Dragana D Sandic´-Stankovi´c

Senior Member, IEEE Dragan D Kukolj

Fellow, IEEE Patrick Le Callet

Quality Assessment of DIBR-Synthesized Views Based on Sparsity of Difference of Closings and Difference of Gaussians

Keywords: Difference of closings, DIBR synthesized view, granulometry, hat-transform, multi-resolution multi-scale image representation, quality prediction

Images synthesized using depth-imagebasedrendering (DIBR) techniques may suffer from complex structural distortions. The goal of the primary visual cortex and other parts of brain is to reduce redundancies of input visual signal in order to discover the intrinsic image structure, and thus create sparse image representation. Human visual system (HVS) treats images on several scales and several levels of resolution when perceiving the visual scene. With an attempt to emulate the properties of HVS, we have designed the no-reference model for the quality assessment of DIBR-synthesized views. To extract a higher-order structure of high curvature which corresponds to distortion of shapes to which the HVS is highly sensitive, we define a morphological oriented Difference of Closings (DoC) operator and use it at multiple scales and resolutions. DoC operator nonlinearly removes redundancies and extracts fine grained details, texture of an image local structure and contrast to which HVS is highly sensitive. We introduce a new feature based on sparsity of DoC band. To extract perceptually important loworder structural information (edges), we use the non-oriented Difference of Gaussians (DoG) operator at different scales and resolutions. Measure of sparsity is calculated for DoG bands to get scalar features. To model the relationship between the extracted features and subjective scores, the general regression neural network (GRNN) is used. Quality predictions by the proposed DoC-DoG-GRNN model show higher compatibility with perceptual quality scores in comparison to the tested state-of-theart metrics when evaluated on four benchmark datasets with synthesized views, IRCCyN/IVC image/video dataset, MCL-3D stereoscopic image dataset and IST image dataset.

I. INTRODUCTION

IEW synthesis using depth-image-based rendering (DIBR) algorithms is important for 3D immersive imaging technologies such as multi-view and free-viewpoint video ith application in various fields: medicine, education, remote surveillance and entertainment. 3D video applications need a large number of views at different viewpoints. At the viewpoints, where video sequences captured by cameras are missing, new video sequences are synthesized from captured video and from their associated depth maps using DIBR algorithms. DIBR techniques introduce new types of distortions that are non-uniformly spread over the image, mainly in the disoccluded areas [START_REF] Bosc | Towards a new quality metric for 3-D synthesized view assessment[END_REF]. The disoccluded areas appear as black holes in image regions that become visible from the position of synthesized view, while invisible from the reference view position. Although the holes are filled synthetically by inpainting or interpolation technique, a texture distortion may appear. Beside the rendering process, distortions in synthesized images appear due to the distortions of texture images and depth maps, which may occur during the process of acquisition, compression and transmission. Distortions in the depth map may cause pixel projection to a wrong position and induce object shifting. The depth compression and view synthesis processes cause obvious distortions at object's edges. The distortions in texture cause intensity variations, while blurring and blocking distortions may appear all over the synthesized image.

The image quality assessment (IQA) of the synthesized view is of great importance for the development of view synthesis algorithms and 3D imaging applications. Traditional IQA metrics designed for natural scene images are unable to capture the non-uniform distortions introduced by synthesis algorithms [START_REF] Bosc | Towards a new quality metric for 3-D synthesized view assessment[END_REF]. The IQA models designed to evaluate distortions of synthesized images due to the rendering process only may not be effective in the evaluation of the complex distortions due to combined sources when texture images and depth maps are also distorted. In multi-view video systems, reference views are not available for all viewpoints. Therefore, referenceless quality metrics are highly desirable. In this paper, we propose a noreference QA model which successfully evaluates distortions due to only a rendering process and complex distortions of synthesized images, which appeared due to the influence of the rendering process and the compromised quality of texture images and depth maps.

The Human Visual System (HVS) uses structural information from the viewing field for cognitive understanding. The disruption structure leads to the reduction of subjective image quality. The receptive fields of simple cells in mammalian primary visual cortex can be characterized as being selective to structure at different spatial scales, spatially localized and oriented [START_REF] Olshausen | Emergence of simple-cell receptive field properties by learning a sparse code for natural images[END_REF]. Visual perception treats images on several levels of resolution simultaneously and this fact must be important in the study of perception [START_REF] Koenderink | The structure of images[END_REF]. By integrating information across spatial scales and resolutions, the HVS builds image representation. Local curvature measurements maybe performed by end-stopped neurons in the visual cortex [START_REF] Dobbins | Endstopped neurons in the visual cortex as a substrate for calculating curvature[END_REF]. The visual cortex is trying to produce an efficient representation in terms of extracting the statistically independent structure in images [START_REF] Olshausen | Sparse coding with an overcomplete basis set: A strategy employed by V1?[END_REF]. The goal of the primary visual cortex and other parts of the brain is to reduce redundancies of input visual signals in order to discover the intrinsic image structure, and thus create sparse image representation. Sparse image representations that capture the salient structures in line with the human perception are reported to be a processing strategy of the nervous system [START_REF] Olshausen | Sparse coding of sensory inputs[END_REF]. Sparsity is meant to extract "higher orders of statistical V dependencies", which cannot be simply described at the image pixel level [START_REF] Olshausen | Sparse coding with an overcomplete basis set: A strategy employed by V1?[END_REF]. We have taken into account the HVS characteristics in the design of the proposed IQA model.

To process the image and extract features from various scales, multi-scale approaches could be suitable. When the operator for generating image copies from fine to coarse scale is convolution by the Gaussian kernel, a linear scale-space can be built where the scale parameter is standard deviation of the Gaussian distribution [START_REF] Witkin | Scale-space filtering: A new approach to multi-scale description[END_REF]. By calculating the difference between the low-pass image copies of neighbor scales, the Difference of Gaussians (DoG) scale space of band-pass detail images is created. After convolution with the Gaussian kernel, the image is uniformly blurred, its edges are blurred and thus their localization may be imprecise. To preserve edges, unblurred and with precise localization, nonlinear morphological filters can be successfully used. Morphological filters are suitable for the construction of scale spaces since they remove structure from an image. The size of the structuring element is a scale parameter. Opening and closing multi-scale morphological filters have been used to create the scale-space of onedimensional gray-scale signals [START_REF] Park | Scale-space using mathematical morphology[END_REF]. Opening and closing are computationally simpler than the Gaussian filter [START_REF] Chen | A multiscanning approach based on morphological filtering[END_REF]. Families of openings and closings, called granulometries, have been used in the morphological multi-scale approaches for shape-size distribution [START_REF] Maragos | Pattern spectrum and multiscale shape representation[END_REF]. Using multi-scale opening and closing, some information is lost from one scale to the next: openings remove peaks of signals, while closings remove valleys. Peaks and valleys, which correspond to the critical points of a contour, are useful for visual perception. The residual of the closing compared to the original represents a bottom-hat transformation [START_REF] Maragos | Morphological systems for multidimensional signal processing[END_REF]. A sequence of hat transforms using structuring elements of increasing size has been used for the creation of the morphological curvature scale-space of 1D signals used for shape analysis [START_REF] Leymarie | Shape features using curvature morphology[END_REF]. Multi-scale hat transforms have been used for extraction of curvature extrema (as a representation for contour) through scales.

The multi-scale image representation at multiple resolutions using the DoG operator for extracting low-order structures has been used in the IQA model of 3D synthesized images [START_REF] Sandic´-Stankovi´c | Blind DIBRsynthesized image quality assessment using multi-scale DoG and GRNN[END_REF]. In this paper, we introduce a novel QA model of DIBRsynthesized views which uses multi-scale image decomposition at multiple resolutions using both the DoG operator to extract low-order structures, and a newly proposed Difference of Closings (DoC) operator to extract higher-order structures (fine grained details, texture) of high curvature, which corresponds to the distortion of shapes to which the HVS is highly sensitive. DoC operator nonlinearly removes redundancies and extracts local structure and contrast to which HVS is highly sensitive. The proposed image decomposition using DoC operator can be described using granulometry by closing or using a multi-scale bottom-hat transform at multiple resolutions. The family of morphological closings with an array of line-shaped verticallyoriented structuring elements of increasing sizes is used to create a set of simplified signals by successively removing image structures (valleys) throughout scales. The size of a SE plays an important role in noise removal and geometrical detail preservation. We define the Difference of Closings (DoC) operator as the difference between morphologically smoothed images of neighboring scales. We believe that features extracted using both DoC and DoG operators at multiple scales and resolutions play complementary roles in characterizing view quality. The complex distortions of DIBR-synthesized images cause changes of both low and high-order structure at DoG and DoC bands, and their sparsities change. We measure sparsity of DoC and DoG bands through scales and resolutions using the Hoyer index [START_REF] Hoyer | Non-negative matrix factorization with sparseness constraints[END_REF] as a scalar feature. To model the relationship between the extracted features and subjective scores, a general regression neural network (GRNN) [START_REF] Specht | A general regression neural network[END_REF] is used. The trained network is then used to map extracted features to the quality score. The proposed DoC-DoG-GRNN model, tested on four datasets of 3D synthesized views, IRCCyN/IVC image dataset [START_REF] Bosc | Towards a new quality metric for 3-D synthesized view assessment[END_REF], MCL-3D stereoscopic image dataset [START_REF] Song | MCL-3D: A database for stereoscopic image quality assessment using 2D-image-plus-depth source[END_REF], IST image dataset [START_REF] Rodrigues | Blind quality assessment of 3-D synthesized views based on hybrid feature classes[END_REF] and IRCCyN/IVC video dataset [START_REF] Bosc | Visual quality assessment of synthesized views in the context of 3D-TV," in 3D-TV System with Depth-Image-Based Rendering[END_REF], shows high compatibility with perceptual quality scores, better than tested state-of-theart metrics.

