape and turbulent motions of DTs in local regions. Contrary to the complicated models of deep-learning-based methods, our proposal can have competitive performances by just using the shallow analysis for encoding spatio-temporal oriented magnitudes.

Generally, our proposed framework takes the following steps for DT description in effect. Firstly, k-order gradients of 2D (resp. 3D) Gaussian filtering kernels are taken into account video analysis as a preprocessing to obtain corresponding gradient-filtered images I • To the best of our knowledge, it is the first time that the oriented features of high-order Gaussian-gradient magnitudes are exploited to make DT representation more robust against above typical issues. • Multi-order of gradients and multi-scale analysis of Gaussian-based filtering are also addressed to forcefully investigate benefits of informative magnitudes. • A modified soft-assignment is introduced to efficiently quantize the Gaussian-gradient magnitudes subject to a pre-defined orientation in comprehensive comparison with the traditional quantification models of oriented features. • Based on the oriented features of Gaussian gradients, discriminative IOM/VOM-based descriptors are structured by just using a simple operator. Furthermore, our experiments have also proved the advantages of the oriented magnitudes for DT representation using LBP-based variants compared to non-oriented ones involved in. • In reasonable dimension, our proposed descriptors perform well in comparison with all non-deep-learning approaches, while being very close to performances of deeplearning models.

∂x k i σ (resp. volumes V ∂x k i σ

Related works

A brief of LBP and its completed model

Let I denote a gray-scale textural image. In consideration of relationships between a center pixel q ∈ I and its local neighbors, Ojala et al. [START_REF] Ojala | Multiresolution gray-scale and rotation invariant texture classification with local binary patterns[END_REF] introduced a LBP pattern as a binary string by measuring differences of their intensities as LBP P,R (q) = s I(p i ) -I(q) P i=1 [START_REF] Ghodsi | Simultaneous joint and object trajectory templates for human activity recognition from 3-d data[END_REF] where I(.) points out the gray-value of a pixel; {p i } P i=1 (P ∈ Z + ) is a collection of P neighbors which are interpolated by a circle sample with center q and radius R; and s(.) is defined as

s(x) =        1, if x ≥ 0 0, otherwise. (2)
As a result, a histogram of 2 P bins is structured to describe a textural image. This leads to a restriction for real implementations. In practice, two following mapping techniques are usually addressed to overcome this curse of dimension: u2 mapping with P(P -1) + 3 bins for uniform patterns, riu2 mapping with P + 2 bins for rotation-invariant uniform patterns. Furthermore, other mappings can be also considered: T AP A mapping [START_REF] Nguyen | Topological attribute patterns for texture recognition[END_REF] for topological features, LBC [START_REF] Zhao | Completed Local Binary Count for Rotation Invariant Texture Classification[END_REF] -an alternative of riu2.

In order to conduct the LBP encoding in diversity, Guo et al. [START_REF] Guo | A completed modeling of local binary pattern operator for texture classification[END_REF] presented completed model of LBP (CLBP). Principally, CLBP consists of three components integrated into various ways to enhance the performance: CLBP S is identical to the typical LBP; CLBP M captures informative magnitudes; and CLBP C measures the global gray-differences of center pixels. In practice, two following integration types are often used due to their better performances: "S M/C" means jointing CLBP M and CLBP C patterns before concatenating with CLBP S ; "S/M/C" denotes a 3D joint of those components.

Gaussian filtering kernel and its derivatives

A well-known Gaussian filtering is a process of convolving a µ-dimensional Gaussian kernel on a spatial domain. Its results agree with the Gaussian distribution. The Gaussian filtering kernel is defined in general as

G µ σ (γ µ ) = 1 (σ √ 2π) µ exp - x 2 1 + x 2 2 + ... + x 2 µ 2σ 2 (3)
in which γ µ = {x i } µ i=1 denotes a collection of µ spatial directions, σ is a pre-defined standard deviation. Appropriately, a k-order

(k ∈ Z + ) partial derivative of G µ σ (γ µ ) is calculated with respect to a direction x i ∈ γ µ as G µ σ,∂x k i (γ µ ) = ∂ k G µ σ (γ µ ) ∂x k i (4)
in which "∂" denotes a gradient operation.

Exploiting oriented features

Oriented features play an important role in representation of local features. Gabor filter [START_REF] Jain | Unsupervised texture segmentation using gabor filters[END_REF] has been early used to extract oriented features in textural analysis. Dalal and Triggs [54] presented histograms of oriented gradient on each local patch to form HoG descriptor for pedestrian detection. Inspired by this well-known descriptor, many other works have been introduced to deal with different problems. The oriented features have been also utilized for key-point description in different detectors, e.g., SIFT detector [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF]. Exploiting this kind of features from local patch around keypoints leads to a powerful description of detected keypoints making these detectors be effective in various applications of computer vision in the years of 2000s.

Proposed method

In our prior works [START_REF] Nguyen | Volumes of blurredinvariant gaussians for dynamic texture classification[END_REF][START_REF] Nguyen | Smooth-invariant gaussian features for dynamic texture recognition[END_REF][START_REF] Nguyen | Rubik gaussian-based patterns for dynamic texture classification[END_REF], we have indicated that taking Gaussian-based filtering kernels into account DT representation could improve the discrimination power of local DT encoding. This is thanks to mitigating the negative impacts the typical problems on DT encoding. However, the achieved improvements are still at a moderate level since those problems may not be dealt with thoroughly. Instead of exploiting Gaussianbased filtered features as in [START_REF] Nguyen | Volumes of blurredinvariant gaussians for dynamic texture classification[END_REF][START_REF] Nguyen | Smooth-invariant gaussian features for dynamic texture recognition[END_REF], this work is motivated by HoG descriptor [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] where oriented information has been successfully exploited for local feature representation. We propose an efficient framework for DT representation based on highorder oriented magnitudes that are decomposed from Gaussiangradient outcomes, as graphically illustrated in Fig. 1. Accordingly, high-order Gaussian-gradient kernels are used to filter a given video for noise reduction. Magnitude features are then extracted from the gradient-filtered outcomes. Different decomposing models are then addressed to separate these obtained magnitude features into oriented magnitudes subject to a given orientation range (see Section 3.1). Finally, robust descriptors are structured by using a simple local operator to encode the oriented magnitudes (see Section 3.2). Experiments for DT classification have validated the good performance of oriented magnitudes compared to Gaussian-based filtered features in [START_REF] Nguyen | Volumes of blurredinvariant gaussians for dynamic texture classification[END_REF][START_REF] Nguyen | Smooth-invariant gaussian features for dynamic texture recognition[END_REF] (see Section 4.5). Hereafter, we express above proposed processes in detail.

Oriented magnitudes of Gaussian gradients

In order to compute Gaussian-oriented magnitudes, we conduct the kernel G µ σ,∂x k i in 2D and 3D filtering dimensions, i.e., G 2D/3D σ,∂x k i . Appropriately, for a given image I, a pixel q ∈ I is filtered by the 2D filtering kernel with respect to spatial coordinates (x, y) as

       I ∂x k σ (q) = G 2D σ,∂x k (x, y) * I(q) I ∂y k σ (q) = G 2D σ,∂y k (x, y) * I(q) (5)
in which "*" denotes a convolving operator; I ∂x k σ and I ∂y k σ are k-order Gaussian-filtered images. Similarly, for a given video V, a voxel u ∈ V is filtered by the 3D filtering kernel with respect to spatial coordinates (x, y) and temporal direction z as

             V ∂x k σ (u) = G 3D σ,∂x k (x, y, z) * V(u) V ∂y k σ (u) = G 3D σ,∂y k (x, y, z) * V(u) V ∂z k σ (u) = G 3D σ,∂z k (x, y, z) * V(u) (6)
where V ∂x k σ , V ∂y k σ , and V ∂z k σ are k-order filtered volumes. Based on above k-order Gaussian-filtered images/volumes, we correspondingly propose 2D/3D oriented magnitudes which are decomposed subject to a direction range. In order to thoroughly investigate the influences of the decomposing process, the following quantification strategies are addressed as Quantification strategies: In consideration of an uniform quantification of an oriented feature f , which is defined at an arbitrary pixel q as f (q), into n bins, it can be decomposed into two components: orientation f (q) ∈ [0, 2π) and magnitude f (q) . Let us suppose that (i -1)λ ≤ f (q) < iλ, where i ∈ {1, 2, .., n} and λ = 2π n . We investigate hereunder 3 following quantification modes for decomposition of an image of oriented features. The two first modes are often used in the literature of feature quatification, while the last one is our proposal for this task. Traditional methods address two possible strategies for decomposition of f into n images of oriented features: {m i } n i=1 . • Hard assignment: f (q) is totally assigned to pixel q of image m i with value f (q) . It means as

       m i (q) = f (q) m j (q) = 0 ∀ j i (7)
• Soft assignment: f (q) is partially assigned to pixel q of image m i with value iλ-f (q) λ f (q) and to pixel q of image m i+1 with value f (q)-(i-1)λ λ f (q) , where m n+1 ≡ m 1 . It means as

             m i (q) = iλ-f (q) λ f (q) m i+1 (q) = f (q)-(i-1)λ λ f (q) m j (q) = 0, where j {i, i + 1} (8)
We introduce in this work an another version of soft assignment, called modified soft assignment, which allows to quantize f (q) into 2n bins {m + i , m - i } n i=1 as follows.

• Modified soft assignment: f (q) is partially assigned to pixel q of image m + i with value iλ-f (q) λ f (q) and to pixel q of image m - i+1 with value f (q)-(i-1)λ λ f (q) , where m - n+1 ≡ m - 1 . It means as

                   m + i (q) = iλ-f (q) λ f (q) m + j (q) = 0 ∀ j i m - i+1 (q) = f (q)-(i-1)λ λ f (q) m - j (q) = 0 ∀ j i + 1 (9)
The main difference between the soft assignment and our modified model is that for n ranges of orientations, the first one produces n bins while the second one generates 2n bins. In other words, each bin m i in the typical approach is now separated into 2 components (m + i and m - i ) to express the quantized feature with more discriminative power in the new approach 1 .

