Felipe C Belém
email: felipe.belem@ic.unicamp.br@esiee.fr

Isabela B Barcelos
email: isabelaborlido@hotmail.com

Leonardo M João
email: leonardo.joao@ic.unicamp.br

Benjamin Perret
email: benjamin.perret@esiee.fr

Jean Cousty
email: jean.cousty@esiee.fr

Silvio J F Guimarães

Alexandre X Falcão
email: afalcao@ic.unicamp.br

Novel Arc-Cost Functions and Seed Relevance Estimations for Compact and Accurate Superpixels

Keywords: Superpixel, Saliency, Multiscale, Segmentation

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Introduction

In order to minimize workload and handle more meaningful data than simple pixels, medical applications [START_REF] Dhore | Chest x-ray segmentation using watershed and super pixel segmentation technique[END_REF][START_REF] Liu | A fully automatic segmentation algorithm for ct lung images based on random forest[END_REF][START_REF] Zhou | Superpixel segmentation of breast cancer pathology images based on features extracted from the autoencoder[END_REF], semantic segmentation methods [START_REF] Yi | Weakly-supervised semantic segmentation with superpixel guided local and global consistency[END_REF], and video segmentation algorithms [START_REF] Liang | Robust video object segmentation via propagating seams and matching superpixels[END_REF] often recur to superpixel segmentation. In brief, superpixels are disjoint groups of connected pixels that present similar characteristics, like color. Consequently, superpixel segmentation methods aim to represent any object by its comprising parts.

Despite the diversity of the superpixel literature, it is possible to list a set of crucial properties for any algorithm [START_REF] Stutz | Superpixels: An evaluation of the state-of-the-art[END_REF]: (i) boundary adherence; (ii) compactness; and (iii) efficiency. However, achieving simultaneously all three is a challenging task [START_REF] Stutz | Superpixels: An evaluation of the state-of-the-art[END_REF]. For instance, the popular Simple Linear Iterative Clustering (SLIC) [START_REF] Achanta | SLIC superpixels compared to state-of-the-art superpixel methods[END_REF] and the Iterative Boundary Implicit Identification (IBIS) [START_REF] Bobbia | Iterative boundaries implicit identification for superpixels segmentation: A real-time approach[END_REF] are two fast approaches for generating highly compact superpixels, but with moderate delineation performance. On the other hand, Superpixel Hierarchy (SH) [START_REF] Wei | Superpixel hierarchy[END_REF] and Dynamic and Iterative Spanning Forest (DISF) [START_REF] Belém | Superpixel segmentation using dynamic and iterative spanning forest[END_REF] efficiently produce superpixels with high boundary adherence, but highly irregular as well. One may perceive such irregularity as an overfitting behavior [START_REF] Schick | An evaluation of the compactness of superpixels[END_REF]. Either way, these methods, by aiming the delineation of every possible border, are unaware of the user's desire to segment a specific object, nor might they be adaptable to do such a task. By acknowledging the complex problem of detecting all borders, several recent methods exploit deep-learning strategies for learning a pattern (e.g., relevant borders) for superpixel generation. Still, such algorithms often present moderate delineation performance and moderate compactness, demanding high computational power. Moreover, both supervised [START_REF] Zhu | Learning the superpixel in a non-iterative and lifelong manner[END_REF][START_REF] Yang | Superpixel segmentation with fully convolutional networks[END_REF] and unsupervised [START_REF] Suzuki | Superpixel segmentation via convolutional neural networks with regularized information maximization[END_REF][START_REF] Yu | Edge-aware superpixel segmentation with unsupervised convolutional neural networks[END_REF] approaches are dependable on data, specially annotated for the former.