In the next Section, we review metrics designed for quality assessment of DIBR-synthesized views which use multi-scale or/and multi-resolution image representation. In Section 3, the proposed model is described in more details. The datasets, evaluation criteria, model's parameters, overall performances and comparison to other metrics, as well as discussion and analysis, are presented in Section 4. The last Section concludes the paper.

II. RELATED WORKS

In this Section, we highlight some QA models designed for DIBR-synthesized images based on multi-resolution and/or multi-scale image representations in order to mimic the multiresolution and multi-scale character of HVS. More complete overview of quality models designed for QA of DIBRsynthesized view is presented at [START_REF] Tian | Quality assessment of DIBR-synthesized views: An overview[END_REF].

A Laplacian type morphological pyramid with band-pass detail images with extracted edges has been used in the MP-PSNR metric [START_REF] Sandic´-Stankovi´c | DIBR synthesized image quality assessment based on morphological pyramids[END_REF], as well as its reduced version [START_REF] Sandic´-Stankovi´c | Multi-scale synthesized view assessment based on morphological pyramids[END_REF]. The difference between the appropriate detail images of the reference and the synthesized image pyramid at different resolutions is measured using the mean squared error to emphasize areas around edges that are prone to synthesis artifacts.

A Laplacian pyramid has been used for the extraction of luminance-based features to assess the naturalness of a 3D synthesized view in the newly proposed no-reference LVGC metric [START_REF] Yan | No reference quality assessment for 3D synthesized views by local structure variation and global naturalness change[END_REF]. In order to capture local distortion, structure features extracted by second-order Gaussian derivatives and chromatic features are employed.

Low-pass images of a Gaussian type pyramid have been used for edge extraction in the Edge Intensity Similarity metric, EIS [START_REF] Zhou | Reduced-reference quality assessment of DIBR-synthesized images based on multi-scale edge intensity similarity[END_REF]. The similarity of edge intensity between appropriate edge images of the reference and the synthesized image pyramid through resolutions has been calculated. The Gaussian pyramid of depth image has been used for the extraction of edge profiles in the no-reference QA metric proposed in [START_REF] Li | No-reference quality index of depth images based on statistics of edge profiles for view synthesis[END_REF]. Quality features have been extracted from the statistical distribution of edge regions. The first-order statistical features have been extracted in the gradient magnitude domain and the secondorder statistical features have been extracted in the Laplacianof-Gaussian domain. The extracted features have been used for building a random forest regression-based quality model. A multi-resolution low-pass image representation using Gaussian filters has been used in the reference-based LMS metric developed to evaluate the distortions in the whole synthesis process [START_REF] Zhou | Quality assessment for view synthesis using low-level and mid-level structural representation[END_REF]. A low-level structural representation is calculated using the statistics of gradient intensity and orientation, while a mid-level structural representation is calculated using bag of words for contour description based on sparse coding. Gaussian filter has been applied for the generation of low-pass image copies at two scales for the extraction of features calculated from statistics of wavelet-based fusion of color and depth images in the QA model proposed to be applied before DIBR synthesis [START_REF] Huang | No-reference quality prediction for DIBR-synthesized images using statistics of fused color-depth images[END_REF].

A multi-resolution image representation using downsampling has been used in the blind IQA model, designed using the philosophy that the DIBR-introduced geometry distortions damage the self-similarity characteristic of natural images and the damage degree tends to decrease as the resolution reduces [START_REF] Gu | Using multiscale analysis for blind quality assessment of DIBR-synthesized images[END_REF]. In the following work, presenting a blind IQA model based on Multiscale Natural Scene Statistical analysis (MNSS) [START_REF] Gu | Multiscale natural scene statistical analysis for no-reference quality evaluation of DIBR-synthesized views[END_REF], it was found that the statistical regularity is destroyed in the DIBR-synthesized views. Estimating the deviation of degradations in main structures at different resolutions between a DIBR-synthesized image and the statistical model, the main structure damage can be quantified.

A morphological wavelet transformation has been used in the MW-PSNR metric, as well as its reduced version [START_REF] Sandic´-Stankovi´c | DIBR synthesized image quality assessment based on morphological wavelets[END_REF], [START_REF] Sandic´-Stankovi´c | DIBR-synthesized image quality assessment based on morphological multi-scale approach[END_REF]. Mean squared error has been used to measure the difference between appropriate subbands of the synthesized and the reference image. The high-high wavelet subband has been used in the computationally extremely efficient No-Reference Morphological Wavelet with a Threshold metric guided by the fact that DIBR-synthesized images are characterized by increased high frequency content [START_REF] Sandic´-Stankovi´c | Fast blind quality assessment of DIBR-synthesized video based on high-high wavelet subband[END_REF].

Haar wavelet transformation has been performed on image blocks and image degradation has been measured using histograms of horizontal detail subbands in the reference-based IQA metric [START_REF] Bosc | A wavelet-based image quality metric for the assessment of 3D synthesized views[END_REF]. Discrete Wavelet Transform using CDF 9/7 filters has been used in the blind IQA model which quantifies geometric distortions, global sharpness, and image complexity [START_REF] Wang | Blind quality metric of DIBR-synthesized images in the discrete wavelet transform domain[END_REF].

A multi-scale image representation using the DoG operator at multiple resolutions has been used to extract a low-order structure, edges, and sparsity measure of DoG bands is calculated as a scalar feature in the general regression neural network-based DoG-GRNN model [START_REF] Sandic´-Stankovi´c | Blind DIBRsynthesized image quality assessment using multi-scale DoG and GRNN[END_REF]. A multi-scale image DoG-representation has been used for extraction of two groups of features: the orientation selective statistics features and the texture naturalness features in the random forest regression based SET model designed to capture distortions in the whole view-synthesis process [START_REF] Zhou | No-reference quality assessment for view synthesis using DoG-based edge statistics and texture naturalness[END_REF].

Multiscale representation has been used in the elastic metric and multi-scale trajectory based video quality metric which quantify the amount of temporal structure inconsistencies and unsmooth viewpoint transitions [START_REF] Ling | Quality assessment of free-viewpoint videos by quantifying the elastic changes of multi-scale motion trajectories[END_REF].

III. THE PROPOSED QA MODEL

The proposed model is designed considering the properties of HVS: its high sensitivity to structure with a goal to reveal the intrinsic structure by reducing redundancies and creating sparse representation, but also its multi-resolution and multi-scale property when perceiving the visual scene. To extract a higherorder structure (fine grained details, texture) of high curvature, which corresponds to the distortion of shapes to which the HVS is highly sensitive, and to nonlinearly remove redundancies, we propose the morphological oriented Difference of Closings (DoC) operator and use it to create the sparse overcomplete multi-resolution and multiscale image representation (MR-MS-DoC) of non-negative integer valued DoC bands. We introduce the new feature based on sparsity of the DoC band. To extract a perceptually important low-order structure, edges, and to linearly reduce redundancies, we use non-oriented Difference of Gaussians (DoG) operator to create multi-resolution and multi-scale image representation (MR-MS-DoG). We believe that both low-order and high-order structure play complementary roles in characterizing the view quality. Since the DIBR-synthesized images are characterized by a structurerelated distortion, their representations are with changed loworder and high-order structures and the sparsity of DoC bands and DoG bands are changed. The sparsity of DoC and DoG bands is measured using the Hoyer index as a scalar feature. The extracted features and subjective scores are used to train the general regression neural network (GRNN). The trained GRNN is then used to predict the quality score. The framework of the proposed model is shown on Fig. 1. Before explaining the proposed model in more details, we shortly review the basics of mathematical morphology used in model development.

A. Morphological Filters

Mathematical morphology is a powerful tool in image analysis mainly due to its nonlinearity and shape description properties. Morphological operations use a small pattern characterized by shape and size, called structuring element, to extract useful information. Morphological operators remove For grayscale image function f (x, y) on Z 2 , the basic morphological operators, dilation f ⊕ SE(1) as expanding operation, and erosion f SE (2) as shrinking operation, perform locally on the area defined by the structuring element SE using operators maximum and minimum [START_REF] Haralick | Image analysis using mathematical morphology[END_REF]. u, yv)}

(1)

( f SE)( u, y + v)} (2)
The morphological filter closing C (3) locally modifies geometric signal features iteratively applying dilations and erosions, smooths signal eliminating specific image detail smaller than the structuring element without the global geometric distortion of unsuppressed features [START_REF] Haralick | Image analysis using mathematical morphology[END_REF].

CSE : ( f • SE)(x, y) = (( f ⊕ SE) SE)(x, y) (3)
Visually, the closing filter smooths contours of image dark regions, tends to fuse narrow breaks, eliminates small holes and fills gaps in the contours. Closing is invariant to translations of the structuring element and spatial invariance holds. The closing is extensive,

( f • SE)(x, y) > f (x, y).
The residual produced after the application of closing filter to the image function f (x, y) is Close-Top-Hat (CTH )(4), also called bottom-hat, or black-top-hat [START_REF] Maragos | Morphological systems for multidimensional signal processing[END_REF].

CT H : ( f • SE)(x, y) -f (x, y) ( 4
)
The CTH is region extraction operator which extracts dark details, valleys of the signal. The hat transformation provides an excellent tool for extracting features smaller than a given size from an uneven background.

B. Structuring Elements for Multi-Scale Morphology

For the creation of the morphological multi-scale image representation, an array of structuring elements of increasing size, SE j, 1 ≤ j ≤ N is needed. The structuring element at scale j, SE j, is created using the dilation of the structuring element at the first scale SE1 with itself (5) [START_REF] Maragos | Pattern spectrum and multiscale shape representation[END_REF]. The definition of scale corresponds to the spatial size of the structuring element.

SE j = SE1 ⊕SE1 ... ⊕ SE1, 2 ≤ j ≤ N (5) j times
The shape of SE j is determined by the shape of the primary pattern SE1, and j controls the size.