Decomposition of gradient-filtered images I ∂x k σ and I ∂y k σ : Following the quantification strategies presented in the previous section, we introduce herefater the decomposition of gradientfiltered images. The high-order oriented magnitude of a pixel q ∈ I is determined so that its gradient direction is agreed with a given range of direction d = [α, β) = [(i -1)λ, iλ), where λ = 2π n , α = (i -1)λ, and β = iλ, i ∈ {1, 2, .., n}. Let us suppose that θ ∂x k ,∂y k σ (q) ∈ d. Accordingly, a feature of Image of Oriented Magnitudes (IOM) could be quantified by the hard-assignment principle (also see Eq. 7) as

HIOM ∂x k ,∂y k σ,i (q) = ||∇I ∂x k ,∂y k σ (q)||, so that θ ∂x k ,∂y k σ (q) ∈ d (10)
by the soft-assignment (also see Eq. 8) as [START_REF] Xu | Scale-space texture description on sift-like textons[END_REF] and by the modified soft-assignment (also see Eq. 9) as

         SIOM ∂x k ,∂y k σ,i (q) = ||∇I ∂x k ,∂y k σ (q)|| × β-θ ∂x k ,∂y k σ (q) β-α SIOM ∂x k ,∂y k σ,i+1 (q) = ||∇I ∂x k ,∂y k σ (q)|| × θ ∂x k ,∂y k σ (q)-α β-α
         pMSIOM ∂x k ,∂y k σ,i (q) = ||∇I ∂x k ,∂y k σ (q)|| × β-θ ∂x k ,∂y k σ (q) β-α nMSIOM ∂x k ,∂y k σ,i+1 (q) = ||∇I ∂x k ,∂y k σ (q)|| × θ ∂x k ,∂y k σ (q)-α β-α (12)
where SIOM ∂x k ,∂y k σ,n+1 (q) ≡ SIOM ∂x k ,∂y k σ,1

(q), nMSIOM ∂x k ,∂y k σ,n+1 (q) ≡ nMSIOM ∂x k ,∂y k σ,1 (q), and ||∇I ∂x k ,∂y k σ (q)|| denotes the k-order magnitude information of q and is calculated as follows.

||∇I ∂x k ,∂y k σ (q)|| = I x k σ (q) 2 + I y k σ (q) 2 (13)
In the meanwhile, θ ∂x k ,∂y k σ (q) denotes the gradient direction of pixel q and is inferred as

θ ∂x k ,∂y k σ (q) = arctan I ∂y k σ (q)/I ∂x k σ (q) (14)
Let us consider an intuitive example of decomposition in Fig. 2 which graphically illustrates an instance of decomposing the magnitudes of two Gaussian-gradient images I ∂x 1 0.5 and I ∂y 1 0.5 in order to obtain 4 HIOM images subject to a set of 4 equal ranges of direction

D 4 = {[0, π/2), [π/2, π), [π, 3π/2), [3π/2, 2π)}.
Decomposition of gradient-filtered volumes V ∂x k σ , V ∂y k σ , and

V ∂z k σ :
The high-order oriented magnitudes of a voxel u ∈ V

             HVOM ∂x k ,∂y k σ,i (u) = ||∇V ∂x k ,∂y k σ (u)||, so that φ ∂x k ,∂y k σ (u) ∈ d HVOM ∂y k ,∂z k σ,i (u) = ||∇V ∂y k ,∂z k σ (u)||, so that φ ∂y k ,∂z k σ (u) ∈ d HVOM ∂z k ,∂x k σ,i (u) = ||∇V ∂z k ,∂x k σ (u)||, so that φ ∂z k ,∂x k σ (u) ∈ d (15) by the soft-assignment as                                          SVOM ∂x k ,∂y k σ,i (u) = ||∇V ∂x k ,∂y k σ (u)|| × β-φ ∂x k ,∂y k σ (u) β-α SVOM ∂x k ,∂y k σ,i+1 (u) = ||∇V ∂x k ,∂y k σ (u)|| × φ ∂x k ,∂y k σ (u)-α β-α SVOM ∂y k ,∂z k σ,i (u) = ||∇V ∂y k ,∂z k σ (u)|| × β-φ ∂y k ,∂z k σ (u) β-α SVOM ∂y k ,∂z k σ,i+1 (u) = ||∇V ∂y k ,∂z k σ (u)|| × φ ∂y k ,∂z k σ (u)-α β-α SVOM ∂z k ,∂x k σ,i (u) = ||∇V ∂z k ,∂x k σ (u)|| × β-φ ∂z k ,∂x k σ (u) β-α SVOM ∂z k ,∂x k σ,i+1 (u) = ||∇V ∂z k ,∂x k σ (u)|| × φ ∂z k ,∂x k σ (u)-α β-α (16) where SVOM ∂z k ,∂x k σ,n+1 (u) ≡ SVOM ∂z k ,∂x k σ,1 (u), SVOM ∂y k ,∂z k σ,n+1 (u) ≡ SVOM ∂y k ,∂z k σ,1 (u), and SVOM ∂x k ,∂y k σ,n+1 (u) ≡ SVOM ∂x k ,∂y k σ,1 (u)
. In the meanwhile, a feature of VOM can be quantified to two bins by the modified soft-assignment as

                                         pMSVOM ∂x k ,∂y k σ,i (u) = ||∇V ∂x k ,∂y k σ (u)|| × β-φ ∂x k ,∂y k σ (u) β-α nMSVOM ∂x k ,∂y k σ,i+1 (u) = ||∇V ∂x k ,∂y k σ (u)|| × φ ∂x k ,∂y k σ (u)-α β-α pMSVOM ∂y k ,∂z k σ,i (u) = ||∇V ∂y k ,∂z k σ (u)|| × β-φ ∂y k ,∂z k σ (u) β-α nMSVOM ∂y k ,∂z k σ,i+1 (u) = ||∇V ∂y k ,∂z k σ (u)|| × φ ∂y k ,∂z k σ (u)-α β-α pMSVOM ∂z k ,∂x k σ,i (u) = ||∇V ∂z k ,∂x k σ (u)|| × β-φ ∂z k ,∂x k σ (u) β-α nMSVOM ∂z k ,∂x k σ,i+1 (u) = ||∇V ∂z k ,∂x k σ (u)|| × φ ∂z k ,∂x k σ (u)-α β-α (17) in which nMSVOM ∂z k ,∂x k σ,n+1 (u) ≡ nMSVOM ∂z k ,∂x k σ,1 (u), nMSVOM ∂y k ,∂z k σ,n+1 (u) ≡ nMSVOM ∂y k ,∂z k σ,1
(u), and

nMSVOM ∂x k ,∂y k σ,n+1 (u) ≡ nMSVOM ∂x k ,∂y k σ,1 (u).
Here, the k-order magnitudes

||∇V ∂x k ,∂y k σ (u)||, ||∇V ∂y k ,∂z k σ (u)||, and ||∇V ∂z k ,∂x k σ (u)|| are computed as                  ||∇V ∂x k ,∂y k σ (u)|| = V x k σ (u) 2 + V y k σ (u) 2 ||∇V ∂y k ,∂z k σ (u)|| = V y k σ (u) 2 + V z k σ (u) 2 ||∇V ∂z k ,∂x k σ (u)|| = V z k σ (u) 2 + V x k σ (u) 2 (18)
In order to illustrate the decomposition of gradient-filtered volumes, Fig. 3 shows an example of computing magnitude volumes of Gaussian gradients. Gradient directions φ ∂x k ,∂y k σ (u), φ ∂y k ,∂z k σ (u), and φ ∂z k ,∂x k σ (u) are inferred as

             φ ∂x k ,∂y k σ (u) = arctan V ∂y k σ (u)/V ∂x k σ (u) φ ∂y k ,∂z k σ (u) = arctan V ∂z k σ (u)/V ∂y k σ (u) φ ∂z k ,∂x k σ (u) = arctan V ∂x k σ (u)/V ∂z k σ (u) (19)
In the meanwhile, Fig. 5 graphically illustrates a general model of decomposing a volume of magnitude features.

It can be seen that for a given direction range, the modified soft decomposition has produced a double number of oriented magnitude outcomes compared to the hard-assignment and the classic soft-assignment. For convenience in further presentation, we could generally refer the above decomposing results: HIOM/SIOM/MSIOM as IOM-based images, HVOM/SVOM/MSVOM as VOM-based volumes.

DT representation based on oriented magnitudes

In order to generally investigate oriented magnitudes for DT representation, we address the IOM and VOM computations in n (n ∈ Z + ) equal ranges of direction as

D n = [(i -1)λ, iλ) n i=1 , where λ = 2π
n denotes an angle coefficient for decomposing the k-order image/volume magnitudes. For example, with respect to λ = π/2, we have n = 4 direction ranges in equality (i.e., Proposed IOM-based descriptors: To be compliant with the k-order 2D Gaussian-gradient filtering, a given video V is separated subject to its three orthogonal planes {XY, XT, YT } to obtain corresponding collections of plane-images f XY , f XT , and f YT . For the plane-image collection f XY , its spatial HIOM, SIOM, MSIOM features of DTs are respectively encoded as

D 4 = {[0, π/2), [π/2, π), [π, 3π/2), [3π/2, 2π)}). In
Γ k,D n σ ( f XY ) = 1 N XY I∈ f XY ξ HIOM ∂x k ,∂y k σ,1 (I) , ..., ξ HIOM ∂x k ,∂y k σ,n (I) (20)
and

Υ k,D n σ ( f XY ) = 1 N XY I∈ f XY ξ SIOM ∂x k ,∂y k σ,1 (I) , ξ SIOM ∂x k ,∂y k σ,2 (I) , ..., ξ SIOM ∂x k ,∂y k σ,n (I) (21) and Ω k,D n σ ( f XY ) = 1 N XY I∈ f XY ξ pMSIOM ∂x k ,∂y k σ,1 (I) , ξ nMSIOM ∂x k ,∂y k σ,1 (I) , ..., ξ pMSIOM ∂x k ,∂y k σ,n (I) , ξ nMSIOM ∂x k ,∂y k σ,n (I) (22)
in which N XY means a number of plane-images in f XY ; ξ(.) denotes a simple function using a local operator (e.g., LBP [START_REF] Ojala | Multiresolution gray-scale and rotation invariant texture classification with local binary patterns[END_REF],

CLBP [START_REF] Guo | A completed modeling of local binary pattern operator for texture classification[END_REF], etc.) in order to figure out the corresponding histograms. Fig. 4 illustrates a graphical view of filtering an input image, hard-decomposing its filtered magnitudes, and encoding the obtained HIOM outcomes correspondingly. In similarity, these encodings could be used for the remaining plane-image collections f XT and f YT to capture temporal IOM-based features for DT representation. As a result, robust local descriptors are structured in simplicity by concatenating the probability distributions of Γ k,D n σ (.), Υ k,D n σ (.), and Ω k,D n σ (.) as

HIOMF k,D n σ (V) = Γ k,D n σ ( f XY ), Γ k,D n σ ( f XT ), Γ k,D n σ ( f YT ) (23)
and

SIOMF k,D n σ (V) = Υ k,D n σ ( f XY ), Υ k,D n σ ( f XT ), Υ k,D n σ ( f YT ) (24)
and

MSIOMF k,D n σ (V) = Ω k,D n σ ( f XY ), Ω k,D n σ ( f XT ), Ω k,D n σ ( f YT ) (25)
Proposed VOM-based descriptors: As mentioned in Section 3.1 for the hard decomposition (refer to Eq. 15), three filtered volumes of oriented magnitudes are pointed out corresponding to three pairs of spacial domains convolved on a given video V. Those volumes are taken into account local analysis to construct a robust descriptor as follows. For an obtained volume HVOM ∂x k ,∂y k σ,i

, (i ∈ {1, 2, .., n}), it is firstly split into collections of filtered plane-images ( f XY , f XT , and f YT ) subject to its three orthogonal planes {XY, XT, YT }. The simple operator ξ(.) is then utilized to capture local spatio-temporal features of DTs as

Ψ(HVOM ∂x k ,∂y k σ,i ) = I∈ f XY ξ(I) N XY , I∈ f XT ξ(I) N XT , I∈ f YT ξ(I)
N YT [START_REF] Wang | Chaotic features for dynamic textures recognition[END_REF] in which N XY , N XT , and N YT are numbers of plane-images f XY , f XT , and f YT of HVOM ∂x k ,∂y k σ,i respectively. Fig. 5 illustrates a graphical view of encoding a HVOM volume. This encoding is similarly deployed for the remaining volumes HVOM ∂y k ,∂z k σ,i and HVOM ∂z k ,∂x k σ,i . As a result, a discriminative descriptor based on the k-order HVOM features is constructed by concatenating these obtained histograms as

HVOMF k,D n σ (V) = Ψ(HVOM ∂x k ,∂y k σ,i ), Ψ(HVOM ∂y k ,∂z k σ,i
),

Ψ(HVOM ∂z k ,∂x k σ,i ) n i=1 (27)
in which denotes a concatenating function of histograms.