Using a different strategy, new algorithms exploit the probable object location, usually computed through deep learning and termed saliency, for adapting the segmentation to the user's needs. Thus, by maximizing the delineation of a specific object, these saliency-based approaches dismiss the segmentation of the remaining ones. To the best of our knowledge, the Saliency-based Superpixels (SS) [START_REF] Xu | Saliency-based superpixels[END_REF] is the first method to adopt this paradigm, generating superpixels through a saliency-driven bottom-up merging process. The Object-based Iterative Spanning Forest (OISF) [START_REF] Belém | Superpixel segmentation by object-based iterative spanning forest[END_REF][START_REF] Belém | Superpixel generation by the iterative spanning forest using object information[END_REF], another example, can alter the superpixel morphology with respect to the given information, but is highly dependable on its quality. Conversely, the Object-based DISF (ODISF) [START_REF] Belém | Towards a simple and efficient object-based superpixel delineation framework[END_REF] offers a lessdependable strategy with higher efficiency and more accurate delineation performance. However, ODISF is unable to produce compact superpixels and cannot further improve its segmentation for more accurate object information.

Another saliency-based framework, named Superpixels through Iterative CLEarcutting (SICLE) [START_REF] Belém | Efficient Multiscale Objectbased Superpixel Framework[END_REF], generalizes both DISF and ODISF for computing superpixels more efficiently. SICLE is composed of three independent steps: (i) seed oversampling; (ii) path-based superpixel generation; and (iii) seed removal. Briefly, it starts from (i) and, for several iterations, applies (ii) and (iii) until obtaining a desired number of superpixels. Even though it inherits the drawbacks of both DISF and ODISF, like their irregularity, its architecture with separated and self-contained steps favors modifications and improvements with reduced effort while maintaining its state-of-the-art efficiency and efficacy.

In this work, we propose several functions for the SICLE framework for effective object delineation through compact or irregular superpixels, exploiting the quality of the saliency estimation. That is, we introduce two novel arc-cost functions for step (ii), which provides a twofold property: for poor-quality saliency information (e.g., inaccurate object boundary definition), our proposals sustain the effective delineation of ODISF; for state-ofthe-art quality, the novel functions improve the border detection significantly without reinforcing erroneous estimations in the segmentation, differently from OISF. For step (iii), we present a novel seed criterion for spreading superpixels and three new penalizations for a more accurate seed removal by focusing on the object's salient borders. Furthermore, we performed an extensive study for building both irregular and compact SICLE variants which, as shown by the experimental results, have surpassed several stateof-the-art methods in delineation (see Figure 1) and speed. For the latter, SICLE variants require only two iterations for segmentation due to our proposed functions. Finally, since saliency usually depicts a single object, we debate the challenges of saliency-based superpixel segmentation using SICLE and, thus, using our proposals with multiple objects of interest.

In brief, we can list the following contributions:

• novel arc-cost functions which incorporate prior object information and produce compact and irregular superpixels robust to saliency errors; • novel seed relevance criterion for superpixel spreading and three novel seed relevance saliency-based penalization for improving delineation and assuring overall compactness; • thorough optimization of SICLE and an extensive performance evaluation in five datasets, considering well-known metrics and seven stateof-the-art methods as baselines, resulting in two SICLE variants for effective delineation and compact superpixel generation; • debate over multiobject handling in SICLE. This article extends and completes a previous work published at a conference [START_REF] Belém | Fast and effective superpixel segmentation using accurate saliency estimation[END_REF]. Differently from it, the saliency importance in our arc-cost functions is not a binary parameter (thus, may be optimized), and the seed penalization is no longer associated with such importance, being independently defined. The remaining contributions are original and do not appear in [START_REF] Belém | Fast and effective superpixel segmentation using accurate saliency estimation[END_REF].

This work is organized as follows. In Section 2, we present a brief overview of state-of-theart superpixel segmentation algorithms. Subsequently, in Section 3, we introduce the theoretical framework necessary for describing our proposed functions in Section 4. Then, the experimental results and a debate over multiobject handling are presented in Sections 5 and 6. Finally, we conclude this work in Section 7 drawing possible future work.