We have explored the simplest case using an array of linear structuring elements with increasing length. The structuring

C. Image Representation Using Granulometry by Closing

The granulometry is a useful tool for morphological multiscale image analysis and have been used for object size estimation, feature extraction for image segmentation and texture classification. A granulometry quantifies an amount of detail in an image at different scales.

We present the morphological multi-scale multi-resolution image decomposition to extract higher-order structure (fine grained details, texture) of high curvature which corresponds to the distortion of shapes to which the HVS is highly sensitive. The image decomposition can be described using the granulometry by closing at multiple resolutions. The multiscale closings are useful smoothing filters because they preserve the shape and the location of vertical abrupt signal discontinuities (edges) [START_REF] Maragos | Pattern spectrum and multiscale shape representation[END_REF]. At each resolution level i, i = 1,... M, a series of low-pass image copies, si,j , j = 1,..., N (6) with decreasing details is created from the initial image, si,0, by applying a morphological filter closing, using an array of linear structuring elements with increasing length SE j (5), as shown in Fig. 3.

si,j = si,0 • SE j, j = 1,..., N (6)
The morphological filter closing fills in the gulfs and the small holes of si,0 relative to SE j. The size of the structuring element plays an important role in noise removal and geometrical detail preservation. With the scale increase, the length of the structuring element increases, more and more details are filtered out and each filtered image si,j loose more and more structure.

The monotonically increasing sequence of closings, {si,j }, j = 1,..., N, constitutes a granulometry by closing. A granulometry by closing captures image structure darker than neighborhood.

A fast algorithm has been proposed for efficient computation of linear granulometry [START_REF] Vincent | Fast grayscale granulometry algorithms[END_REF]. We define Difference of Closings (DoC) operator [START_REF] Witkin | Scale-space filtering: A new approach to multi-scale description[END_REF] as the difference between morphologically filtered images of adjacent scales.

DoC : di,j = si,j -si,j-1, j = 1,..., N (7)
The sequence of details removed between nearby closing filters, {di,j}, j = 1,..., N, creates a multi-scale image representation at resolution level i. Since the closing is an extensive operator, si,0 < si,1 < ••• < si,N-1 < si,N , the detail images, di,j, contain only non-negative integer values. The DoC band at resolution level 3 and scale 4 created from the image Balloon synthesized from the texture images and depth maps with JPEG distortion is shown on Fig. [START_REF] Dobbins | Endstopped neurons in the visual cortex as a substrate for calculating curvature[END_REF]. The details extracted in DoC bands correspond to perceptually important building blocks of higher-order structure.

Morphological size distribution isolates features from noise at different scales. The difference of closings/openings at successive scale has been used in pattern spectrum as shape-size descriptor for multi-scale shape representation and description [START_REF] Maragos | Pattern spectrum and multiscale shape representation[END_REF]. Morphological size distribution has been used for image noise reduction in the algorithm proposed for improved JPEG compression [START_REF] Peters | A new algorithm for image noise reduction using mathematical morphology[END_REF].

The multi-scale image transformation using DoC operator is reversible without information loss. The initial image at resolution level i, si,0, can be perfectly reconstructed from the detail images of all scales, di,j, j = 1,..., N (all DoC bands) and the low-pass approximation at the highest scale, si,N [START_REF] Park | Scale-space using mathematical morphology[END_REF].

N si,0 = si,N -di,j (8)
j=1

To create the initial image for the next resolution level, si+1,0, the initial image from the current resolution level, si,0, is filtered using morphological filter closing with the structuring element SER and downsampled by a factor of two (σ↓) by rows and by columns [START_REF] Chen | A multiscanning approach based on morphological filtering[END_REF]. The initial image at the first resolution level, s1,0, is the image under test f . Initial images of all resolution levels, si,0, i = 1,..., M + 1, constitute the morphological pyramid of Gaussian type.

s1,0 = f si+1,0 = σ↓(si,0 • SER) i = 1,..., M (9)
Since the morphological operators used in multiresolution decomposition schemes involve only integers and only max, min and addition in their computation the calculation of morphological multiresolution decompositions have low computational complexity.

D. Image Representation Using Multi-Scale Hat-Transform

The close-top-hat (CTH) transform can be used to build the basic element of image semantic structure, and it is closely related to the perceptual quality of images. The hat transform can effectively capture image local structure and contrast to which the HVS is highly sensitive. The CTH operator enhances variations of pixel intensity in a neighborhood determined by the structuring element while preserving edges non-blurred. The hat transforms give important information about the fineness of variations of the gray level, with no need for image decomposition into harmonic frequencies. The hat transform has been used for extraction curvature, as one of the most powerful approaches for the representation and interpretation of objects in an image [START_REF] Leymarie | Shape features using curvature morphology[END_REF]. There is psychophysical, physiological and mathematical support in favor of using curvature as a representation for contours.

We can also describe MR-MS-DoC image decomposition using multi-scale hat transform at multiple resolutions. To create multi-scale close-top-hat transform at resolution level i,{cthi,j }, j = 1,..., N, [START_REF] Maragos | Pattern spectrum and multiscale shape representation[END_REF] we also use the granulometry by closing, {si,j}, j = 1,..., N given in [START_REF] Olshausen | Sparse coding of sensory inputs[END_REF].

cth i,j = si,j -si,0 j = 1,..., N, ( 10
)
where the initial images at different resolutions, si,0, i = 1,..., M+1, are created as defined in [START_REF] Chen | A multiscanning approach based on morphological filtering[END_REF]. Since the closing is an extensive operator, the CTH transform creates detail image that contains pixels of non-negative integer values.

We can also define the Difference of Closings (DoC) operator as the difference between close-top-hat transforms of nearby scales [START_REF] Maragos | Morphological systems for multidimensional signal processing[END_REF], as shown in Fig. 5.

DoC : di,j = cthi,j -cthi,j-1 j = 2,..., N di,1 = cthi,1 (11)
where di,j is the detail image (DoC band) at resolution level i and scale j. The sequences of details, {di,j}, at scales j = 1,..., N, and resolution levels i = 1,..., M create multiresolution multiscale DoC-based image representation.

E. Image Representation Using DoG Operator

Edges are important for visual perception playing a major role in the recognition of image content. The Difference of Gaussians (DoG) operator has been proposed as fast approximation of LoG operator used for edge extraction [START_REF] Marr | Theory of edge detection[END_REF]. The DoG operator has been used in the DoG-GRNN model designed for the QA of the DIBR-synthesized images to extract low-order structure (edges) at multiple scales and resolutions [START_REF] Sandic´-Stankovi´c | Blind DIBRsynthesized image quality assessment using multi-scale DoG and GRNN[END_REF].

In this work, DoG-based image representation of low-order structure (edges) is used, together with the DoC-based image representation of higher-order structure (fine details, texture) to improve the IQA model's performances. At each resolution level i, i = 1,... P, an array of Gaussian smoothed images, gi,j, j = 1,..., Q is generated by convolution of the input image gi,0 with the Gaussian function G(x, y,σj) using an array of standard deviations of the Gaussian distribution σj, j = 1,..., Q as scale parameters [START_REF] Leymarie | Shape features using curvature morphology[END_REF]. The Difference of Gaussians (DoG) operator is defined as the difference of two nearby Gaussian smoothed images [START_REF] Sandic´-Stankovi´c | Blind DIBRsynthesized image quality assessment using multi-scale DoG and GRNN[END_REF].

DoG : ei,j = gi,j-1 -gi,j (13)
The DoG band ei,j , is a band-pass filtered image which contains all frequencies between the cut-off frequencies of the two Gaussians which correspond to edge lines. Edges of different fineness are detected at different scales as well as the noise due to synthesis distortion. The DoG bands at resolution levels 1 and 5, at scale 2, created from the image Balloon synthesized from the texture images and depth maps with JPEG distortion are shown on Fig. 6. DoG bands of higher resolution contain enhanced edges while DoG bands of lower resolution remind saliency maps.

The initial image for the next resolution level, gi+1,0 is generated from the last Gaussian smoothed image of the current resolution level gi,Q by downsampling it with a factor of two by rows and by columns [START_REF] Hoyer | Non-negative matrix factorization with sparseness constraints[END_REF].

g1,0 = f gi+1,0 = σ↓(gi,Q), i = 1,..., P -1 ( 14
)
The multi-resolution multi-scale DoG-based image representation consists of detail images ei,j at resolution levels i, i = 1,... P, and scales j, j = 1,..., Q.

F. Measure of Sparsity Sparsity of signal representation has been important concept in signal analysis, compression, sampling used in diverse areas such as image processing, medical imaging and face recognition. The Hoyer index, calculated as normalized relationship between l1 and l2 norms, has been proposed as a measure of sparseness of the image representation obtained using non-negative matrix factorization [START_REF] Hoyer | Non-negative matrix factorization with sparseness constraints[END_REF]. The Hoyer index quantifies how much energy is packed into only a few components. It takes value 0 when all coefficients are equal and it is 1 when only a single coefficient is non-zero. The It satisfies five heuristic criteria (out of six) used for comparing sparseness measures [START_REF] Hurley | Comparing measures of sparsity[END_REF]. Hoyer index has been used in the IQA models of DIBR-synthesized images based on morphological pyramid, morphological wavelets [START_REF] Bokan | Blind DIBR-synthesized image quality assessment based on sparsity features in morphological multiscale domain[END_REF] and DoG-based image representation [START_REF] Sandic´-Stankovi´c | Blind DIBRsynthesized image quality assessment using multi-scale DoG and GRNN[END_REF]. We use the Hoyer index to measure sparsity of DoC and DoG bands in the proposed model. The Hoyer indices of DoC and DoG bands from the interval [0, 1] are concatenated and form the neural network input vector to GRNN.