Similarly, this HVOMF encoding could be applied to 3 SVOM (resp. 6 MSVOM) outcomes extracted by the soft decomposition (refer to Eq. 16) subject to the direction range D n . Accordingly, other robust descriptors based on the k-order SVOM (resp. MSVOM) features are formed by concatenating the corresponding histograms as

SVOMF k,D n σ (V) = Ψ(SVOM ∂x k ,∂y k σ,i ), Ψ(SVOM ∂y k ,∂z k σ,i
),

Ψ(SVOM ∂z k ,∂x k σ,i ) n i=1 (28)
and

MSVOMF k,D n σ (V) = Ψ(pMSVOM ∂x k ,∂y k σ,i
),

Ψ(nMSVOM ∂x k ,∂y k σ,i
),

Ψ(pMSVOM ∂y k ,∂z k σ,i
),

Ψ(nMSVOM ∂y k ,∂z k σ,i
),

Ψ(pMSVOM ∂z k ,∂x k σ,i
),

Ψ(nMSVOM ∂z k ,∂x k σ,i ) n i=1 (29)
Our proposed IOM/VOM-based descriptors take the following benefits to improve the performance compared to other local Gaussian-based descriptors (also see Sections 4.3, 4.5 for comprehensive evaluations):

• Different from exploiting Gaussian-based filtered features to construct local descriptors FoSIG [START_REF] Nguyen | Smooth-invariant gaussian features for dynamic texture recognition[END_REF] and V-BIG [START_REF] Nguyen | Volumes of blurredinvariant gaussians for dynamic texture classification[END_REF], in this work, the high-order oriented magnitudes are taken into account DT representation. Thanks to the decomposing models presented in Section 3.1, the magnitudes of Gaussian-gradient-filtered outcomes are addressed in diversity of invariant features to enhance the robustness against the well-known issues in more effect. In the mean while, exploiting oriented features makes those outcomes still more discriminative for texture description. • The Gaussian-gradient filterings allow to produce more filtered outcomes for the DT encoding. In the meanwhile, just one DoG-based element was used in FoSIG [START_REF] Nguyen | Smooth-invariant gaussian features for dynamic texture recognition[END_REF] and V-BIG [START_REF] Nguyen | Volumes of blurredinvariant gaussians for dynamic texture classification[END_REF] due to taking the Different of Gaussians (DoG) kernel into account the filterings. • To enhance the discrimination power, it is possible to address the IOM/VOM-based descriptors for a multianalysis of high-orders along with different Gaussian filtering scales, while keeping their representation in reasonable dimensions thanks to the tiny size of single-scale ones (see Table 2). In the meantime, just single-scale of Gaussian filtering was addressed in FoSIG [START_REF] Nguyen | Smooth-invariant gaussian features for dynamic texture recognition[END_REF] and V-BIG [START_REF] Nguyen | Volumes of blurredinvariant gaussians for dynamic texture classification[END_REF]. • It should be noted that the 2D-magnitude information (i.e., non-decomposition applied to) is also exploited in [START_REF] Song | Texture representation using local binary encoding across scales, frequency bands and image domains[END_REF] for structuring textual images. However, taking it into account DT representation is not more adaptive than taking its oriented properties (see Table 3 for a fact of this statement). It has proved the interest of our proposed framework.

Experiments and evaluations

Datasets and protocols

Hereafter, benchmark datasets and protocols for evaluating our proposal are detailed. A brief of their properties is then shown at a glance in Table 1.

UCLA dataset: It has 200 DT videos fixed in 110 × 160 × 75 dimension [START_REF] Saisan | Dynamic texture recognition[END_REF]. Those mainly characterize disorder motions of waterfall, plant, flower, fountain, fire, boiling water, etc. (see Fig. 6 for several instances of them). For DT classification task, UCLA is often organized in challenging scenarios as follows:

... ... 8)", "fire(8)", "flowers( 12)", "fountains(20)", "plants(108)", "sea(12)", "smoke(4)", "water( 12)", and "waterfall( 16)", where the numbers in parentheses denote quantities for the corresponding classes. Because of the dominance of "plants(108)", it is removed to form 8-class scheme with more challenges [START_REF] Ravichandran | View-invariant dynamic texture recognition using a bag of dynamical systems[END_REF][START_REF] Xu | Dynamic texture classification using dynamic fractal analysis[END_REF]. In order to evaluate DT classification in two schemes, following to the experimental protocol in [START_REF] Ghanem | Maximum margin distance learning for dynamic texture recognition[END_REF][START_REF] Tiwari | A novel scheme based on local binary pattern for dynamic texture recognition[END_REF][START_REF] Nguyen | Smooth-invariant gaussian features for dynamic texture recognition[END_REF], a half of DT sequences in each class is randomly addressed for testing and the remain for training. The average of 20 trials for each scheme is reported as a final rate. 7 for several DT samples). For DT classification, it is often arranged into the following subsets. The LOO protocol [START_REF]Dynamic texture representation using a deep multi-scale convolutional network[END_REF][START_REF] Dubois | Characterization and recognition of dynamic textures based on the 2d+t curvelet transform[END_REF][START_REF] Nguyen | Volumes of blurredinvariant gaussians for dynamic texture classification[END_REF] is utilized to evaluate the performances.

• DynTex35 is composed by splitting from 35 DynTex videos as follows. Each video is split into 8 non- Note: LOO and 4fold are leave-one-out and four cross-fold validation respectively. 50%/50% denotes a protocol of taking randomly 50% samples for training and the remain (50%) for testing.

overlapping sub-DTs at random cutting points with respect to axes X, Y, and T, but not half of those. For instance, cutting coordinates can be x = 170, y = 130, and t = 100 as in [START_REF] Zhao | Dynamic texture recognition using local binary patterns with an application to facial expressions[END_REF][START_REF] Arashloo | Dynamic texture recognition using multiscale binarized statistical image features[END_REF][START_REF] Tiwari | A novel scheme based on local binary pattern for dynamic texture recognition[END_REF][START_REF] Nguyen | Completed statistical adaptive patterns on three orthogonal planes for recognition of dynamic textures and scenes[END_REF][START_REF] Nguyen | Smooth-invariant gaussian features for dynamic texture recognition[END_REF]. In addition, two more sub-DTs are also collected subject to the T axis of the splitting process. As a result, 10 sub-videos for each of 35 videos are obtained in different spatio-temporal dimensions to form a challenging scheme with 35 categories. • Alpha includes 60 DT videos which are grouped into 3 classes: "grass", "sea", and "trees". Each category has 20 sequences. • Beta includes 162 DT videos which are grouped into 10 categories: "sea(20)", "vegetation(20)", "trees( 20)", flags(20)", "calm water(20)", "fountains(20)", "traffic( 9)", "smoke( 16)", "escalator( 7)", and "rotation(10)", in which the numbers in parentheses mean quantities of videos in the corresponding categories. • Gamma includes 264 DT videos which are also arranged into 10 categories: "flowers(29)", "sea(38)", "naked trees(25)", "foliage(35)", "escalator(7)", "calm water(30)", "flags(31)", "grass(23)", "traffic(9)", and "fountains(37)", where the numbers in parentheses denote quantities of sequences in the corresponding categories. DynTex++ dataset: It is composed as follows. 345 Dyn-Tex's videos are split and filtered so that only major textural motions are captured [START_REF] Ghanem | Maximum margin distance learning for dynamic texture recognition[END_REF]. The obtained sub-videos are then grouped into 36 classes with 100 sub-videos for each of them, i.e., 3600 sub-videos in total. Be similar to experimental protocol in [START_REF] Ghanem | Maximum margin distance learning for dynamic texture recognition[END_REF][START_REF] Arashloo | Dynamic texture recognition using multiscale binarized statistical image features[END_REF][START_REF] Tiwari | Dynamic texture recognition using multiresolution edge-weighted local structure pattern[END_REF], a half of samples in each class is randomly taken out for training, and the remain for testing. The average of 20 trials is reported as a final result.

Experimental settings

For computing high-order Gaussian-gradient-based responses: We investigate 2D/3D Gaussian filtering kernels in high-order gradients of k ∈ {1, 2, 3, 4}. Therein, standard deviation σ ∈ {0.5, 0.7, 1, 1.3, 1.5, 2} and spatio-temporal coordinates of convolution x, y, z ∈ [-3σ, 3σ] could be empirically conducted for each Gaussian-gradient kernel in order to compute corresponding filtered outcomes.

For the decomposition of oriented magnitudes: With respect to addressing direction ranges for decomposing these obtained responses to achieve IOM-based images and VOM-based volumes, it can take into account various numbers of equal direction ranges, e.g., n ∈ {4, 6, 8} respectively corresponding to λ ∈ {π/2, π/3, π/4}. Furthermore, as mentioned in Section 3.1 (refer to Eqs. ( 10), ( 11), ( 12), ( 15), ( 16), ( 17)), the modified soft-assignment decomposition has produced a double number of oriented magnitude outcomes than the others. To take an objective evaluation in effectiveness of these decomposing models, we address n = 8 (i.e., D 8 ) for the traditional models (i.e., hard and soft) and n = 4 (i.e., D 4 ) for our modified soft assignment in order to obtain the same numbers of outcomes. This could be appropriate since for a direction range [0, π/2), the soft model and its modified version respectively decompose a magnitude image into 2 SIOMs (refer to Eq. ( 11)) and 4 MSIOMs (refer to Eq. ( 12)) by adopting the pixels which their gradient directions are close to π/4. It is nearly the same that the hard model is addressed in two ranges [0, π/4) and [π/4, π/2) to obtain 2 HIOMs (refer to Eq. ( 10)) correspondingly.

For structuring IOM-based and VOM-based descriptors: In order to encode the obtained outcomes of oriented magnitudes, we use a simple operator CLBP [START_REF] Guo | A completed modeling of local binary pattern operator for texture classification[END_REF], one of the most popular local operator, with riu2 mapping and local supporting region {(P, R)} = {(8, 1)}, i.e., ξ = CLBP riu2 8,1 . To structure our proposed descriptors in reasonable dimension, the integration of "S M/C" should be utilized for jointing CLBP's components. That means it generally needs ω = 3(P + 2) × 3 × |∇| bins for representing the oriented magnitudes decomposed by a direction range. Therein, |∇| denotes a number of Gaussian-gradient magnitudes fed into a decomposing model. As a result, the final dimension to describe a DT video is subject to which the decomposing model is taken into account. For instance, using D 8 for the traditional decomposition (i.e., n = 8), dimension of single-scale HIOMF k,D σ is also the same as those above, i.e., ω×2×4 = 720 and ω×2×4 = 2160 bins respectively. Table 2 shows the dimensions of our descriptors in comparison with those of current local methods. Due to these tiny bins, it is possible to take advantage of the IOM/VOM-based outcomes in multi-oriented magnitudes by addressing multi-scale of standard deviations and multi-order of Gaussian-gradient kernels. This analysis is to enrich more discriminative information for improvement of their performances.

For DT classification: In order to evaluate performances of our IOM/VOM-based descriptors in classifying DTs, we use the linear multi-class SVM classifier of LIBLINEAR [START_REF] Fan | LIBLINEAR: A library for large linear classification[END_REF], which the default parameters are involved in.

Method

#bins P = 8 LBP-TOP u2 [START_REF] Zhao | Dynamic texture recognition using local binary patterns with an application to facial expressions[END_REF] 3(P(P -1) + 3) 177 VLBP [START_REF] Zhao | Dynamic texture recognition using local binary patterns with an application to facial expressions[END_REF] 2 3P+2 -CVLBP [START_REF] Tiwari | Dynamic texture recognition based on completed volume local binary pattern[END_REF] 3 × 2 3P+2 -HLBP [START_REF] Tiwari | A novel scheme based on local binary pattern for dynamic texture recognition[END_REF] 6 × 2 P 1536 CLSP-TOP riu2 [START_REF] Nguyen | Completed local structure patterns on three orthogonal planes for dynamic texture recognition[END_REF] 6(P + 2) 2 600 WLBPC [START_REF] Tiwari | Improved weber's law based local binary pattern for dynamic texture recognition[END_REF] 6 × 2 P 1536 MEWLSP [START_REF] Tiwari | Dynamic texture recognition using multiresolution edge-weighted local structure pattern[END_REF] 6 × 2 P 1536 CVLBC [START_REF] Zhao | Dynamic texture recognition using volume local binary count patterns with an application to 2d face spoofing detection[END_REF] 2(3P + 3) 2 1458 CSAP-TOP riu2 [START_REF] Nguyen | Completed statistical adaptive patterns on three orthogonal planes for recognition of dynamic textures and scenes[END_REF] 12(P + 2) 2 1200 FDT u2 [START_REF] Nguyen | Directional beams of dense trajectories for dynamic texture recognition[END_REF] 216P((P -1) + 3) 12744 FD-MAP u2 L=2 [START_REF] Nguyen | Directional beams of dense trajectories for dynamic texture recognition[END_REF] 216P((P -1) + 3) + 16 12760

HIOMF k,D 8 σ , SIOMF k,D 8 σ , MSIOMF k,D 4 σ 72(P + 2) 720 HVOMF k,D 8 σ , SVOMF k,D 8 σ , MSVOMF k,D 4 σ 216(P + 2) 2160
Note: P denotes the concerned neighbors. "-" means "not available".