Related Works

In this section, we present and debate the most notable state-of-the-art superpixel segmentation algorithms. We can broadly classify them based on their behavior towards prior object information: (i) classical; (ii) deep-learning-based; and (iii) saliency-based. Unfortunately, most surveys do not encompass (ii) and (iii), being recently developed strategies. Still, for a more detailed discussion on this topic, we recommend several distinguished works [START_REF] Stutz | Superpixels: An evaluation of the state-of-the-art[END_REF][START_REF] Wan | Spherical superpixels: Benchmark and evaluation[END_REF][START_REF] Wang | Superpixel segmentation: A benchmark[END_REF].

The majority of superpixel methods are classical and can be further divided into three subgroups based on their strategy: (i) clusteringbased; (ii) graph-based; and (iii) path-based. As the label suggests, the algorithms in (i) model the segmentation task as a pixel-clustering problem, and the most well-known work is the Simple Linear Iterative Clustering (SLIC) [START_REF] Achanta | SLIC superpixels compared to state-of-the-art superpixel methods[END_REF]. In SLIC, superpixels are generated through an adapted K-means in a 5-dimensional feature space comprising the pixels' color and spatial coordinates. Given its simplicity and efficiency, several works extended or were inspired by SLIC, producing highly effective and efficient algorithms. One of such is the Linear Spectral Clustering (LSC) [START_REF] Li | Superpixel segmentation using linear spectral clustering[END_REF], which produces superpixels using a weighted Kmeans in a 10-dimensional feature space. Another example is the Iterative Boundaries Implicit Identification (IBIS) [START_REF] Bobbia | Iterative boundaries implicit identification for superpixels segmentation: A real-time approach[END_REF], an efficient approach for generating highly compact superpixels by minimizing the number of pixel operations without any performance degradation. Although certain methods classify themselves as content-sensitive [START_REF] Liu | Intrinsic manifold SLIC: A simple and efficient method for computing content-sensitive superpixels[END_REF], content-adaptive [START_REF] Xiao | Contentadaptive superpixel segmentation[END_REF], or even texture aware [START_REF] Wu | Texture-aware and structure-preserving superpixel segmentation[END_REF], they often rely on clustering techniques, specially K-means. However, such approaches, including those that use Gaussian Mixture Models (GMM) [START_REF] Ban | Superpixel segmentation using gaussian mixture model[END_REF] and Density-based Spatial Clustering of Applications with Noise (DBSCAN) [START_REF] Shen | Real-time superpixel segmentation by DBSCAN clustering algorithm[END_REF] for clustering, tend not to ensure the desired number of superpixels [30]. Moreover, although they generate compact superpixels, their delineation performance is usually moderate.

Those termed as graph-based usually generate superpixels through edge operations in the image graph. For instance, the Entropy Rate Superpixels (ERS) [START_REF] Liu | Entropy rate superpixel segmentation[END_REF] performs such task by optimizing the entropy of a random walk in the graph's topology. However, ERS tends to present high computational time. On the other hand, the Superpixel Hierarchy (SH) [START_REF] Wei | Superpixel hierarchy[END_REF] builds a hierarchical superpixel segmentation through the Borůvka algorithms, achieving linear time complexity. In contrast to clustering-based approaches, both ERS and SH provides accurate object delineation at the expense of generating highly irregular superpixels. Furthermore, for SH, such delineation may be compromised by errors propagated from lower levels to upper ones in the hierarchy.