G. Feature Pooling

In the field of IQA, machine learning techniques are widely used to model the human perception achieving excellent performance as they build functional relationship between the image features and subjective quality scores. A Generative Adversarial Network has been used in the no-reference quality metric of free-viewpoint images which uses masks to mimic dis-occluded regions [START_REF] Ling | Re-visiting discriminator for blind free-viewpoint image quality assessment[END_REF]. A General Regression Neural Network (GRNN) as a powerful regression tool based on statistical principles shows good results even with smaller learning datasets and is particularly advantageous with sparse data in real-time environment [START_REF] Specht | A general regression neural network[END_REF]. GRNN has been used to model the relationship between perceptually relevant features and subjective scores in IQA model which handle multiple distortion [START_REF] Li | Blind image quality assessment using a general regression neural network[END_REF]. It has been used in the IQA models of DIBRsynthesized images [START_REF] Bokan | Blind DIBR-synthesized image quality assessment based on sparsity features in morphological multiscale domain[END_REF], [START_REF] Sandic´-Stankovi´c | Blind DIBRsynthesized image quality assessment using multi-scale DoG and GRNN[END_REF]. GRNN has been compared to support vector regression algorithm (SVR) in the models proposed for IQA of synthesized images based on morphological pyramid decomposition and morphological wavelet [START_REF] Bokan | A no-reference synthesized view quality assessment using statistical features in morphological multiscale domain[END_REF]. It has been shown that SVR-based model shows lower performances than GRNN-based model. We have used GRNN to model the relationship between the extracted features and subjective scores. The learned model is then used to predict the image/video quality. The only parameter is the spread, that represents the width of radial basis function of GRNN. The GRNN has been implemented using Matlab function newgrnn.

IV. EXPERIMENTAL RESULTS

In this section, used datasets and evaluation criteria are shortly described. We've described the choise of model's parameters and show the performances of the proposed model calculated using datasets. We've also presented the parameter set of the proposed model for the prediction of images synthesized from compressed content with improved computational efficiency. The proposed DoC-DoG-GRNN model is compared to other IQA models and their performances are analyzed.

A. Datasets and Evaluation Criteria

The proposed metric is evaluated using publicly available datasets of DIBR-synthesized images and videos, namely the IRCCyN/IVC DIBR image dataset [START_REF] Bosc | Towards a new quality metric for 3-D synthesized view assessment[END_REF], [START_REF]IRCCyN/IVC DIBR Image Quality Dataset[END_REF], the MCL-3D stereoscopic image dataset [START_REF] Song | MCL-3D: A database for stereoscopic image quality assessment using 2D-image-plus-depth source[END_REF], [START_REF]MCL-3D Stereoscopic Image Quality Dataset[END_REF] and the IRCCyN/IVC DIBR video dataset [START_REF] Bosc | Visual quality assessment of synthesized views in the context of 3D-TV," in 3D-TV System with Depth-Image-Based Rendering[END_REF], [START_REF]IRCCyN/IVC DIBR Video Quality Dataset[END_REF].

IRCCyN/IVC image/video dataset contains 12 reference frames/videos from three multi-view video plus depth sequences and 84 frames/videos synthesized using seven DIBR-synthesis algorithms. The frames are synthesized by a single-view synthesis (a frame at the "virtual viewpoint" is synthesized using captured texture frame and its associated depth frame from single viewpoint). The images/videos of these two datasets are synthesized from undistorted data, so the focus is on rendering artifacts. Video sequences BookArrival is of 15 frames/s rate while Lovebird and Newspaper are of 30 frames/s rate. The frames resolution is 1024 × 768 pixels. Mean Opinion Scores (MOS) are provided for both the synthesized and for the reference frames/videos. The Difference Mean Opinion Scores (DMOS) is calculated by measuring the difference between the reference and the synthesized image MOS.

The MCL-3D dataset contains 684 synthesized stereoscopic image pairs synthesized from nine image-plus-depth sources and associated mean opinion score (MOS) values. One third of images are of resolution 1024 × 728 and two thirds are of resolution 1920 × 1080. The majority of images (648 stereo images) are synthesized using both left and right views distorted by one of six distortion types: Gaussian blur, additive white noise, down-sampling blur, JPEG and JPEG-2000 compression and transmission error. Distortions have been symmetrically applied to left and right viewpoints to either the texture images or the depth maps or both texture and depth maps at four distortion levels before stereoscopic image rendering using the View Synthesis Reference Software [START_REF] Tanimoto | View Synthesis Algorithm in View Synthesis Reference Software 3[END_REF]. The minority of images (36 stereo images) have been synthesized from undistorted texture and depth maps of single view using four DIBR synthesis algorithms. They contain only the distortions caused by imperfect rendering.

In order to evaluate performances of the proposed model and other metrics, three evaluation criteria were used: Root Mean Squared Error (RMSE) to compute the prediction error, Pearson Linear Correlation Coefficient (PLCC) to compute prediction accuracy and Spearman Rank-order Correlation Coefficient (SROCC) for prediction monotonicity. The performances of the proposed model and all tested metrics are calculated using the Differential Mean Opinion Score (DMOS) for the IRCCyN/IVC image/video dataset and using the Mean Opinion Score (MOS) for the MCL-3D dataset. A novel methodology for performance evaluation which takes into account statistical significance of subjective scores, with the ability to analyse the data statistically after merging results from multiple datasets, can be used when the standard deviation of subjective scores is provided [START_REF] Krasula | On the accuracy of objective image and video quality models: New methodology for performance evaluation[END_REF].

B. Performance Calculation Using Train-Test Process

In the evaluation of the proposed model, the k-fold cross validation strategy is used to split the dataset with D images to k disjoint test subsets of similar size, each with D/k images (for k = 5 it is 20% images of the dataset), and to select the train subsets with D-D/k images (for k = 5 it is 80% images of the dataset) such that there is not overlap between the test and the train subsets. GRNN has been trained by mapping the features calculated from the train subset to the subjective scores. Then, the trained model has been used to predict the quality of the images from the test subset. The k-fold cross validation procedure is performed 1000 times to avoid over-fitting. We've analyzed the 5 ways for the performance calculation marked as Case1, Case1A, Case 2, Case 2A and Case 2B.

Case 1: At each iteration of the k-fold cross validation strategy, GRNN is used to predict the scores of the test subset. Then, the performances of the test subset predicted scores are calculated. The median value of the performances through cross-validation iterations and through 1000 repetitions is calculated as the final model's performances.

Case 1A: The only difference to Case 1 is that the test subset predicted scores are additionally nonlinearly mapped to the predicted DMOSp using a five-parameter function ( 15) before performance calculation.

DMOSp = a • score 4 + b • score 3 + c • score 2 +d • score + e ( 15
)
The parameters a,b,c,d,e are obtained through regression to minimize the difference between DMOS and DMOSp. Case 2: At each iteration of the k-fold cross-validation strategy, GRNN is used to predict the scores of the test subset. Then, we concatenate the predicted scores from k iterations of single cross-validation process to get the array of the whole dataset predicted scores. We calculate the median of the whole dataset predicted scores through 1000 repetitions. Finally, we calculate the performances on the median of the whole dataset predicted scores as the final model's performances.

Case 2A: The only difference to Case 2 is that median of the whole dataset predicted scores are additionally nonlinearly mapped using a function ( 15) before performance calculation.

Case 2B: The only difference to Case 2 is that the test subset predicted scores are additionally nonlinearly mapped using a function [START_REF] Specht | A general regression neural network[END_REF] before concatenation to the whole dataset scores.

C. Model's Parameters and Performances

In this section, parameters selection is described. Parameters of the DoC-DoG-GRNN model are the number of resolution levels and the number of scales of DoC-based and DoG-based decompositions and the spread that represents the width of radial basis function of GRNN. For more efficient calculation, we've first explored the parameters of DoC-based decomposition using the DoC-GRNN model. Then, using the selected parameters of DoC-based decomposition, we've explored the parameters of DoG-based decomposition using DoC-DoG-GRNN model. The selected parameters allow the highest model's performances. We have selected parameter set 1 using IRCCyN/IVC image dataset. Then, using the selected parameters, we've calculated the performances of the proposed model for three datasets.

We've explored performances of the DoC-GRNN model using different number of resolution levels (1-7), scales [START_REF] Bosc | Towards a new quality metric for 3-D synthesized view assessment[END_REF][START_REF] Olshausen | Emergence of simple-cell receptive field properties by learning a sparse code for natural images[END_REF][START_REF] Koenderink | The structure of images[END_REF][START_REF] Dobbins | Endstopped neurons in the visual cortex as a substrate for calculating curvature[END_REF][START_REF] Olshausen | Sparse coding with an overcomplete basis set: A strategy employed by V1?[END_REF][START_REF] Olshausen | Sparse coding of sensory inputs[END_REF][START_REF] Witkin | Scale-space filtering: A new approach to multi-scale description[END_REF] and different values of spread parameter (0.001-0.05). The best performances are achieved when 5 resolution levels and 5 scales are used. Performances are further improved when the DoC bands from the first scale, di,1, and the second scale, di,2, are omitted, and three DoC bands at scales 3-5, di,3 -di,5 [START_REF] Witkin | Scale-space filtering: A new approach to multi-scale description[END_REF], i = 1,...,5, of each resolution level are used. To create initial images for multi-scale image decomposition at different resolutions, morphological close filter with lineshaped vertically oriented structuring element of length 2, SER = SE1, is used. The sparsity of 15 DoC bands is calculated using the Hoyer index as a scalar feature. The lowpass image copy of the lowest resolution level, s6,0, is also used for the feature extraction and the total number of features of DoC-GRNN model is [START_REF] Song | MCL-3D: A database for stereoscopic image quality assessment using 2D-image-plus-depth source[END_REF]. In order to improve model's performances, we've also used image representation generated using the DoG operator to extract low-order structure at multiple scales and resolutions. For the generation of DoG bands at scales j, j = 1,..., Q, Gaussian function using an array of standard deviations, σj, is applied. Standard deviations at different scales, σj, are calculated such that σj = k j -1 , where constant multiplicative factor k is determined by the number of low-pass images at one resolution level, Q: k = 2 1/Q . The selected size of the Gaussian kernel window is integer(6 * σj). We have explored the performances of the DoC-DoG-GRNN model using DoC-based decomposition with 5 resolution levels and scales 3-5 at each level and DoG-based decomposition with different number of resolution levels (1-7) and scales (1-7), with different values of spread parameter (0.001-0.05). Although high performances of the combined DoC-DoG-GRNN model, with PLCC higher than 0.88, is achieved using different number of resolution levels and scales of DoG-based decomposition, we've selected P = 5 resolution levels and Q = 6 scales to achieve the highest value of SROCC. From DoG bands, ei,1-ei,6, i = 1,...,5, 30 scalar features are calculated using Hoyer index as a measure of sparsity.