Assessments of effectiveness of decomposing models

As mentioned in Sections 3.1 and 3.2, corresponding to the decomposing models, we address the proposed IOM/VOMbased descriptors for DT classification task on the challenging schemes, i.e., Beta, Gamma, and DynTex++. For an objective comparison, we also take non-oriented Gaussian-gradient magnitudes into account DT representation with the same encoding parameters (i.e., ξ = CLBP riu2 8,1 ) in order to construct corresponding descriptors of image/volume non-oriented magnitude features (IMF k σ and VMF k σ ). Experimental results in Table 3 have shown classification rates of these descriptors in various scale analyses. Based on those, it could be pointed out two crucial statements as follows.

• In general, it can be seen from Table 3 that the ability of the basic soft-assignment does not perform well in decomposing Gaussian-gradient magnitudes for DT encoding compared to the hard one. Even, it is inferior to the non-decomposing model (i.e., exploiting IMF and VMF features of non-oriented magnitudes) in some cases, e.g., DT recognition on Beta as shown in Fig. 8. It may be due to the intensified textural appearances caused by quantizing oriented magnitudes in adjacent orientation ranges instead of softly separating as in our modified model. • As expected, our modified soft-assignment has much improved the performance compared to its original model (see classification rates in columns "3D-S" and "3D-B" of Table 3). Furthermore, its discriminative power is significantly better than that of the non-decomposing and hard ones (see Table 3). This is thanks to the adjusted voting strategy as proposed in Section 3.1. It has appropriately adopted the magnitude features subject to a given direction range to obtain filtered outcomes in more robustness for DT encoding (refer to Eqs. ( 12) and ( 17) for detail). Due to the good discrimination in the extraction of oriented magnitudes, the modified soft decomposition should be recommended for processing Gaussian-gradient magnitudes in practice. Accordingly, in the rest of this work, we mainly discuss the performances of the MSIOMF and MSVOMF descriptors in comprehensive comparison with those of recent approaches. Fig. 8. (Best viewed in color) Performances (%) on Beta of descriptors based on the 4 th -order 3D Gaussian-gradient magnitudes using both decomposing and non-decomposing models.

Complexity of IOM/VOM-based descriptors

In general, it could be seen from the construction in Section 3.2 that it takes three main stages to structure our IOM/VOMbased descriptors: i) the filtering using Gaussian-gradient kernels; ii) the processes of decomposition; iii) the local feature extraction from the obtained IOM/VOM outcomes. Hereunder, we thoroughly discuss the complexity of encoding our proposed descriptors as well as measure the corresponding runtimes compared to other LBP-based ones.

Let Q LBP = O(P × H × W) be the computational cost of the basic LBP [START_REF] Ojala | Multiresolution gray-scale and rotation invariant texture classification with local binary patterns[END_REF] operator for encoding an image with H × W dimension, in which P denotes a number of concerned neighbors. For encoding a video V, Zhao et al. [START_REF] Zhao | Dynamic texture recognition using local binary patterns with an application to facial expressions[END_REF] addressed LBP on three orthogonal planes {XY, XT, YT } of V to form LBP-TOP patterns with the cost of Q LBP-TOP = O(P × H × W × T ), where T denotes the quantity of V's frames. As mentioned in Section 4.2, ξ = CLBP [START_REF] Guo | A completed modeling of local binary pattern operator for texture classification[END_REF] was addressed in this work to encode the IOM/VOM outcomes. The CLBP's complexity is approximately estimated as Q ξ ≈ 3 × Q LBP because its complementary components (i.e., CLBP S , CLBP M , CLBP C ) could be computed independently (refer to [START_REF] Guo | A completed modeling of local binary pattern operator for texture classification[END_REF] for more details). In addition, it can be deduced from Eqs. ( 10), [START_REF] Xu | Scale-space texture description on sift-like textons[END_REF], [START_REF] Ji | Wavelet domain multifractal analysis for static and dynamic texture classification[END_REF] that the cost of the decomposition for the gradient-filtered images is estimated as

Q IOM = Q ∇ I + Q θ ,
where Q ∇ I and Q θ denote the cost of computing the magnitude image and the gradient direction respectively. Due to Eqs. ( 13) and ( 14),

Q ∇ I = Q θ = O(H × W), i.e., Q IOM ≈ O(H × W) in general.
Similarly, referring to Eqs. ( 15), ( 16), ( 17), we also have the cost of decomposing the gradient-filtered volumes

Q VOM = Q ∇ V + Q φ , where Q ∇ V
and Q φ mean the cost of computing magnitude volumes and the gradient directions respectively. Due to Eqs. ( 18) and ( 19),

Q ∇ V = Q φ = O(H × W × T ), i.e., Q VOM ≈ O(H × W × T ) in general.
Based on those above, the complexity of our proposed descriptors can be deduced as follows.

Complexity of MSIOMF descriptor: According to Eq. ( 22), it can be deduced that the computational cost of encoding planeimages

I ∈ f XY is Q Ω f XY = 2n × N XY × (Q ξ + Q IOM + Q G 2D ).
Therein, Q G 2D denotes the cost of the 2D Gaussian-gradient filtering; N XY = T means the number of plane-images in f XY . Because of the much smaller value of n (e.g., n = 4 for the modified soft-assignment (see Section 4.2)), as well as the separable property of the 2D Gaussian-gradient filtering, they can be disregarded. It means

Q Ω f XY = T × (Q ξ + Q IOM ) ≈ O(P × H × W × T ).
Since MSIOMF is structured on the separate collections of plane-images f XY , f XT , and f YT (see Eq. ( 25)), its complexity is estimated as

Q MSIOMF ≈ max {Q Ω f XY , Q Ω f XT , Q Ω f YT } . Consequently, Q MSIOMF ≈ O(P × H × W × T ).
Complexity of MSVOMF descriptor: It can be seen from Eq. ( 26) that the cost for encoding a VOM-based volume can be estimated as Q Ψ ≈ T × Q ξ . Subject to Eq. ( 29), the complexity of MSVOMF is formed as

Q MSVOMF = 6n × (Q Ψ + Q VOM + Q G 3D ),
where Q G 3D is the cost of the 3D Gaussian-gradient filtering. Due to the much smaller value of n as well as the separable property of the 3D Gaussian-gradient filtering, they can be disregarded. Consequently,

Q MSVOMF ≈ O(P × H × W × T ).
As a result, the complexity of encoding IOM/VOM-based features is the same simple order as that of other LBP-based methods: CVLBC [START_REF] Zhao | Dynamic texture recognition using volume local binary count patterns with an application to 2d face spoofing detection[END_REF], CSAP-TOP [START_REF] Nguyen | Completed statistical adaptive patterns on three orthogonal planes for recognition of dynamic textures and scenes[END_REF], CVLBP [START_REF] Tiwari | Dynamic texture recognition based on completed volume local binary pattern[END_REF], VLBP [START_REF] Zhao | Dynamic texture recognition using local binary patterns with an application to facial expressions[END_REF], V-BIG [START_REF] Nguyen | Volumes of blurredinvariant gaussians for dynamic texture classification[END_REF], FoSIG [START_REF] Nguyen | Smooth-invariant gaussian features for dynamic texture recognition[END_REF], etc. (refer to these works for more detail). In the meantime, the performance of our proposed descriptors on DT recognition is significantly better than theirs, as thoroughly discussed in Sections 4.5, 4.6, and 4.7. With respect to the processing time, we measure runtime of encoding the IOM/VOM-based descriptors in comparison with the LBPbased others implemented by our prior work [START_REF] Nguyen | Rubik gaussian-based patterns for dynamic texture classification[END_REF]. It can be seen from Table 4 that our runtimes are as similar as the others. It should be emphasized that all those executions have been implemented by raw MATLAB codes and run in single-threading on a 64-bit Linux desktop with a configuration of CPU Core i7 Scalability of the proposed method: We consider hereunder the scalability of our method. Table 5 shows the necessary time for descriptor construction of a video of size 352 × 288 × 250 by using different numbers of threads of processing cores in the CPU to evaluate its hardware scalability. Let ω denote the sequential coefficient. According to the Amdahl's law [START_REF] Amdahl | Validity of the single processor approach to achieving large-scale computing capabilities[END_REF], the maximal speedup which can be achieved by using C threads is determined as follows: ω + 1-ω C -1 (refer to [START_REF] Gunther | Scalability-A Quantitative Approach[END_REF] for more detail). This allows to deduce the sequential coefficients when the number of threads is changed, as presented in the last row of Table 5. Accordingly, the proposed method can be highly parallelized since the sequential coefficient is relatively small (it only varies around 0.08). This beneficial property is an advantage of the proposed method for hardware scalability since the calculation of descriptor can be effectively sped up thanks to the parallelizing mechanism with the involved threads. We thoroughly discuss the significant effectiveness of taking high-order oriented magnitudes into account DT representation in comparison with other Gaussian-based filtered features. Based on the experimental results in Tables 6, 7, 8, and 9, it could be stated the following crucial assessments:

• Firstly, to prove the validation of our proposal, we have also implemented other local DT descriptors, named IMF k σ and VMF k σ , that are correspondingly based on the 2D/3D non-oriented magnitudes of Gaussian gradients (i.e., nondecomposing models involved in). It can be seen from Tables 3, 8, and 9 that IMF k σ and VMF k σ are not generally efficient compared to taking advantage of their oriented ones.

• Instead of exploiting Gaussian-based filtered characteristics as done in FoSIG [START_REF] Nguyen | Smooth-invariant gaussian features for dynamic texture recognition[END_REF] and V-BIG [START_REF] Nguyen | Volumes of blurredinvariant gaussians for dynamic texture classification[END_REF], taking the high-order oriented magnitudes into account DT representation has significantly improved the discrimination power (see Tables 10 and11). σ descriptors (represented by 2D-S k and 3D-S k respectively) are sharply decreased when the higher level of standard deviation σ is used for the gradient filterings.

• The higher level of standard deviation σ is used for the Gaussian-gradient filterings, the less robustness of our MSIOMF k,D 4 σ and MSVOMF k,D 4 σ descriptors is mostly achieved. Absolutely, it can be verified in Fig. 9 that with an increase of σ from 0.5 to 2, their performances on Dyn-Tex++ dataset are decreased about from 1% to 3% in general. This is due to lack of appearance features caused by the Gaussian-gradient filterings with large levels of σ.