Similarly to SLIC and K-means, most pathbased approaches base themselves on the Image Foresting Transform (IFT) framework for generating superpixels. In such category, superpixels are built by path searching from a subset of vertices (i.e., seeds) to the remaining ones based on a similarity criterion. Aside from guaranteeing the desired number of superpixels, IFT-based methods often achieve top delineation performance. We may cite state-of-the-art methods, such as the Iterative Spanning Forest (ISF) [START_REF] Vargas-Muñoz | An iterative spanning forest framework for superpixel segmentation[END_REF] framework, which inspired later and more effective approaches. The Recursive ISF (RISF) [START_REF] Galvão | RISF: recursive iterative spanning forest for superpixel segmentation[END_REF] method, for instance, produces a sparse hierarchical superpixel segmentation by executing several ISFs at each level. As a different example, the Dynamic ISF (DISF) [START_REF] Belém | Superpixel segmentation using dynamic and iterative spanning forest[END_REF] uses a strategy of seed oversampling and iterative seed removal for computing a superpixel segmentation using a dynamic arc-cost estimation. Lastly, although the most common drawback of such category is the difficulty in producing compact superpixels [START_REF] Stutz | Superpixels: An evaluation of the state-of-the-art[END_REF], the Waterpixels [30] algorithm manages to produce compact superpixels using the watershed transformation.

Classical methods usually produce accurate object delineation but are unaware of the user's expectations towards the objects. On the other hand, in recent years, several novel approaches have tried to overcome such difficulty by exploiting deep-learning techniques to generate superpixels. One possible approach is to extend existing methods for incorporating features learned through deep-learning, such as the one proposed in [START_REF] Awaisu | Fast superpixel segmentation with deep features[END_REF]. However, the authors in [START_REF] Tu | Learning superpixels with segmentation-aware affinity loss[END_REF] argue that directly incorporating deep features is insufficient for proper delineation. Thus, they propose the construction of pixel-affinities maps through deep learning using a new loss function, named SEgmentation Aware Loss (SEAL), and modify preexistent superpixel algorithms (e.g., ERS) for considering such information, resulting in the SEAL-ERS approach. Similarly, the Deep Affinity Learning for Hierarchical ERS (DAL-HERS) [START_REF] Peng | Hers superpixels: Deep affinity learning for hierarchical entropy rate segmentation[END_REF] also computes pixel-affinity maps and generates superpixels hierarchically through the Borůvka method considering the entropy of the graph. Another strategy consists of building end-to-end supervised [START_REF] Zhu | Learning the superpixel in a non-iterative and lifelong manner[END_REF][START_REF] Yang | Superpixel segmentation with fully convolutional networks[END_REF][START_REF] Jampani | Superpixel sampling networks[END_REF] or unsupervised [START_REF] Suzuki | Superpixel segmentation via convolutional neural networks with regularized information maximization[END_REF][START_REF] Yu | Edge-aware superpixel segmentation with unsupervised convolutional neural networks[END_REF] networks for superpixel generation. However, although promising, both approaches demand more research [START_REF] Belém | Efficient Multiscale Objectbased Superpixel Framework[END_REF][START_REF] Tu | Learning superpixels with segmentation-aware affinity loss[END_REF] since they present: (a) high data dependency; (b) moderate delineation; and (c) high computational cost.

In order to achieve the best of both strategies (i.e., effectiveness and incorporating object information), current saliency-based algorithms, apart from [START_REF] Xu | Saliency-based superpixels[END_REF], generalize classical algorithms for incorporating saliency information, often estimated through deep-learning. As an example, the Object-based ISF (OISF) [START_REF] Belém | Superpixel segmentation by object-based iterative spanning forest[END_REF][START_REF] Belém | Superpixel generation by the iterative spanning forest using object information[END_REF] is a generalization of ISF that allows user control over the superpixel's morphology and displacement with respect to an object saliency map. However, given that OISF is slow and highly dependable on the map's quality, the authors in [START_REF] Belém | Towards a simple and efficient object-based superpixel delineation framework[END_REF] propose Objectbased DISF (ODISF), a faster and more robust method for achieving accurate object delineation. Still, for both DISF and ODISF, it is difficult to establish the number of iterations for segmentation, and they do not produce compact superpixels. Moreover, although robust, they cannot improve their delineation for better saliency maps. Finally, in order to solve the iteration problem, the authors in [START_REF] Belém | Efficient Multiscale Objectbased Superpixel Framework[END_REF] proposed Superpixels through Iterative CLEarcutting (SICLE), a generalization of both DISF and ODISF, in which the user can define a maximum number of iterations, producing a saliency-based multiscale segmentation.