The GRNN input vector of the combined DoC-DoG-GRNN model is created by concatenation of 16 features calculated from DoC-based decomposition and 30 features calculated from DoG bands. The SROCC of the combined DoC-DoGGRNN model, DoC-GRNN and DoG-GRNN models with fluctuation of the GRNN spread parameter for three datasets, IRCCyN/IVC image/video and MCL-3D dataset, are shown on Fig. 7. High performances of DoC-GRNN model are achieved mostly thanks to the capacity of morphological series to capture higher order properties of spatial random processes. The combined model, DoC-DoG-GRNN, shows better performances than the DoC-GRNN and DoG-GRNN models, for all datasets. From the results we conclude that both loworder structure and high-order structure extracted at multiple scales and resolutions should be considered for improved results in quality assessment of DIBR-synthesized views. We've selected the spread parameter 0.014 as the value for the best performances of IVC image dataset. Performances of the proposed DoC-DoG-GRNN model for three datasets, IRCCyN/IVC image/video and MCL-3D dataset, calculated on five ways, Case 1, Case 1A, Case 2, Case 2A and Case 2B are shown in Table I. The best performances for all datasets are achieved in Case 2B.

D. The Sensitivity on the Cross-Validation Partition Ratio

We've tested the performances of the proposed QA model using three partition ratio, k = 10, 5, 4 of the k-fold cross-validation strategy, and the results calculated according to the Case 1 procedure are shown in Table II. With decrease of the training data partition size, the prediction performances decrease, although the partition ratio, k, has small influence on performance. For all tested partitions, SROCC is higher than 91% for MCL-3D dataset and higher than 82% for IRCCyN/IVC image/video dataset. Performances shown in all other tables are calculated using k = 5.

E. Comparison of the Proposed Model to Other IQA Models

To validate the effectiveness of the proposed model, it is compared to the models designed to evaluate the quality of DIBR-synthesized images and to the models designed to evaluate the quality of natural images. In the evaluation of the training-free models, their scores are nonlinearly mapped to the predicted scores using a cubic logistic function [START_REF]Test Plan for Evaluation of Video Quality Models for Use With High Definition TV Content[END_REF]. In the calculation of video sequence score using training-free metrics, median time pooling of frame's scores is applied, using all frames. Performances of the proposed and other tested models evaluated on IRCCyN/IVC image/video dataset and MCL-3D stereo image dataset are presented in Table III. Table IV contains performances of the metrics designed for QA of DIBRsynthesized images with rendering distortions only, taken from the papers where they were proposed.

The first group of metrics used for comparison designed to evaluate quality of DIBR-synthesized images, which use machine learning-based techniques, contains no-reference models, LVGC [START_REF] Yan | No reference quality assessment for 3D synthesized views by local structure variation and global naturalness change[END_REF], SET [START_REF] Zhou | No-reference quality assessment for view synthesis using DoG-based edge statistics and texture naturalness[END_REF], CSC-NRM [START_REF] Ling | How to learn the effect of non-uniform distortion on perceived visual quality? Case study using convolutional sparse coding for quality assessment of synthesized views[END_REF] and referencebased models, Q [START_REF] Ling | Perceptual representations of structural information in images: Application to quality assessment of synthesized view in FTV scenario[END_REF], Q [START_REF] Wang | Measuring coarseto-fine texture and geometric distortions for quality assessment of DIBRsynthesized images[END_REF], ST-IQM [START_REF] Ling | Image quality assessment for free viewpoint video based on mid-level contours feature[END_REF] and LMS [START_REF] Zhou | Quality assessment for view synthesis using low-level and mid-level structural representation[END_REF]. The proposed DoC-DoG-GRNN model is a no-reference machine learning-based technique. In the evaluation of the DoC-DoG-GRNN model, cross-validation process described in III B is applied and the performances are calculated as it is described in Case 2B. In the calculation of video sequence score, we've used non-linear time pooling of frame's features. The best performances of the proposed DoC-DoGGRNN model are achieved when median time pooling is used for DoC-based features and max time pooling is used for DoG-based features. We've calculated the features of all frames. The performances of the random-forest-based metrics designed to evaluate quality of DIBR-synthesized images with distortions due to whole synthesis process, LVGC using 310 features and SET using 51 features, are taken from [START_REF] Yan | No reference quality assessment for 3D synthesized views by local structure variation and global naturalness change[END_REF] and [START_REF] Zhou | No-reference quality assessment for view synthesis using DoG-based edge statistics and texture naturalness[END_REF]. The Convolutional Sparse Coding-based No-Reference Model, CSC-NRM (shown in Table IV), uses convolutional sparse coding to learn from local regions and computes a sparse representation of an image [START_REF] Ling | How to learn the effect of non-uniform distortion on perceived visual quality? Case study using convolutional sparse coding for quality assessment of synthesized views[END_REF].

The performances of the reference-based models designed to evaluate the quality of DIBR-synthesized images using machine learning techniques, Q [START_REF] Ling | Perceptual representations of structural information in images: Application to quality assessment of synthesized view in FTV scenario[END_REF], Q [START_REF] Wang | Measuring coarseto-fine texture and geometric distortions for quality assessment of DIBRsynthesized images[END_REF], ST-SIQM [START_REF] Ling | Image quality assessment for free viewpoint video based on mid-level contours feature[END_REF] and LMS [START_REF] Zhou | Quality assessment for view synthesis using low-level and mid-level structural representation[END_REF] are taken from the papers where they were introduced. Q [START_REF] Ling | Perceptual representations of structural information in images: Application to quality assessment of synthesized view in FTV scenario[END_REF] quantifies the impact of structure-related distortion on perceived quality of synthesized images using the low-level contour descriptor, the mid-level contour category descriptor and the task-oriented non-natural structure descriptor. Q [START_REF] Wang | Measuring coarseto-fine texture and geometric distortions for quality assessment of DIBRsynthesized images[END_REF] estimates the degree of texture and geometric distortions using coarse-scale and fine-scale features using the fact that the HVS first produces a coarse perception of the global image, and then focuses on specific local areas for a fine perception of image quality. Sketch Token-based Synthesized Image Quality Model, ST-SIQM [START_REF] Ling | Image quality assessment for free viewpoint video based on mid-level contours feature[END_REF], measures how classes of contours change after synthesis from higher semantic level using a mid-level contour descriptor.

The metrics from the first group can reliably predict the quality of DIBR-synthesized images from IRCCyN/IVC image dataset. The proposed model DoC-DoG-GRNN outperforms all other methods from all groups, achieving high performances, PLCC = 0.93, SROCC = 0.889, in the evaluation of IRCCyN/IVC image dataset. The metrics DoC-DoG-GRNN, LVGC, SET, Q [START_REF] Wang | Measuring coarseto-fine texture and geometric distortions for quality assessment of DIBRsynthesized images[END_REF] and LMS [START_REF] Zhou | Quality assessment for view synthesis using low-level and mid-level structural representation[END_REF] can also reliably evaluate the MCL-3D dataset. DoC-DoG-GRNN, LVGC and SET models achieve high performances in the evaluation of MCL-3D dataset. The proposed DoC-DoG-GRNN model is comparable to the best state of the art LVGC model in the evaluation of MCL-3D datasets. The proposed DoC-DoG-GRNN model can also evaluate the IRCCyN/IVC video dataset with high performances, PLCC = 0.9130, SROCC = 0.9035, outperforming all other methods from all groups.