• Decomposing the Gaussian-gradient filtered outcomes in the same ranges of direction, the obtained MSVOM features are more discriminative than the MSIOM ones (see Fig. 10 for a graphical view of those in settings of D 4 and σ = 1.3, see Table 6 for other circumstances in general). This is because there are complements from the intensification of pairs of gradients in the MSVOM decomposition. • It can be found out that for the challenging datasets (i.e., DynTex35, Beta, Gamma), the proposed descriptors with the odd derivatives often give better effectiveness of DT classification (see Tables 6 and7). Therefore, they should be nominated for applications in practice. • As expected in Section 3.2, the multi-analysis has significantly improved the discrimination power. Indeed, it can be seen from Tables 6 and7 that using 2-scale of Gaussian filterings with different standard deviations, the abilities of MSIOMF k,D 4 {σ} and MSVOMF k,D 4 {σ} are enhanced and more "stable" than those of the single-scale. Also, the 2-order descriptors are better than the single-order ones (see Tables 6 and 8). Furthermore, an incorporation of 2-scale and 2-order features points out the best (see Table 9). Consequently, based on the effectiveness of MSIOMF {k},D should be recommended for real applications as well as for comprehensive comparison with recent methods due to their best performances. In further evaluations, if parameters of the MSIOM/MSVOMbased descriptors are not explicit, these default settings are mentioned.

Comprehensive comparison to shallow methods

Classification on UCLA: It can be seen from Tables 6, 7, 8, and 9 that our MSIOM/MSVOM-based descriptors achieve very good rates on the schemes of UCLA. Therein, thanks to exploiting more oriented magnitudes from pairs of Gaussiangradients (see Section 3.1), the MSVOM-based ones have the performances in more "stability". With respect to the settings for comparison, our results are mostly the best in comparison with all current methods (see Table 10). Particularly, both MSIOMF and MSVOMF obtain the best rate of 100% on both schemes 50-LOO and 50-4fold. In the meantime, MSIOMF {1 st ,2 nd } {0.5,1.0} obtains 99.00% and 98.59% for DT classification on 9-class and 8-class breakdowns respectively, while MSVOMF {1 st ,4 th } {0.7,1.0} achieves better rates of 99.35% for both these breakdowns (see Fig. 11 for its specific confusions on 9-class, and Fig. 12 on 8-class). In addition, the highest rate on 9-class is 99.80% obtained by MSVOMF 4 th {1.5} (see Table 6); on 8-class (see Tables 7 and8). It is noteworthy that FD-MAP [START_REF] Nguyen | Directional beams of dense trajectories for dynamic texture recognition[END_REF] (99.35%, 99.57%), DNGP [START_REF] Rivera | Spatiotemporal directional number transitional graph for dynamic texture recognition[END_REF] (99.6%, 99.4%), and CVLBC [START_REF] Zhao | Dynamic texture recognition using volume local binary count patterns with an application to 2d face spoofing detection[END_REF] (99.2%, 99.02%) nearly have the same our abilities on these two breakdowns. However, CVLBC and FD-MAP are inferior to ours in classifying DTs on 50-LOO and 50-4fold of UCLA (see Table 10), as well as not better than ours on subsets of DynTex and on DynTex++ (see Table 11). Moreover, CVLBC and DNGP have not been verified on other challenging subsets, i.e., Alpha, Beta, and Gamma. In respect of comparing with Gaussian-based descriptors (i.e., V-BIG [START_REF] Nguyen | Volumes of blurredinvariant gaussians for dynamic texture classification[END_REF] and FoSIG [START_REF] Nguyen | Smooth-invariant gaussian features for dynamic texture recognition[END_REF]), our proposal has prominent results (see Table 10). This has proved the interest of oriented magnitudes instead of purely exploiting Gaussian-filtered features for DT representation.

Classification on DynTex: It can be verified from Table 11 that in general, our MSIOMF and MSVOMF descriptors mostly have the best performances in comparison with all non-deeplearning approaches. Specifically, our MSVOMF descriptor just reaches at 99.71% rate of DT recognition on DynTex35 due to a mutual confusion between two classes of very similar DT motions, as highlighted in red in Fig. 13. This result is just a little lower than CSAP-TOP's [START_REF] Nguyen | Completed statistical adaptive patterns on three orthogonal planes for recognition of dynamic textures and scenes[END_REF] (100%). However, beside a larger dimension (13200 bins), CSAP-TOP is also not better than ours on the other sub-sets of this schema (i.e., Alpha, Beta, and Gamma), as well as on UCLA (see Table 10). In terms of classifying DTs on other challenging schemes, due to two confusions in Fig. 14, our MSVOMF obtains 96.67% on Alpha, about 3.3% lower than V-BIG [START_REF] Nguyen | Volumes of blurredinvariant gaussians for dynamic texture classification[END_REF] with rate of 100%. However, in the other schemes, V-BIG [START_REF] Nguyen | Volumes of blurredinvariant gaussians for dynamic texture classification[END_REF] does not perform in stability (see Tables 10 and11). In the meanwhile, performances of 96.3% on Beta and 95.08% on Gamma are the very good rates in comparison with all shallow methods (see Tables 10 and11). It is noteworthy that recently, two local-feature-based methods RUBIG (95.68%) [START_REF] Nguyen | Rubik gaussian-based patterns for dynamic texture classification[END_REF] and MEMDP (96.91%) [START_REF] Nguyen | Momental directional patterns for dynamic texture recognition[END_REF] are in the nearly same order with MSVOMF on Beta but they are not on the others (see Tables 10 and11). In addition, the highest rates of our proposed descriptors are 98.33% on Alpha and 95.83% on Gamma (see Tables 6,8, and 9). Also recommended for mobile applications, MSIOMF {1 st ,2 nd } {0.5,1.0} obtains the promising rates, 99.14% on DynTex35 and 95.68% on Beta just in a small dimension of 2880 bins.

Classification on DynTex++: It can be seen from Table 9 that the multi-analysis of deviations and gradients could significantly improve the performance of our proposed descriptors in DT classification on DynTex++, mostly about 97% compared to the other analyses (see Tables 6,[START_REF] Srivastava | Salient object detection using background subtraction, gabor filters, objectness and minimum directional backgroundness[END_REF][START_REF] Dehghan | Binary quadratic programing for online tracking of hundreds of people in extremely crowded scenes[END_REF]. With respect to settings chosen for comparison, our MSIOMF and MSVOMF descriptors achieve the highest rates in comparison with the shallow methods, except MEWLSP [START_REF] Tiwari | Dynamic texture recognition using multiresolution edge-weighted local structure pattern[END_REF] with less about 0.6% (see Table 11). Nevertheless, MEWLSP has not been verified on the challenging subsets of DynTex (i.e., Alpha, Beta, and Gamma), as well as not better than ours on UCLA (see Table 10). For improvement in further context, the challenging categories in red rates in Fig. 17 should be concentrated on.

Comprehensive comparison to deep-learning methods

Classification on UCLA: It can be seen from Table 10 that our shallow framework has good performances in understanding turbulent motions of DTs in UCLA videos compared to deep-learning methods. For instance, MSVOMF {1 st ,4 th } {0.7,1.0} obtains rates of 100% for 50-4fold; 99.35% for both 9-class and 8-class. These are about 0.5∼1% better than rates of DT-CNN [START_REF] Andrearczyk | Convolutional neural network on three orthogonal planes for dynamic texture classification[END_REF] that utilizes GoogleNet learning framework to achieve rates of 99.5%, 98.35%, and 99.02% respectively (see Table 10).

Classification on DynTex: On the challenging subsets of DynTex, the deep-learning techniques [START_REF] Qi | Dynamic texture and scene classification by transferring deep image features[END_REF][START_REF] Andrearczyk | Convolutional neural network on three orthogonal planes for dynamic texture classification[END_REF][START_REF] Hong | D3: recognizing dynamic scenes with deep dual descriptor based on key frames and key segments[END_REF] have shown their effectiveness in learning features of DTs (see Table 11). However, they take a tremendous number of parameters for complicated learning algorithms. In the meanwhile, just using a shallow analysis, our proposal also has results being close to those of the deep-learning methods. More particularly, our MSVOMF {1 st ,4 th } {0.7,1.0} obtains 96.67% on Alpha, 3.3% lower than st-TCoF's [START_REF] Qi | Dynamic texture and scene classification by transferring deep image features[END_REF], D3's [START_REF] Hong | D3: recognizing dynamic scenes with deep dual descriptor based on key frames and key segments[END_REF], and DT-CNN's [START_REF] Andrearczyk | Convolutional neural network on three orthogonal planes for dynamic texture classification[END_REF]. This is due to just two confusions between "Grass" and "Tree" (see Fig. 14). In regard to classifying DTs on Beta and Gamma, our proposed framework also obtains promising rates, 96.30% on Beta by MSVOMF {1 st ,4 th } {0. Tables 7, 8, and 9); while 95.83% on Gamma by many of the SIOMF and SVOMF descriptors (see Tables 8 and9). In terms of settings chosen for comparison, the obtained performances are 96.30% on Beta and 95.08%, a little lower on Gamma. Those are due to confusions of similar DT motions in categories shown in Fig. 15 for Beta and Fig. 16 for Gamma. Classification on DynTex++: Just using a simple framework, performances of our proposed descriptors are nearly the same as those of deep-learning methods. Indeed, it can be verified from Table 11 that results of deep model DT-CNN [START_REF] Andrearczyk | Convolutional neural network on three orthogonal planes for dynamic texture classification[END_REF] are 98.18% with AlexNet framework and 98.58% with GoogleNet. These are just 0.3∼0.6% better than our MSVOMF {1 st ,4 th } {0.7,1.0} with 97.87%. It should be noted that AlexNet and GoogleNet used complicated algorithms and ∼61M and ∼6.8M learned parameters respectively for learning DT features on different datasets. For further improvement, the challenging categories expressed in red rates in Fig. 17 and MSVOMF {k},D 2 {σ} ) be in inferior performances (see Table 12) due to lack of complements of micro-oriented information. In spite of that, MSIOMF {1 st ,2 nd },D 2 {0.5,1.0} , just 1400 bins, obtains noticeable rates on Beta (95.68%) and Gamma (95.98%). It may be a potential solution for mobile applications on edge devices having limited resources.

• It can be verified from Table 12 that addressing the decomposing models in smaller angles (e.g., D 8 ) makes a sharp increase of dimensions while the performance seems not to be improved, except a little of MSVOMF {1 st ,4 th },D 8 {0.7,1.0} on Gamma. This can be due to the weakness of appearance features caused by the smallness of direction ranges. Recently, the deep-learning trend has become one of the main streams of computer vision community. Deep-learning methods have often achieved good results in recognizing DTs on challenging schemes (see Table 11). Nevertheless, they have spent much computational cost in learning millions of parameters through deep neural networks (DNN). In addition, because the inference phase is based on a huge volume of learned parameters, a deployment of DNN models on edge devices is challenging. For instance, it takes ∼61M parameters for AlexNet [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF] and ∼6.8M for GoogleNet [START_REF] Szegedy | Going deeper with convolutions[END_REF] implemented in DT-CNN [START_REF] Andrearczyk | Convolutional neural network on three orthogonal planes for dynamic texture classification[END_REF] for DT representation. As a result, it is restricted to deploy the deep-learning for real applications in mobile devices as well as embedded sensor systems due to a strict requirement of tiny resources for their executions. Contrary to the complicated models of the deep-learning-based methods, our proposed framework have obtained the competitive performances but just using shallow analysis, expected to be one of potential solutions for mobile implementations. Indeed, just utilizing a simple operator to capture the IOM/VOM-based features from the Gaussian-gradient magnitudes, our MSIOMF and MSVOMF descriptors have the significant performance compared to that of all non-deep-learning methods, while being close to that of the deep-learning ones (see Tables 10 and11). Furthermore, CLBP [START_REF] Guo | A completed modeling of local binary pattern operator for texture classification[END_REF] at the period of local encoding could be replaced by other robust operators (e.g., CLBC [START_REF] Zhao | Completed Local Binary Count for Rotation Invariant Texture Classification[END_REF], MRELBP [START_REF] Liu | Median robust extended local binary pattern for texture classification[END_REF], LRP [START_REF] Nguyen | Rubik gaussian-based patterns for dynamic texture classification[END_REF], LDP-based [START_REF] Zhang | Local derivative pattern versus local binary pattern: Face recognition with high-order local pattern descriptor[END_REF][START_REF] Nguyen | Momental directional patterns for dynamic texture recognition[END_REF], LVP-based [START_REF] Fan | A novel local pattern descriptor -local vector pattern in high-order derivative space for face recognition[END_REF][START_REF] Nguyen | Directional dense-trajectorybased patterns for dynamic texture recognition[END_REF], etc.) in order to investigate the IOM/VOM-based features in different circumstances for potential enhancements.