Theoretical Background

In this section, we present the theoretical background of our proposed framework.

While Section 3.1 introduces basic concepts on images, saliency maps, and image digraphs, Section 3.2 presents the Image Foresting Transform (IFT) [START_REF] Falcão | The image foresting transform: Theory, algorithms, and applications[END_REF], the core algorithm for object delineation for the Superpixels through Iterative CLEarcutting (SICLE) [START_REF] Belém | Efficient Multiscale Objectbased Superpixel Framework[END_REF] framework (described in Section 3.3).

Image Digraph

We may define an image I as a pair ⟨I, F⟩ such that every p ∈ I ⊂ Z 2 denotes a picture element (i.e., pixel) whose feature vector (e.g., CIELAB color in this work) is represented by F(p) ∈ R m , for m ∈ N >0 . Thus, I is a grayscale or a color image whenever m = 1 or m > 1, respectively. For example, an object saliency map O = ⟨I, O⟩ is a grayscale image in which O(p) ∈ [0, 1] defines a proportional object likelihood, often referred as (object) saliency, ∀p ∈ I. Intuitively, brighter is the intensity O(p) of p, higher is its probability of belonging to the object of interest.

From I, we can build an image digraph G = ⟨V, A⟩ in which V ⊆ I is the vertex set and A ⊂ V 2 is the arc set. A popular approach for establishing the arcs of G is through A r = {⟨x, y⟩ : ∥x -y∥ 2 ≤ r}, given x, y ∈ V and r ∈ R ≥0 . One may note that A √ 2 represents the 8adjacency relation. If ∃ ⟨x, y⟩ ∈ A then we term x as adjacent of y. Finally, in this work, every ⟨x, y⟩ ∈ A is unique and x ̸ = y (i.e., G is simple).

A (directed) path ρ = ⟨v 1 , . . . , v k ⟩ is a sequence of distinct vertices such that ⟨v i , v i+1 ⟩ ∈ A, for i, k ∈ N >0 and i < k, and it is trivial when k = 1. Whenever ρ is non-trivial, we name v i as predecessor of v i+1 and v i+1 as sucessor of v i in ρ. Moreover, v 1 is the root of ρ while v k is its terminus, and we may explicitly present both by ρ v1⇝v k . However, for simplicity, we may omit the root from such notation. For instance, ρ s⇝x ⊙ ⟨x, y⟩ = ρ x ⊙ ⟨x, y⟩ = ⟨v 1 = s, . . . , v k = x, y⟩ denotes the concatenation of the path ρ x , irrespective of its origin s ∈ V, with an arc ⟨x, y⟩, merging the two instances of x into one.

Image Foresting Transform

In [START_REF] Falcão | The image foresting transform: Theory, algorithms, and applications[END_REF], the authors proposed the Image Foresting Transform (IFT), a framework highly effective for object delineation [START_REF] Bragantini | Graph-based image segmentation using dynamic trees[END_REF][START_REF] Borlido | Towards interactive image segmentation by dynamic and iterative spanning forest[END_REF]. Briefly, the IFT, in this work, finds a path with optimum cost from a seed s ∈ S to every non-seed vertex v ∈ V \ S, being S ⊂ V the seed set.