The second group of metrics contains training-free metrics designed to evaluate DIBR-synthesized images with rendering distortions, namely blind metrics: Q [START_REF] Wang | Blind quality metric of DIBR-synthesized images in the discrete wavelet transform domain[END_REF], MNSS [START_REF] Gu | Multiscale natural scene statistical analysis for no-reference quality evaluation of DIBR-synthesized views[END_REF], Q [START_REF] Jakhetiya | Kernelridge regression-based quality measure and enhancement of threedimensionalsynthesized images[END_REF], OMIQA [START_REF] Jakhetiya | A highly efficient blind image quality assessment metric of 3-D synthesized images using outlier detection[END_REF], APT [START_REF] Gu | Modelbased referenceless quality metric of 3D synthesized images using local image description[END_REF], NIQSV [START_REF] Tian | NIQSV: A no reference image quality assessment metric for 3D synthesized views[END_REF], NIQSV+ [START_REF] Tian | NIQSV+: A noreference synthesized view quality assessment metric[END_REF] and reference-based metrics Q [START_REF] Wang | Quality assessment for DIBR-synthesized images with local and global distortions[END_REF], LoGS [START_REF] Li | Quality assessment of DIBR-synthesized images by measuring local geometric distortions and global sharpness[END_REF], PSNR' [START_REF] Tian | A full-reference image quality assessment metric for 3D synthesized views[END_REF], EIS [START_REF] Zhou | Reduced-reference quality assessment of DIBR-synthesized images based on multi-scale edge intensity similarity[END_REF], EM-IQA [START_REF] Ling | Image quality assessment for DIBR synthesized views using elastic metric[END_REF], MP-PSNR [START_REF] Sandic´-Stankovi´c | Multi-scale synthesized view assessment based on morphological pyramids[END_REF], CT-IQA [START_REF] Ling | Quality assessment for synthesized view based on variable-length context tree[END_REF], MW-PSNR [START_REF] Sandic´-Stankovi´c | DIBR synthesized image quality assessment based on morphological wavelets[END_REF], BF-M [START_REF] Ling | The role of structure and textural information in image utility and quality assessment tasks[END_REF]. The performances of MNSS are taken from [START_REF] Gu | Multiscale natural scene statistical analysis for no-reference quality evaluation of DIBR-synthesized views[END_REF]. The joint blind quality assessment and enhancement algorithm for 3D synthesized images Q [START_REF] Jakhetiya | Kernelridge regression-based quality measure and enhancement of threedimensionalsynthesized images[END_REF] use a global predictor, Kernel Ridge Regression, to identify the surface of the regions with geometric distortions with low prediction error and integrates the Natural Image Quality Evaluator to judge the structural distortions. Efficient OMIQA [START_REF] Jakhetiya | A highly efficient blind image quality assessment metric of 3-D synthesized images using outlier detection[END_REF] capture the geometric and structural distortions by identifying and removing geometrically distorted pixels, outliers, using nonlinear median filtering. Autoregression Plus Threshold (APT) model [START_REF] Gu | Modelbased referenceless quality metric of 3D synthesized images using local image description[END_REF] captures the geometry distortion by calculating the error between the DIBR-synthesized image and its autoregression-based local prediction, and uses the threshold to highlight the most important regions. The five parameter nonlinear logistic function has been applied to APT scores before performances calculation as in [START_REF] Gu | Modelbased referenceless quality metric of 3D synthesized images using local image description[END_REF]. NIQSV [START_REF] Tian | NIQSV: A no reference image quality assessment metric for 3D synthesized views[END_REF] and NIQSV+ [START_REF] Tian | NIQSV+: A noreference synthesized view quality assessment metric[END_REF] calculate the error between the DIBR-synthesized image and its morphologically filtered approximation and use morphological gradient to highlight the most important edge regions. NIQSV+ additionally detects black holes and stretching distortion. NIQSV+ is calculated with the following parameters: color weight K1 = 0.5, black hole weight k_z = 200 and C = 1.

Reference-based Q [START_REF] Wang | Quality assessment for DIBR-synthesized images with local and global distortions[END_REF] captures holes and strip distortions to characterize the local quality of DIBR-synthesized image by analyzing local similarity and estimates the global sharpness using the Just Noticed Difference. LoGS [START_REF] Li | Quality assessment of DIBR-synthesized images by measuring local geometric distortions and global sharpness[END_REF] quantifies local geometric distortions in disoccluded regions and global sharpness has been quantified using a reblurring-based strategy. The simple weighted PSNR, PSNR' [START_REF] Tian | A full-reference image quality assessment metric for 3D synthesized views[END_REF], compensates the object shift and uses a disparity map as a mask to weight the final distortion. The performances of multi-scale edge intensity similarity metric, EIS, are taken from [START_REF] Zhou | Reduced-reference quality assessment of DIBR-synthesized images based on multi-scale edge intensity similarity[END_REF]. The training-free metrics of the second group can successfully predict quality of images from IRCCyN/IVC image dataset although there is a need for the performance improvement. Blind metrics Q [START_REF] Wang | Blind quality metric of DIBR-synthesized images in the discrete wavelet transform domain[END_REF], Q [START_REF] Jakhetiya | Kernelridge regression-based quality measure and enhancement of threedimensionalsynthesized images[END_REF], MNSS and reference-based metrics, Q [START_REF] Wang | Quality assessment for DIBR-synthesized images with local and global distortions[END_REF], LoGS and PSNR' show higher performances when evaluating IRCCyN/IVC image dataset. Blind models OMIQA, APT and NIQSV+ show low performances when they evaluate MCL-3D dataset. Referencebased metrics, Q [START_REF] Wang | Quality assessment for DIBR-synthesized images with local and global distortions[END_REF], LoGS, MP-PSNR and MW-PSNR, can be used for the evaluation of MCL-3D dataset but with limited performances. Metrics from this group show lower performances in prediction of the video quality. Among training-free metrics, OMIQA [START_REF] Jakhetiya | A highly efficient blind image quality assessment metric of 3-D synthesized images using outlier detection[END_REF] achieves the best performances in the evaluation of IRCCyN/IVC video dataset.

The third group of metrics contains reference-based metrics designed for the evaluation of natural images. We've tested classic PSNR, structure-based metrics, SSIM, MS-SSIM [START_REF] Wang | Multiscale structural similarity for image quality assessment[END_REF], IW-SSIM [START_REF] Wang | Information content weighting for perceptual image quality assessment[END_REF], GSM [START_REF] Liu | Image quality assessment based on gradient similarity[END_REF], GMSD [START_REF] Xue | Gradient magnitude similarity deviation: A highly efficient perceptual image quality index[END_REF], PSIM [START_REF] Gu | A fast reliable image quality predictor by fusing micro-and macro-structures[END_REF], saliencybased metric VSI [START_REF] Zhang | VSI: A visual saliency-induced index for perceptual image quality assessment[END_REF] and sparseness-based SSRM model [START_REF] Ahar | From sparse coding significance to perceptual quality: A new approach for image quality assessment[END_REF]. MS-SSIM [START_REF] Wang | Multiscale structural similarity for image quality assessment[END_REF] combines similarity measures calculated at different resolutions of low-pass pyramids. Information content Weighting SSIM measure, IW-SSIM [START_REF] Wang | Information content weighting for perceptual image quality assessment[END_REF], combines weighting maps of information content extracted from a Laplacian pyramid and multi-scale structural simularity measure.

The Gradient Structure Metric (GSM) [START_REF] Liu | Image quality assessment based on gradient similarity[END_REF] uses gradient similarity to measure the change in contrast and structure. The Gradient Magnitude Similarity Deviation (GMSD) model [START_REF] Xue | Gradient magnitude similarity deviation: A highly efficient perceptual image quality index[END_REF] predicts image quality using gradient magnitude similarity (GMS) between the reference and the distorted images combined with the pooling using standard deviation of GMS map. The Perceptual SIMilarity (PSIM) model [START_REF] Gu | A fast reliable image quality predictor by fusing micro-and macro-structures[END_REF] uses the gradient magnitude maps at two scales, at a small scale to measure the variations in macrostructure in order to emulate the human's basic perception, and at a large scale to reflect the variations in microstructure and to emulate the human's detailed perception. Visual Saliency-based Index (VSI) [START_REF] Zhang | VSI: A visual saliency-induced index for perceptual image quality assessment[END_REF] use visual saliency map (it can reflect how "salient" a local region is to the HVS) as a feature map to characterize the quality of local image regions and as a weighting function at the scorepooling stage. Sparseness Significance Ranking Measure model (SSRM) [START_REF] Ahar | From sparse coding significance to perceptual quality: A new approach for image quality assessment[END_REF] is based on the sparsity hypothesis that the visual quality assessment should be compatible with sparse coding, which is one of the main strategies of redundancy reduction implemented in our brain. The metrics of the third group can be successfully used to assess the quality of images from MCL-3D dataset. Among these models, SSRM shows the best performances in the evaluation of MCL-3D dataset. The metrics of this group show low performances when they are used to predict the quality of images/videos from the IRCCyN/IVC image/video dataset.

F. Generalization Ability

We've explored the generalization ability of the proposed DoC-DoG-GRNN model using the IRCCyN/IVC dataset for training and subsets of IVY dataset [START_REF] Jung | Critical binocular asymmetry measure for the perceptual quality assessment of synthesized stereo 3D images in view synthesis[END_REF] for testing. IVY dataset contains 84 stereo images synthesized from undistorted content of 7 sources using 4 DIBR synthesis algorithms [START_REF]Reference Software for Depth Estimation and View Synthesis[END_REF]- [START_REF] Yoon | Inter-view consistent hole filling in view extrapolation for multi-view image generation[END_REF] by view extrapolation using only one reference view. IVY dataset has been created to study the effects of binocular asymmetry caused by mismatch between left and right image on quality of synthesized stereo images.

Four subsets of IVY dataset are created considering four DIBR synthesis algorithm. The best performances are achieved in evaluation of subset 1, which contains images synthesized using VSRS algorithm [START_REF]Reference Software for Depth Estimation and View Synthesis[END_REF], as shown in Table V. The lowest performances are achieved in evaluation of subset 4, which contains images synthesized using algorithm [START_REF] Yoon | Inter-view consistent hole filling in view extrapolation for multi-view image generation[END_REF]. Algorithm [START_REF] Yoon | Inter-view consistent hole filling in view extrapolation for multi-view image generation[END_REF] achieves consistency between the virtual views without distortion of depth values and allows generation of high quality image. The proposed model fails in the evaluation of synthesized images of high quality (without shapes and edges distortion). The performances of the proposed model evaluated on IVY dataset, when it is used for both training and testing calculated as case1A, are also shown in the Table V.

G. QA of Images Synthesized From Distorted Content

In Section IV C, we've selected model's parameters (parameter set 1) using IRCCyN/IVC image dataset which contains images synthesized from undistorted content. It has been shown that the proposed model using parameter set 1 shows high performances when predicting the quality of images of MCL-3D dataset, mainly synthesized from distorted content. For MCL-3D dataset, slightly better performances are achieved when the proposed model uses parameter set 2: DoC-based decomposition at 5 resolution levels with scales 2-7 at each resolution level, di,2-di,7, i = 1,...5, DoG-based decomposition at 5 resolution levels with scales 3-6 at each resolution We analyze performances of the proposed DoC-DoGGRNN model using parameter set 2 by individual distortion types applied to texture images or/and depth maps. We've used six subsets of the MCL-3D dataset, each with 108 images, synthesized from the content distorted by one of six distortion types: JPEG, JPEG-2000 compression, Gaussian blur, additive white noise, down-sampling blur and transmission error applied at four distortion levels. The proposed model achieves the best performance in evaluation of the images synthesized from the blurred content and compressed content, as shown in Table VII.