In respect of real-world applications of our proposed framework, it can be used for early-warning fire monitoring systems or for computing devices of ubiquitous smart home [START_REF] Lai | Multi-appliance recognition system with hybrid SVM/GMM classifier in ubiquitous smart home[END_REF] in order to detect fire-flame as investigated in former work [START_REF] Dimitropoulos | Spatio-temporal flame modeling and dynamic texture analysis for automatic video-based fire detection[END_REF]. It is thanks to the effectiveness of the proposed IOM/VOM-based features in the shallow analysis. In addition, ours may be considered in other applications based on DT analysis as done in several former works: facial expression [START_REF] Zhao | Dynamic texture recognition using local binary patterns with an application to facial expressions[END_REF][START_REF] Huang | Spatiotemporal local monogenic binary patterns for facial expression recognition[END_REF][START_REF] Shao | Joint discriminative learning of deep dynamic textures for 3d mask face anti-spoofing[END_REF], segmentation [START_REF] Gonc ¸alves | Dynamic texture segmentation based on deterministic partially self-avoiding walks[END_REF][START_REF] Gonc ¸alves | Dynamic texture analysis and segmentation using deterministic partially self-avoiding walks[END_REF][START_REF] Chen | Automatic dynamic texture segmentation using local descriptors and optical flow[END_REF][START_REF] Chen | Unsupervised dynamic texture segmentation using local descriptors in volumes[END_REF], lipreading [START_REF] Zhao | Lipreading with local spatiotemporal descriptors[END_REF], iris recognition [START_REF] De Melo Langoni | Evaluating dynamic texture descriptors to recognize human iris in video image sequence[END_REF], etc.

Conclusions

In this paper, we have proposed a simple and efficient framework in which the high-order oriented magnitudes of Gaussian gradients are exploited for DT representation. Accordingly, the decomposing models of hard and soft-based assignments have been investigated in different direction ranges (i.e., D 2 , D 4 , D 8 ) for extracting IOM/VOM-based features from the Gaussiangradient magnitudes. Therein, the modified soft-assignment model of D 4 has pointed out the best performances. The experimental results for DT classification issue have validated that local descriptors MSIOMF {k},D 4 {σ} and MSVOMF {k},D 4 {σ} based on these extracted features have significant enhancement of discrimination power in comparison with the Gaussian-based descriptors (i.e., FoSIG [START_REF] Nguyen | Smooth-invariant gaussian features for dynamic texture recognition[END_REF] and V-BIG [START_REF] Nguyen | Volumes of blurredinvariant gaussians for dynamic texture classification[END_REF]) as well as the others in state-of-the-art methods. Also, those have confirmed the interest of our approach based on the oriented magnitudes of Gaussian gradients rather than based on the non-oriented ones.

For perspectives, the problems of zero-pixels/voxels in the IOM/VOM-based outcomes, which are caused by the decomposing models, can negatively affect the discriminative power when using CLBP [START_REF] Guo | A completed modeling of local binary pattern operator for texture classification[END_REF] for the local encoding stage. To overcome those, CLBP can be modified in the future work to adapt this encoding context. Moreover, in case of treating the curse of expansive dimension, it is able to address a multi-scale analysis of supporting regions (e.g., {(P, R)} = {(8, 1), (8, 2)}) to explore more local relationships of IOM/VOM-based features in larger neighborhoods for further improvements.

Filtering

  the video using high-order Gaussian-gradient kernels to point out Gaussian-filtered outcomes Decomposing oriented magnitudes subject to a set of given direction ranges Encoding oriented magnitudes using a simple local operator to form proposed descriptors Computing magnitudes based on these filtered outcomes An input video

Fig. 1 .

 1 Fig. 1. (Best viewed in color) A proposed framework for encoding a video in general. Therein, the blue arrows denote progresses of pre-processing, the black one denotes progress of encoding features of oriented magnitudes.

Fig. 2 .

 2 Fig. 2. (Best viewed in color) A hard-assignment model for decomposing the magnitudes of two Gaussian-gradient images I ∂x 1 0.5 and I ∂y 1 0.5 into 4 HIOM images subject to a set of 4 equal ranges of direction D 4 = {[0, π/2), [π/2, π), [π, 3π/2), [3π/2, 2π)}.

Fig. 3 .

 3 Fig. 3. An instance of 3D Gaussian-gradient filtering and computing volumes of magnitude features.

Fig. 2 ,

 2 we respectively used direction ranges of D 4 to decompose a magnitude image ||∇I ∂x k ,∂y k σ ||. Hereunder, we propose robust descriptors structured corresponding to the IOM-based and VOMbased outcomes.

Fig. 4 .

 4 Fig. 4. (Best viewed in color) Flowchart of HIOM model subject to direction ranges d i = [(i -1)λ, iλ) in D n . Therein, the black arrows are noted for preprocessing while the blue ones are for encoding.

Fig. 5 .

 5 Fig. 5. (Best viewed in color) Flowchart of HVOM model subject to direction ranges d i = [(i -1)λ, iλ) in D n . Therein, the black arrows are preprocessing steps while the blue ones are for encoding.

Fig. 6 .

 6 Fig. 6. Several DT sequences of UCLA DynTex dataset: It is a challenging dataset for DT classification [16]. DynTex has 679 DT videos recorded in AVI format with dimension of 352 × 288 × 250, in which turbulent motions of DTs are captured in different conditions of environmental changes (see Fig.7for several DT samples). For DT classification, it is often arranged into the following subsets. The LOO protocol[START_REF]Dynamic texture representation using a deep multi-scale convolutional network[END_REF][START_REF] Dubois | Characterization and recognition of dynamic textures based on the 2d+t curvelet transform[END_REF][START_REF] Nguyen | Volumes of blurredinvariant gaussians for dynamic texture classification[END_REF] is utilized to evaluate the performances.• DynTex35 is composed by splitting from 35 DynTex videos as follows. Each video is split into 8 non-

Fig. 7 .

 7 Fig. 7. Several samples of DynTex dataset

8 σ

 8 (i.e., |∇| = 1) is ω × 8 = 720 bins, while that of single-scale HVOMF k,D 8 σ (i.e., |∇| = 3) is ω × 8 = 2160 bins. Those are the same bins for SIOMF k,D 8 σ and SVOMF k,D 8 σ respectively. Due to addressing D 4 for the modified soft-assignment, the dimensions in single-scale analysis of MSIOMF k,D 4 σ and MSVOMF k,D 4

4. 5 .

 5 Assessments of MSIOMF k,D 4 σ and MSVOMF k,D 4 σ

Fig. 9 . 4 σ

 94 Fig. 9. (Best viewed in color) Performances on DynTex++ of high-order MSIOMF k,D 4 σ and MSVOMF k,D 4

4

 4

  {σ} and MSVOMF {k},D 4 {σ} in classifying DTs, the settings of those: MSIOMF {1 st ,2 nd },D 4 {0.5,1.0} and MSVOMF {1 st ,4 th },D 4 {0.7,1.0}

Fig. 11 .

 11 Fig. 11. Confusion matrix (%) for MSVOMF {1 st ,4 th },D 4 {0.7,1.0} on 9-class. Fig. 12. Confusion matrix (%) for MSVOMF {1 st ,4 th },D 4 {0.7,1.0} on 8-class.

Fig. 13 .

 13 Fig. 13. (Best viewed in color) Classification rates of MSVOMF {1 st ,4 th } {0.7,1.0} on specific categories of DynTex35.

Fig. 14 .

 14 Fig. 14. Confusions of MSVOMF {1 st ,4 th } {0.7,1.0} on Alpha.

Fig. 15 .

 15 Fig. 15. Confusions of MSVOMF {1 st ,4 th } {0.7,1.0} on Beta.

Fig. 16 .

 16 Fig. 16. Confusions of MSVOMF {1 st ,4 th } {0.7,1.0} on Gamma.

  should be addressed in future works.5. Global discussionsBeside the comprehensive evaluations are thoroughly discussed in Section 4.5, it can be derived further statements and experimental results of MSIOMF {k},D n {σ} and MSVOMF {k},D n {σ} as: • It should be noted that high-order oriented magnitudes extracted by a direction range D 2 = {[0, π), [π, 2π)} could make the corresponding descriptors (i.e., MSIOMF {k},D 2

  {σ}

Fig. 17 .

 17 Fig. 17. (Best viewed in color) Specific rates of MSVOMF {1 st ,4 th } {0.7,1.0} on each category of DynTex++. The challenging categories are in red rates. Therein, the numbers in the parentheses denote the numbered class labels correspondingly .

  ||∇V σ ||) with respect to different ranges of gradient directions θ σ and φ σ . Finally, a simple local operator (e.g., CLBP[START_REF] Guo | A completed modeling of local binary pattern operator for texture classification[END_REF]) is utilized for extracting spatio-temporal features from the IOM/VOM-based outcomes. As a result, robust descriptors are then structured with high performances in reasonable dimensions. Experiments for classifying DTs on benchmark datasets have verified the considerable ability of our proposed descriptors in comparison with state of the art. In short, our major contributions can be listed as follows.

). Magnitudes ||∇I σ || (resp. ||∇V σ ||) and gradient directions θ σ (resp. φ σ ) are then referred from these Gaussian-gradient filterings. After that, k-order oriented magnitudes IOM k σ (resp. VOM k σ ) are pointed out by decomposing the magnitudes ||∇I σ || (resp.

Table 1 .

 1 A brief of main properties of DT datasets.

	Dataset	Sub-dataset #Videos	Resolution #Classes Protocol
		50-class	200	110 × 160 × 75	50 LOO and 4fold
	UCLA	9-class	200	110 × 160 × 75	9 50%/50%
		8-class	92	110 × 160 × 75	8 50%/50%
		DynTex35	350 different dimensions	10 LOO
	DynTex	Alpha Beta	60 162	352 × 288 × 250 352 × 288 × 250	3 LOO 10 LOO
		Gamma	264	352 × 288 × 250	10 LOO
	DynTex++		3600	50 × 50 × 50	36 50%/50%

Table 2 .

 2 A comparison of various dimensions of LBP-based descriptors.

Table 3 .