To further study model's performances in the evaluation of images synthesized from compressed content, we've used IST dataset [START_REF] Rodrigues | Blind quality assessment of 3-D synthesized views based on hybrid feature classes[END_REF] which contains images extracted from video sequences synthesized from texture/depth view pairs encoded with 3DV-ATM v10.0 [START_REF]3D-AVC: 3DV-ATM V10[END_REF], which is the 3D-AVC [START_REF] Hannuksela | 3DAVC Draft Text 7[END_REF] reference software for MVD coding. IST dataset contains 180 images from 10 MVD sequences synthesized using both left and right views compressed with different value of quantization parameter. The images have been synthesized by two synthesis algorithms, the VSRS-1D-Fast [START_REF]VSRS-1D-Fast[END_REF] and the VSIM [START_REF] Farid | Depth image based rendering with inverse mapping[END_REF].

Although the synthesized images contain both rendering and compression artifacts, the focus is on the distortion introduced by compressed texture and depth maps. To achieve computationally more efficient prediction of images synthesized from compressed content, we've selected The model achieves high performances using parameter set 3 in the evaluation of images synthesized from compressed content with higher computational efficiency. Performances of the proposed model and other tested metrics evaluated on IST dataset are presented in Table IX. The performances of the proposed model are better than other tested metrics.

V. CONCLUSION Considering the fact that image structures are crucial for visual quality perception and that the primary visual cortex and other parts of the brain reduce redundancies of input visual signals in order to discover the intrinsic image structure, thus creating sparse image representation, we propose the referenceless model to evaluate the perceptual quality of DIBRsynthesized views. We define an oriented morphological Difference of Closings (DoC) operator and use it to nonlinearly remove redundancies and extract the higher-order structure (fine-grained image details-texture) of high curvature, which correspond to distortion of shapes. Using the DoC operator, perceptually important details in image local structure and contrast to which the HVS is highly sensitive are extracted at multiple scales and resolutions. We introduce a new feature based on sparsity of DoC band. To linearly remove redundancies and to extract the low-order structure (edges), the non-oriented Difference of Gaussians (DoG) operator is employed. Both operators, DoC and DoG, are applied at multiple scales and resolutions to mimic a hierarchical, multiresolution and multi-scale character of HVS. Such two types of structural features play a complementary role in the visual quality assessment and they are sensitive to complex distortions of DIBR-synthesized views. Measures of sparsity of DoC and DoG bands are calculated as scalar features. Extracted features are mapped to the final score in a perceptually meaningful way by a trained general regression neural network. Performances of the proposed DoC-DoG-GRNN model that are calculated on four datasets of DIBR-synthesized views, IRCCyN/IVC image/video dataset, MCL-3D dataset and IST dataset demonstrate a high compatibility with perceptual quality scores, better than the tested state-ofthe-art models. The code for feature extraction is available at https://sites.google.com/site/draganasandicstankovic/code/ doc-dog.

Fig. 1 .

 1 Fig. 1. The framework of the proposed DoC-DoG-GRNN model.

  details from the signal without blurring the remaining structure.

Fig. 2 .

 2 Fig. 2.The primary pattern SE1, line-shaped vertically oriented structuring element of length 2.

Fig. 3 .

 3 Fig.3. The multi-scale image decomposition using morphological filter closing (C) with an array of linear structuring elements of increasing size SE j, j = 1,..., N,; si,0 the input image at resolution level i; si,j the low-pass copy at scale j; di,j the detail image (DoC band) at scale j.

Fig. 4 .

 4 Fig. 4. The DoC band at resolution level 3 at scale 4, d3,4, (bottom) of the image Balloon synthesized from the texture images and depth maps with JPEG distortion is created as the difference between smoothed images at scales 4 and scale 3 of resolution level 3, d3,4 = s3,4-s3,3: s3,3 (first row left), s3,4, (first row right).

Fig. 5 .

 5 Fig. 5. The Difference of Closings (DoC) operator can be defined as the difference of close-top-hats of nearby scales, di,j = cthi,j -cthi,j-1: cthi,j closetop-hat at resolution level i and scale j, di,j the detail image (DoC band) at resolution level i and scale j, si,0 the input image at resolution level i.

  gi,j(x, y) = i,0(x, y) * G(x, y,σj), j = 1,..., Q 2+y2 G(x, y,σj) e j (12) j

Fig. 6 .

 6 Fig. 6. The DoG band at scale 2 of resolution level 1, e1,2 (in the middle) and at scale 2 of resolution level 5, e5,2 (right), created from the image Balloon synthesized from JPEG distorted content (left).

Fig. 7 .

 7 Fig. 7. SROCC of the combined DoG-DoC-GRNN model, DoC-GRNN and DoG-GRNN models for IVC datasets (up) and MCL-3D dataset (down).

Fig. 8 .

 8 Fig. 8. SROCC of the proposed DoG-DoC-GRNN model, DoC-GRNN, DoG-GRNN models for IST image dataset.

parameter set 3 :

 3 4 resolution levels and 3 scales for DoC-based decomposition, di,1 -di,3, i = 1,...,4, 4 resolution levels with single scale for DoG-based decomposition, ei,1, i = 1,...,4, and spread parameter 0.022. Sparsity is calculated from 12 DoC bands, low-pass copy s5,0 and 4 DoG bands. The GRNN input vector of the DoC-DoG-GRNN model is created by concatenation of 13 features calculated from DoC-based decomposition and 4 features calculated from DoG bands. The number of operation for calculation of 17 features of DoC-DoG-GRNN model using parameter set 3 is 10 times lower than for calculation of 46 features when parameter set 1 is used in unoptimized case. The SROCC of the composed DoC-DoG-GRNN model, DoC-GRNN and DoG-GRNN models with fluctuation of the GRNN spread parameter using DMOS as subjective scores for IST dataset, is shown on Fig. 8. The performances of the DoC-DoG-GRNN model calculated using parameter set 3 evaluated on IST dataset and on two subsets of MCL-3D dataset, each with 108 stereo images synthesized from the content compressed by JPEG and JPEG 2000, calculated as case1, are shown in the Table VIII.

TABLE I PERFORMANCES

 I OF THE PROPOSED DoC-DoG-GRNN MODEL CALCULATED ON 5 WAYS

TABLE II MODEL

 II 'S PERFORMANCES FOR DIFFERENT DATA PARTITION RATIOS

TABLE III PERFORMANCES

 III OF THE PROPOSED DoC-DoG-GRNN MODEL AND OTHER METRICS

TABLE V CROSS

 V -VALIDATION OF THE PROPOSED MODEL

TABLE VI PERFORMANCES

 VI OF DoC-DoG-GRNN MODEL USING PARAMETER SET 2 FOR MCL-3D SUBSETS BY MVD SOURCES

  The performances slightly vary by sources as shown in TableVI. The parameter set 2 can be used for prediction of images of both sizes, 1024 × 768 and 1920 × 1088.

		TABLE VII
	MODEL'S PERFORMANCES FOR MCL-3D SUBSETS WITH
		DIFFERENT DISTORTION TYPES
	level, ei,3-ei,6	i = 1,...,5, and spread 0.09. The number of features
	using parameter set 2 is 5 * 6 + 1 + 5 * 4 = 51.
	We've analyzed performances of the proposed DoC-
	DoGGRNN model using parameter set 2 by nine MVD sources
	with images of smaller size, 1024 × 768, Baloon, Kendo,
	Love_bird1, and with images of larger size, 1920 × 1088, Shark,
	GT_fly, MicroWorld, Dancer, PoznanStreet, Poznan_hall2,
	each subset with 76 stereo images.

TABLE IX PERFORMANCES

 IX OF THE PROPOSED MODEL AND OTHER METRICS

FOR IST DATASET


Estelle Bongini
email: estelle.bongini@sncf.fr

Stéphane Molla

Pierre-Etienne E Gautier

Dominique Habault

Pierre-Olivier Mattei

F Poisson

P O Mattéi

Synthesis of noise of operating vehicles: development within SILENCE of a tool with listening features

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

SILENCE is an integrated research project funded by the Sixth Framework Programme of the European Commission. SILENCE will provide relevant technologies for efficient control of surface transport noise, innovative strategies for action plan for urban transport noise abatement and practical tools for their implementation, and will bring about a significant reduction of people's exposure to noise, especially in urban conditions.

Within this framework, subproject B is dedicated to the development of a tool for modelling exterior noise of both road and rail vehicles (train, tram, car, etc). The tool will combine a precise estimation of the sound level and sound synthesis. It will be used to assess the impact of the modification of a single source on the pass-by noise, both level and auralisation. This global modelling tool is called VAMPPASS, an acronym for Vehicle Acoustic Modelling for Pass-by Prediction and Audio Synthesis Software. This software will supply standard indicators, such as time domain signature, sound pressure level and third octave band analysis. It will be used to determine the best combination of optimised sources, i.e. to obtain an effective reduction of the overall pass-by noise according to the standards. Moreover, it will help generating listening test samples for psychoacoustic studies.

This paper first deals with the global modelling scope: the output required and the constraints. Then, the synthesis method is described. Finally, the acoustic source characterisation within this scope is presented.