 3 Classification rates (%) on the challenging schemes of descriptors based on non-oriented-magnitude and IOM/VOM-based features. {0.7} 91.36 90.74 90.74 93.21 90.74 91.36 93.83 92.80 93.94 91.29 94.32 91.29 90.91 91.67 95.77 96.08 97.13 97.01 96.66 87.99 93.68 {1.0} 91.36 91.36 91.98 92.59 90.12 91.98 93.21 92.42 92.80 93.18 93.18 92.80 89.39 90.53 94.72 95.73 96.18 96.76 95.57 88.92 93.19 {1.3} 91.98 91.36 91.98 92.59 89.51 89.51 93.83 90.15 92.80 91.67 92.42 92.42 87.50 90.53 94.61 95.05 96.05 96.05 95.47 85.51 91.09 {1.5} 89.51 91.36 91.36 91.98 90.74 91.36 92.59 92.05 92.05 92.42 92.05 93.18 89.02 91.67 93.90 94.98 95.51 95.85 95.08 86.96 91.10 2 nd {0.7} 91.36 93.83 91.36 94.44 91.36 91.36 94.44 93.18 93.56 93.56 93.18 93.56 89.39 91.67 95.66 95.76 96.51 96.82 96.77 85.73 93.09 {1.0} 93.21 93.21 92.59 95.06 90.74 92.59 91.98 92.42 93.18 92.80 93.56 94.32 90.91 93.56 94.88 95.39 96.44 96.23 96.09 86.03 92.10 {1.3} 91.36 91.36 91.36 93.83 91.98 88.27 90.74 90.53 93.56 91.67 93.94 89.77 89.02 91.67 94.10 94.51 95.31 96.28 94.88 84.76 92.17 {1.5} 90.74 92.59 93.21 93.21 91.36 90.74 92.59 92.42 92.80 93.94 93.18 93.18 86.64 91.67 94.19 94.07 95.14 95.93 95.40 83.51 91.35

	Scheme	Beta	Gamma	DynTex++
	Order {σi}	2D-H 2D-S 3D-H 3D-S 3D-B	IMF VMF 2D-H 2D-S 3D-H 3D-S 3D-B	IMF VMF 2D-H 2D-S 3D-H 3D-S 3D-B	IMF VMF
	1 st
		{0.7} 89.51 89.51 91.98 92.59 92.59 89.51 93.83 91.67 93.94 90.53 93.18 93.94 86.74 91.67 95.54 95.67 96.51 96.81 96.23 85.49 92.57
	3 rd	{1.0} 91.36 92.59 93.21 92.59 91.36 88.89 93.83 91.67 93.18 90.53 91.29 92.80 89.39 92.80 93.52 95.34 95.82 96.18 95.04 85.71 91.88 {1.3} 95.06 93.21 95.06 93.83 91.36 88.27 93.21 92.42 91.29 93.18 93.18 92.05 89.77 92.42 93.88 94.34 95.27 96.16 94.63 83.84 92.31
		{1.5} 90.74 91.98 93.21 93.83 88.89 90.12 90.74 90.91 91.29 91.67 91.67 90.53 90.53 92.80 94.20 94.38 94.83 95.66 93.79 85.00 91.26
		{0.7} 92.59 93.83 93.83 93.83 92.59 90.12 93.21 90.91 93.18 93.56 93.94 93.94 86.74 90.53 94.81 95.02 96.39 96.07 95.99 85.62 93.07
	4 th	{1.0} 90.74 91.36 92.59 95.06 89.51 88.89 93.83 89.77 90.53 92.05 94.32 90.91 86.74 94.32 94.27 95.22 95.55 96.57 95.46 85.46 92.47 {1.3} 90.12 90.74 90.74 94.44 90.12 89.51 92.59 91.29 91.29 92.80 93.56 92.42 88.64 92.05 93.58 94.77 95.56 95.82 94.37 86.73 93.68
		{1.5} 89.51 91.98 91.36 94.44 90.74 89.51 93.83 90.53 92.42 92.80 94.32 93.18 90.15 93.56 92.72 93.90 94.89 95.62 94.44 84.19 91.09
	Note: Respectively, 2D-H and 3D-H denote for oriented magnitude descriptors HIOMF k,D 8 σ MSVOMF k,D 4 σ with the modified soft decomposition. 3D-B denotes for SVOMF k,D 8 σ based on the basic soft-assignment model. IMF and VMF stand for non-oriented magnitude ones IMF k and HVOMF k,D 8 σ using the hard-decomposing model, while 2D-S and 3D-S are for MSIOMF k,D 4 σ σ and and VMF k σ , i.e., none of the decomposing models is involved in the DT encoding.
	(%)			=0.7
	Classification rates			=1.0 =1.3 =1.5

Table 4 .

 4 Processing time of several LBP-based methods to structure a 50 × 50 × 50 video of DynTex++.

	Descriptor	Gradient (σ [,σ ]) (P, R) Mapping Runtime(s)
	VLBP [42]	-	-	(4, 1)	-	≈ 0.22
	LBP-TOP [42]	-	-	(8, 1)	u2	≈ 0.15
	CLSP-TOP [46]	-	-	(8, 1)	riu2	≈ 0.27
	CSAP-TOP [38]	-	-	(8, 1)	riu2	≈ 0.50
	FoSIG 2D [40]	-	(0.5, 6) (8, 1)	riu2	≈ 0.37
	V-BIG 3D [39]	-	(0.5, 6) (8, 1)	riu2	≈ 0.35
	RUBIG [49]	-	(0.5, 6) (8, 1)	riu2	≈ 0.56
	Our MSIOMF k,D 4 σ Our MSVOMF k,D 4 σ	1 st 1 st	σ = 0.7 (8, 1) σ = 0.7 (8, 1)	riu2 riu2	≈ 0.48 ≈ 0.62
	Note: "-" means "not available". Runtime of other LBP-based ap-
	proaches is reported by implementations of our former work [49].

Table 5 .

 5 Evaluation of hardware scability.

	Number of threads	1	2	3	4
	Times (in s)	622.74 334.23 242.08 195.46
	Speed-up	1	1.863	2.572	3.186
	Sequential coefficient (ω)	NA	0.0734 0.0831 0.0851
	3.4GHz 16G RAM.

Table 6 .

 6 Classification rates (%) on DT benchmark datasets of MSIOMF k,D 4 .50 98.90 98.55 96.74 98.80 98.29 95.43 96.67 96.67 90.74 95.68 92.42 93.94 95.90 97.12 {0.7} 99.00 99.50 99.00 99.50 99.70 99.60 96.63 97.50 98.29 96.57 95.00 96.67 90.74 93.21 93.94 94.32 96.08 97.01 {1.0} 99.50 99.50 100 99.50 99.30 98.75 97.50 97.17 98.86 98.29 96.67 96.67 91.36 92.59 92.80 93.18 95.73 96.76 {1.3} 98.50 99.00 98.50 99.50 99.05 97.65 95.11 97.39 98.57 98.86 96.67 96.67 91.36 92.59 92.80 92.42 95.05 96.05 {1.5} 99.00 99.00 99.50 99.50 96.85 98.70 96.85 98.04 99.14 98.86 96.67 96.67 91.36 91.98 92.05 92.05 94.98 95.85 .10 99.15 97.61 98.04 98.29 98.29 96.67 98.33 95.06 92.59 92.80 91.29 95.22 97.13 {0.7} 99.00 99.50 99.50 99.50 98.40 98.90 97.72 97.39 98.86 98.86 96.67 96.67 89.51 92.59 93.94 93.18 95.67 96.81 .05 94.67 96.74 98.57 98.57 96.67 96.67 93.21 93.83 91.29 93.18 94.34 96.16 {1.5} 99.00 99.00 99.00 99.50 98.55 98.40 96.30 97.17 98.86 99.43 96.67 96.67 91.98 93.83 91.29 91.67 94.38 95.66 {2.0} 99.50 98.50 99.50 99.50 98.70 98.45 98.49 96.20 98.00 98.86 96.67 96.67 93.21 93.21 92.05 93.18 92.89 93.79 .85 97.83 99.02 96.29 97.43 96.67 96.67 90.74 94.44 91.29 93.56 94.77 95.82 {1.5} 99.50 99.50 99.50 99.50 98.45 99.80 99.35 98.49 98.00 96.57 96.67 96.67 91.98 94.44 92.42 94.32 93.90 95.62 {2.0} 99.50 100 99.50 100 98.50 99.20 98.59 99.24 93.71 97.43 96.67 96.67 91.36 95.06 92.42 95.45 92.53 94.39

	σ	and MSVOMF k,D 4 σ	descriptors.

Note: 2D-S and 3D-S are shortened for MSIOMF k,D 4 σ and MSVOMF k,D 4 σ respectively. 50-LOO and 50-4fold denote results on 50-class breakdown using leave-one-out and four cross-fold validation.

Table 7 .

 7 Classification rates (%) on DT benchmark datasets of MSIOMF k,D 4 {σ} and MSVOMF k,D 4 {σ} descriptors. 0} 99.50 99.50 99.50 99.50 99.20 98.80 99.02 97.50 99.14 99.43 96.67 96.67 93.83 94.44 92.42 93.56 96.67 97.28 {0.7, 1.0} 99.00 99.50 99.50 99.50 99.25 98.10 97.72 98.48 99.14 98.86 95.00 96.67 92.59 93.83 92.80 93.18 96.72 96.78 {1.0, 1.3} 98.50 99.50 99.00 99.50 98.90 98.65 98.04 96.41 99.71 98.86 96.67 96.67 91.98 93.83 92.42 93.18 96.43 96.89 {1.0, 1.5} 99.00 99.50 99.50 99.50 98.25 98.85 97.50 97.50 99.43 98.86 96.67 96.67 91.36 92.59 92.05 92.80 96.27 96.88 {1.5, 2.0} 99.50 99.50 99.50 99.50 98.15 99.15 96.85 96.85 99.43 98.86 98.33 96.67 93.21 91.98 91.67 92.05 95.51 95.81 .25 98.15 97.61 97.72 99.43 99.71 96.67 96.67 94.44 93.21 93.56 91.67 96.72 97.06 {0.7, 1.0} 99.50 99.50 99.50 99.50 98.65 98.70 97.83 97.28 99.14 99.43 96.67 96.67 91.98 92.59 92.80 92.04 96.36 97.25 {1.0, 1.3} 100 100 100 100 99.25 98.20 95.87 98.59 98.86 99.14 96.67 96.67 93.83 93.83 93.56 93.94 96.33 96.76 {1.0, 1.5} 99.00 100 99.50 99.50 98.30 97.90 97.72 97.61 99.43 99.71 96.67 96.67 92.59 93.83 90.91 91.67 96.26 96.48 {1.5, 2.0} 99.50 99.00 99.50 99.50 98.15 99.40 96.63 96.85 98.00 99.14 96.67 96.67 91.98 93.83 90.53 92.80 94.24 95.24

		Dataset				UCLA				DynTex	DynTex++
		Sub-set	50-LOO	50-4fold	9-class	8-class	DynTex35	Alpha	Beta	Gamma
	Order	{σ i }	2D-S 3D-S 2D-S 3D-S 2D-S 3D-S 2D-S 3D-S	2D-S 3D-S 2D-S 3D-S 2D-S 3D-S 2D-S 3D-S	2D-S 3D-S
		{0.5, 0.7} 99.00 99.50 99.50	100 97.95 98.85 95.22 96.41 98.57 98.00 95.00 96.67 93.83 95.06 93.18 93.18 93.77 97.36
	1 st {0.5, 1.2 nd {0.5, 0.7} {0.5, 1.0} {0.7, 1.0} {1.0, 1.3} 99.50 100 100 100	100 100 100 100 99.50 100 100 100	100 98.85 97.75 96.52 97.50 98.57 98.86 96.67 96.67 93.83 93.83 93.56 93.94 96.66 97.37 100 98.45 98.50 97.28 97.39 97.14 98.00 96.67 96.67 91.98 93.21 93.94 93.56 96.63 97.08 100 99.15 98.65 97.17 97.39 98.86 97.43 96.67 96.67 93.83 93.83 92.80 93.56 96.45 97.08 100 99.35 99.00 96.20 97.83 98.29 98.00 96.67 96.67 92.59 93.83 93.18 93.94 95.91 96.45
		{1.0, 1.5} 99.50 99.50 98.50	100 98.80 99.00 98.26 99.02 99.43 92.57 96.67 96.67 93.21 93.21 93.56 93.18 96.39 96.67
		{1.5, 2.0} 98.50	100 98.00	100 98.90 98.95 98.70 98.04 98.29 97.71 96.67 96.67 92.59 91.98 92.80 94.70 94.86 95.50
		{0.5, 0.7} 99.50 99.50 99.50 99.50 98.90 98.70 97.39 98.37 98.57 99.14 96.67 96.67 92.59 92.59 93.56 93.18 96.69 97.32
	3 rd 100 99.50 994 th {0.5, 1.0} 100 100 {0.5, 0.7} 100 100 100 100 97.20 98.30 97.50 97.50 97.14 97.43 96.67 96.67 91.98 91.36 94.32 92.80 96.29 96.80 {0.5, 1.0} 100 100 100 100 98.90 98.40 98.48 97.72 96.86 98.29 96.67 96.67 93.21 93.21 93.18 93.56 94.47 96.88 {0.7, 1.0} 99.50 100 99.50 100 98.20 99.05 97.39 99.57 98.29 97.71 96.67 96.67 93.83 94.44 92.42 94.70 96.19 96.93 {1.0, 1.3} 99.50 100 99.50 100 98.85 98.80 98.70 98.91 95.43 98.57 96.67 96.67 91.98 96.30 92.05 93.94 96.17 96.81
		{1.0, 1.5} 99.50	100	100	100 98.65 99.15 98.26 98.70 97.43 96.86 96.67 96.67 92.59 94.44 92.80 95.08 95.62 96.24
		{1.5, 2.0} 99.50	100 99.50	100 98.90 99.25 97.93 99.15 98.29 96.86 96.67 96.67 91.36 93.83 93.18 94.70 94.68 95.61
	Note: 2D-S and 3D-S are shortened for MSIOMF k,D 4 {σ} and MSVOMF k,D 4 {σ} respectively. 50-LOO and 50-4fold denote results on 50-class breakdown using leave-one-out and
	four cross-fold validation.

Table 8 .

 8 Classification rates (%) on DT benchmark datasets of multi-order MSIOMF {k},D 4 .95 99.70 96.74 99.57 98.57 98.86 96.67 96.67 93.21 95.06 94.70 95.08 97.04 97.18 {1.0} 99.50 99.50 99.50 99.50 99.20 98.25 97.72 98.15 98.86 98.57 96.67 96.67 93.21 94.44 94.70 95.08 97.07 97.28 5} 99.00 99.00 99.50 99.50 98.70 98.25 98.37 98.80 99.71 99.14 96.67 96.67 92.59 93.21 91.29 91.29 95.77 96.32 {2.0} 99.50 99.00 99.50 99.50 98.70 98.90 97.07 97.61 98.29 99.14 96.67 96.67 91.98 92.59 92.80 93.56 94.93 95.07 .55 97.50 97.83 97.71 99.14 96.67 96.67 93.21 93.83 92.05 94.70 95.76 96.50 {1.5} 98.50 99.50 98.50 99.50 99.05 98.25 96.41 97.93 99.43 99.43 96.67 96.67 90.74 91.98 93.56 94.32 95.87 96.25 .95 99.24 97.72 98.00 98.86 96.67 96.67 91.36 93.83 91.29 94.70 95.96 96.66 {1.5} 99.50 100 99.50 100 98.70 98.95 97.17 98.59 99.14 97.14 96.67 96.67 92.59 94.44 93.18 93.94 95.35 96.13 {2.0} 99.50 100 99.50 100 98.20 99.15 97.72 97.39 98.00 96.86 96.67 96.67 91.98 94.44 93.18 95.08 94.74 95.13

	σ	and MSVOMF {k},D 4 σ	descriptors.

Table 9 .

 9 Classification rates (%) on DT benchmark datasets of multi-order MSIOMF {k},D 4 5} 99.00 99.50 99.50 99.50 98.45 98.70 97.61 97.39 99.71 99.71 96.67 96.67 93.21 92.59 91.67 91.67 96.91 97.15 {1.5, 2.0} 99.50 99.50 99.50 99.50 98.20 98.65 96.30 97.07 98.86 99.43 96.67 96.67 90.74 92.59 91.67 92.05 95.68 96.07 .30 97.39 98.70 99.43 99.43 96.67 96.67 92.59 92.59 95.08 95.83 97.20 97.35 {1.5, 2.0} 99.50 99.50 99.50 99.50 98.25 99.35 98.59 98.04 98.86 99.43 96.67 96.67 90.74 93.21 93.94 95.08 96.06 96.53

	{σ}	and MSVOMF {k},D 4 {σ}	descriptors.

Table 10 .

 10 Comparison of recognition rates (%) on UCLA.

	Group Encoding method	50-LOO 50-4fold 9-class 8-class
		FDT [22]	98.50	99.00 97.70 99.35
	A	FD-MAP [22]	99.50	99.00 99.35 99.57
		DDTP [64]	99.00	99.50 98.75 98.04
	B	AR-LDS [15] Chaotic vector [26]	89.90 N -	--85.10 N 85.00 N --
		3D-OTF [11]	-	87.10 97.23 99.50
	C	DFS [65]	-	100 97.50 99.20
		STLS [13]	-	99.50 97.40 99.50
		MBSIF-TOP [35]	99.50 N	-	-	-
	D	DNGP [21]	-	-99.60 99.40
		B3DF SMC [36]	99.50 N	99.50 N 98.85 N 98.15 N
		VLBP [42]	-	89.50 N 96.30 N 91.96 N
		LBP-TOP [42]	-	94.50 N 96.00 N 93.67 N
		CVLBP [44]	-	93.00 N 96.90 N 95.65 N
		HLBP [45]	95.00 N	95.00 N 98.35 N 97.50 N
		CLSP-TOP [46]	99.00 N	99.00 N 98.60 N 97.72 N
		MEWLSP [58]	96.50 N	96.50 N 98.55 N 98.04 N
		WLBPC [60]	-	96.50 N 97.17 N 97.61 N
		CVLBC [61]	98.50 N	99.00 N 99.20 N 99.02 N
	E	CSAP-TOP [38]	99.50	99.50 96.80 95.98
		FoSIG [40]	99.50	100 98.95 98.59
		V-BIG [39]	99.50	99.50 97.95 97.50
		HILOP [66]	99.50	99.50 97.80 96.30
		MMDP D M/C [50]	100	100 98.70 98.70
		MEMDP D M/C [50]	100	100 98.90 98.70
		RUBIG [49]	100	100 99.20 99.13
		Our MSIOMF {1 st ,2 nd },D 4 {0.5,1.0}	100	100 99.00 98.59
		Our MSVOMF {1 st ,4 t h },D 4 {0.7,1.0}	100	100 99.35 99.35
		DL-PEGASOS [17]	-	97.50 95.60	-
		PI-LBP+super hist [48]	-	100 N 98.20 N	-
	F	Orthogonal Tensor DL [33] PCANet-TOP [31]	-99.50 *	99.80 98.20 99.50 ---
		DT-CNN-AlexNet [30]	-	99.50 * 98.05 * 98.48 *
		DT-CNN-GoogleNet [30]	-	99.50 * 98.35 * 99.02

*

Note: "-" means "not available". Superscript "*" denotes results using deep learning methods. "N" is rate with 1-NN classifier.

[START_REF] Nguyen | Momental directional patterns for dynamic texture recognition[END_REF]

-LOO and 50-4fold are results of 50-class using leave-one-out and four cross-fold validation respectively. Group A is optical-flow-based methods, B: model-based, C: geometrybased, D: filter-based, E: local-feature-based, F: learning-based.

Table 11 .

 11 Comparison of rates (%) on DynTex and DynTex++.

	Group Encoding method	Dyn35 Alpha	Beta Gamma Dyn++
		FDT [22]	98.86 98.33 93.21	91.67	95.31
	A	FD-MAP [22]	98.86 98.33 92.59	91.67	95.69
		DDTP [64]	99.71 96.67 93.83	91.29	95.09
		3D-OTF [11]	96.70 83.61 73.22	72.53	89.17
	C	DFS [65] 2D+T [57]	97.16 85.24 76.93 -85.00 67.00	74.82 63.00	91.70 -
		STLS [13]	98.20 89.40 80.80	79.80	94.50
		MBSIF-TOP [35]	98.61 N 90.00 N 90.70 N 91.30 N 97.12 N
	D	DNGP [21]	-	-	-	-	93.80
		B3DF SMC [36]	99.71 N 95.00 N 90.12 N 90.91 N 95.58 N
		VLBP [42]	81.14 N	-	-	-94.98 N
		LBP-TOP [42]	92.45 N 98.33 88.89 84.85 N 94.05 N
		DDLBP with MJMI [47]	-	-	-	-	95.80
		CVLBP [44]	85.14 N	-	-	-	-
		HLBP [45]	98.57 N	-	-	-96.28 N
		CLSP-TOP [46]	98.29 N 95.00 N 91.98 N 91.29 N 95.50 N
		MEWLSP [58]	99.71 N	-	-	-98.48 N
		WLBPC [60]	-	-	-	-95.01 N
	E	CVLBC [61] CSAP-TOP [38]	98.86 N 100 96.67 92.59 --	-91.31 N 90.53 -
		FoSIG [40]	99.14 96.67 92.59	92.42	95.99
		V-BIG [39]	99.43	100 95.06	94.32	96.65
		HILOP [66]	99.71 96.67 91.36	92.05	96.21
		MMDP D M/C [50]	99.43 98.33 96.91	92.05	95.86
		MEMDP D M/C [50]	99.71 96.67 96.91	93.94	96.03
		RUBIG [49]	98.86	100 95.68	93.56	97.08
		Our MSIOMF {1 st ,2 nd },D 4 {0.5,1.0}	99.14 96.67 95.68	94.70	97.29
		Our MSVOMF {1 st ,4 t h },D 4 {0.7,1.0}	99.71 96.67 96.30	95.08	97.87
		DL-PEGASOS [17]	-	-	-	-	63.70
		PCA-cLBP/PI/PD-LBP [48]	-	-	-	-	92.40
		Orthogonal Tensor DL [33]	-87.80 76.70	74.80	94.70
		Equiangular Kernel DL [34]	-88.80 77.40	75.60	93.40
	F	st-TCoF [29]	-	100 *	100 *	98.11 *	-
		PCANet-TOP [31]	-96.67 * 90.74 *	89.39 *	-
		D3 [32]	-	100 *	100 *	98.11 *	-
		DT-CNN-AlexNet [30]	-	100 * 99.38 *	99.62 *	98.18 *
		DT-CNN-GoogleNet [30]	-	100 *	100 *	99.62 *	98.58

*

Note: "-" is "not available". Superscript "*" are results using deep learning algorithms. "N" is rate with 1-NN classifier. Dyn35 and Dyn++ stand for DynTex35 and DynTex++ sub-datasets. Group A denotes optical-flow-based methods, C: geometrybased, D: filter-based, E: local-feature-based, F: learning-based.

Table 12 .

 12 Rates (%) of MSIOMF and MSVOMF in different ranges D n on challenging datasets using the settings chosen for comparison.

	Range Descriptor	#bins Dyn35 Alpha Beta Gamma Dyn++
	D 2	MSIOMF {1 st ,2 nd } {0.5,1.0} MSVOMF {1 st ,4 th } {0.7,1.0}	1400 4320	97.43 96.67 95.68 98.29 96.67 93.83	95.83 94.70	96.96 97.86
	D 4	MSIOMF {1 st ,2 nd } {0.5,1.0} MSVOMF {1 st ,4 th } {0.7,1.0}	2880 8640	99.14 96.67 95.68 99.71 96.67 96.30	94.70 95.08	97.29 97.87
	D 8	MSIOMF {1 st ,2 nd } {0.5,1.0} MSVOMF {1 st ,4 th } {0.7,1.0} 17280 5760	97.71 96.67 95.68 98.57 96.67 94.44	93.18 95.45	97.32 97.43

Note: Dyn35 and Dyn++ stand for DynTex35 and DynTex++ respectively.

A simple MATLAB code of our modified soft assignment to decompose high-order

2D/3D Gaussian gradients subject to a pre-defined orientation range is available at http://tpnguyen.univ-tln.fr/download/MATCodeIVOM

Acknowledgment

We would like to express our deep gratitude to the reviewers and editors, who pointed out the valuable and insightful remarks allowing us to clarify the presentation of this work. Also, we would like to send many thanks to those in Faculty of IT, HCMC University of Technology and Education, Ho Chi Minh City, Vietnam, who gave us several crucial supports.