Global Modelling Scope

The global modelling tool has to handle various requirements:

• It has to manage several operational conditions (mean traffic flow conditions, pass-by acceleration, braking, on different types of infrastructure, and so on) and the impact of pass-by evolution on the acoustic source characteristics, • It has to allow simulation of a pass-by in the time domain to produce listening samples in addition to pass-by indicators such as L Aeq,tp , time signature, • The project being dedicated to global noise reduction with optimisation of each source, the tool has to keep an acoustic representation in which each physical source is represented, identifiable and easily modifiable • Finally, the tool is dedicated to parametric studies which implies low computation time.

The vehicle is acoustically represented as a set of equivalent point sources. Furthermore, the synthesis method has to manage pass-by scenarios with several traffic conditions, thus a time-frequency representation is chosen for the source and vehicle global definition.

Synthesis Method -Acoustic Source Model

The synthesis method has to manage both the vehicle building as a set of acoustic sources and the time evolution defined through a pass-by scenario. Fig. 1 shows the tool architecture.

Since each vehicle is described by a superposition of N equivalent sources {S n } n=1..N , the pressure p at point M and time t, for such a vehicle is:

∑ = = N n Sn t M p t M p 1 ) , ( ) , ( (1)
p Sn (M,t) is the pressure radiated from the source S n , received at the time t at point M. Considering the sources within the scope of the present application, a sinusoid plus broad-band noise approach is chosen for the simulation. So, a source with a given spectrum is split into two contributions: a monofrequency contribution and a broadband one. An additive synthesis approach is used for the monofrequency components. The level, phase and directivity of the sources are the only parameters introduced in the software. For a stationary omnidirectional harmonic source S, with a circular frequency ω, a wavenumber k, emitting at a unit amplitude at the distance d(S,M) from M, the pressure is written:

- ℜ = t i M S ikd S e t A t M p (2) ( ) ) ( ) , ( ) ( ) , ( φ ω +
where Φ is the phase at the origin.

Furthermore, the broadband noise contribution at the receiver point M of each source is described in a time-frequency representation that takes into account its evolution during pass-by. The time-frequency representations of each source are summed to represent the overall signal. The time signal of the overall broadband noise is obtained with an inverse short time Fourier transform:

∑∑ +∞ -∞ = - = - = = m N k N km j m n n n e k P N mR t g t P ISTFT t M P 1 0 / 2 ) ( 1 ) ( ) )( ( ) , ( π (3)
where g is a convenient window, ) ( ˆk P m is the short-time spectrum of the signal at receiver point M, in window m and frequency f k and R controls the overlap. This method allows short computation times.

The Doppler effect and other effects of the vehicle motion can be taken into account directly in the time-frequency representation describing the broadband noise and also implemented in the pure tone components. Moreover, this synthesis method allows to take into account propagation conditions by introducing ground effect with a model specified in [START_REF] Molla | Prediction and audio synthesis of vehicle noise in urban area[END_REF]. The ground reflection is taken into account by introducing image sources. In this case, the same time-frequency patterns are obtained for the source itself and its image, with a temporal shift induced by the reflection.

Source Characterisation

The vehicle is represented as a set of equivalent acoustic point sources. Each physical acoustic source is usually represented by one equivalent source, to ensure an easily identification between equivalent and real sources. To represent one equivalent point source with one real source simplifies vehicle definition, but it implies additional data in the source definition. Particularly, equivalent sources present directivity patterns that must integrate the complexity of the radiation patterns and the extension of the real source.

The sources are thus described by their location, their spectral characteristics and their directivity patterns. In the Silence project, most of sources to be identified are traction noise sources. In urban conditions, aeroacoustic sources are not significant and rolling noise is provided using the TWINS software [START_REF] Thompson | TWINS, Track-Wheel Interaction Noise Software[END_REF] or antenna measurements presented below. Different measurements methods, supplemented with analytical models, can be used to provide traction source characteristics.

Two-dimensional (2D) antenna measurements have been carried out to locate acoustic sources in real operating pass-by. They have required 64 microphones randomly distributed on a plane antenna (see Fig. 2) and these measurements are postprocessed with beam forming algorithms after having removed the Doppler effect. These measurements provide relative levels between identified sources (see Fig. 3).

Fig. 2. 2D antenna measurements

Fig. 3. Example of cartography from plane antenna measurements

A one-dimensional (1D) antenna method has been also developed: it consists on a vertical microphone array (see Fig. 4). The train pass-by very slowly in front of this antenna. Real sources are managed during pass-by to isolate them from each other. Train position relative to the antenna is determined with a system of reflecting strips regularly attached on the train and a photosensitive cell on the vertical array.

The locations of the equivalent sources are determined with the maps that give for each 1/3 rd octave band, the overall level versus the lateral position of the train and the microphones heights (see Fig. 5). The position of the maximum of the sound power level, at one frequency band, provides the position of the equivalent source.

These measurements allow to locate precisely maximum level of each real source, for several frequency ranges. These measurements are also used to characterise the level and the spectrum of each source.

According to the frequency, real sources that are too close cannot be separated with the antenna. The separation of the contributions requires further information. Analytical models can provide spectral characteristics of each source that can help for level train location (reflecting strips) microphones (at various heights)

Normalised power density of the exhaust radiation, around 50Hz, higher microphones For a total identification of close sources, vertical antenna measurements can be used with a microphone very close to each source. Thus, the coherence between the microphone close to the source and antenna measurements indicate if the source contributes to the field recorded by the antenna. This method is valid as long as the coherence between near field and far field is not lost because of the Doppler effect or any propagation effect between source and antenna.

After the separation of contributions of each source from the other, the contribution of each source is separated between broadband noise and pure tone components. Determination of the pure-tone components is realised using the Loudness spectrum. The loudness density takes into account the masking phenomena: when several pure tone signals are present together, some of them are masked by the others and so not heard as a harmonic component (depending on the frequencies and the levels of the signals). For vertical array measurements, noise source signals are considered as stationary signals and loudness can be implemented with Moore's model [START_REF] Boullet | La sonie des sons impulsionnels: Perception, mesures et modèles[END_REF]. The loudness density highlights the frequency ranges of high energy density that will be heard. To determine the frequency and the level of the pure tone components, the Power Spectral Density is used.

For example, on the left of Fig. 6, at least six pure tones can be identified on the power spectral density (100Hz and its harmonics), whereas on the right, the loudness density indicates that only three of them will be heard (100 Hz, 200 Hz and 400 Hz). These three components are modelled by pure tones.

The remaining part of the energy of the signal is represented by a broadband noise source. The third-octave band power spectrum of this source is computed from the total power spectrum from which the contribution of the pure tone components is subtracted.

Finally, the 1D antenna measurements are also used to determine the directivity pattern allocated to each equivalent sources. Evolution of the level during source pass-by provides a horizontal pattern whereas the radiated field measured by the Loudness density for air conditionning Fig. 6. Power spectral density and Loudness spectrum of an air-conditioning system several microphones vertically distributed provides a vertical directivity. These data can be used to identify the measured directivity with a simple model as a monopole or dipole. This identification process is supplemented with analytical models that provide radiation patterns expected for specific sources (fans, pipes) in free-field. For example, in the low frequency range, a fan can be considered as a dipole [START_REF] Roger | Préambule, fondements théoriques et état de l'art, colloque Bruit aérodynamique des ventilateurs[END_REF][START_REF] Neise | Review of fan noise generation mechanisms and control methods, Congrès Bruit des ventilateurs[END_REF]. A baffled pipe at low frequency (kr<3, where r is the radius of the pipe and k the wave number) can be considered as a monopole [START_REF] Hocter | Exact and approximate directivity patterns of the sound radiated from a cylindrical duct[END_REF].

However, the 1D antenna measurements provide very complex radiation patterns, thus identification with simple sources is mostly not satisfactory. It appears that the diffraction due to the source environment (close environment such as the cavity of the diesel powerpack or the ring around a fan, or the more distant environment such as the train body) has a significant impact on the radiated field. To study this impact, either numerical simulations or dedicated measurements can be carried out.

In the case of railway traction noise sources, numerical simulations proved to be unsuitable: on the one hand, the phenomena occurred in the low frequency range that implies that the whole train body has to be modelled (large wavelengths); on the other hand, relatively small boundary elements are required for the reliability of the calculations; these two conditions lead to non-feasible computations in realistic times.

Dedicated measurements have been performed on a 1:14 scale driving-car model, with a diesel powerpack compartment and a ringed fan on the roof. This model is studied to characterize the diffraction effects produced by the car body. The sources (fan or motor) are replaced by loudspeakers with controlled radiation behaviour. The pressure field is measured with two vertical microphones arrays, and four microphones suspended over the roof.

First post-processing consists of maps of the transfer functions between input voltage of loudspeakers and pressure at microphones. These maps show that the radiated field becomes more complex as frequency increases, with the occurrence of shadow/image sources.

In addition to these first results, more measurements have to be carried out to separate the diffraction phenomena. For example, for a laterally embedded source, the effect of the back cavity and the effect of the train body have to be separated.

Conclusions

The global modelling software described is operational. It provides the acoustic indicators to characterize the pass-by and the sound samples to conduct listening tests in the laboratory. A user-friendly interface is in progress. Source characterisation has been performed for a regional train as an example. Various measurement methods are proposed according to the requirements of the software to define the sources. The combination of measurement methods, analytical models and previous analysis on specific phenomena such as diffraction, allows to the positions, spectrum and directivities of all the sources to be characterised and then the train or car model to be assembled.

Fig. 1 .

 1 Fig. 1. Tool architecture

Fig. 4 . 5 .

 45 Fig. 4. 1D antenna measurements Fig. 5. Cartography post-processed from 1D antenna

Fig. 7 .Fig. 8 .

 78 Fig. 7. Experimental 1:14 scale train model Fig. 8. Transfer functions between laterally embedded loudspeaker and microphones, along the train for motor compartment radiation at 3696 Hz, corresponding to 264 Hz at full scale