We can compute the path-cost f * (ρ) ∈ R ≥0 of any path ρ ∈ P, where P is the set of all possible ones in G, through a path-cost function f * . For instance, Equation 1 illustrates two different path-functions f max , f sum effective for superpixel segmentation [START_REF] Belém | Superpixel segmentation using dynamic and iterative spanning forest[END_REF][START_REF] Belém | Superpixel segmentation by object-based iterative spanning forest[END_REF][START_REF] Belém | Towards a simple and efficient object-based superpixel delineation framework[END_REF][START_REF] Vargas-Muñoz | An iterative spanning forest framework for superpixel segmentation[END_REF]:

f * (⟨x⟩) = 0, if x ∈ S ∞, otherwise f max (ρ x ⊙ ⟨x, y⟩) = max {f max (ρ x ), w * (x, y)} f sum (ρ x ⊙ ⟨x, y⟩) = f sum (ρ x ) + w * (x, y) (1)
in which w * (x, y) ∈ R ≥0 denotes the arc-cost function of ⟨x, y⟩ ∈ A. A path ρ x is said to be optimum if, for any other

ϱ x ∈ P, f * (ρ x ) ≤ f * (ϱ x ).
Thus, by exploiting a generalization of the Dijkstra's algorithm, the IFT assigns an optimum path ρ x , ∀x ∈ V, while minimizing a cost map C(x) = min ρx∈P {f * (ρ x )}. Concomitantly, it builds a predecessor map P that maps x to its predecessor in ρ x or to a unique marker when x is its root (i.e., x ∈ S). Furthermore, one can recursively map, through P, x to its root s = R(x) ∈ S and, consequently, to its tree T s = {y : R(y) = s}. Note that, in this work, every tree is a superpixel. Lastly, if f * satisfy certain conditions [START_REF] Ciesielski | Pathvalue functions for which dijkstra's algorithm returns optimal mapping[END_REF], ∀s ∈ S, T s is an optimum-path tree, resulting in an optimum-path forest. Otherwise, T s still exhibits important properties for segmentation [START_REF] Mansilla | Oriented image foresting transform segmentation: Connectivity constraints with adjustable width[END_REF].

Superpixels through Iterative CLEarcutting

Figure 2 presents the SICLE framework for superpixel generation. Therefore, the construction of a SICLE variant involves selecting: (i) a seed oversampling strategy; (ii) a path-cost and an arccost function (for the IFT); and for estimating the importance of a seed, (iii) a seed preservation curve, a seed relevance criterion, and a object penalization. Differently from most methods [START_REF] Achanta | SLIC superpixels compared to state-of-the-art superpixel methods[END_REF][START_REF] Li | Superpixel segmentation using linear spectral clustering[END_REF][START_REF] Liu | Intrinsic manifold SLIC: A simple and efficient method for computing content-sensitive superpixels[END_REF][START_REF] Vargas-Muñoz | An iterative spanning forest framework for superpixel segmentation[END_REF], SICLE starts by selecting a number of N 0 of seeds significantly higher than the final number N f of superpixels (i.e., N 0 ≫ N f ), given N 0 , N f ∈ N >0 . The motivation is based on the seed redundancy premise: if two seeds are similar (i.e., spatial position and features), then they are similarly relevant (i.e., promote accurate 2 The top image shows the SICLE framework for generating 100 superpixels using a deep-learning saliency estimator [START_REF] Qin | U2-net: Going deeper with nested u-structure for salient object detection[END_REF]. Superpixel borders and seeds are depicted by cyan lines and dots, respectively. Whereas the bottom image explicit the inputs and outputs of each process in the pipeline. object delineation). This premise can be empirically perceived since different approaches achieve equivalent delineation [START_REF] Belém | Efficient Multiscale Objectbased Superpixel Framework[END_REF] and, consequently, one may opt for a simple random sampling, especially given the reported loss of efficiency of saliencybased samplings [START_REF] Belém | The importance of object-based seed sampling for superpixel segmentation[END_REF].

In the subsequent step, superpixels are generated using the seed-restricted version of the IFT and, so far, it has been implemented considering the f max path-cost function. For estimating the arc-cost ⟨x, y⟩ ∈ A, SICLE considered a dynamic arc-cost estimation [START_REF] Bragantini | Graph-based image segmentation using dynamic trees[END_REF]

w dyn (x, y) = F(T s ) -F(y) 2 ,
given that s = R(x), which considers the mean features of the tree T s growing during the IFT execution. Due its instability [START_REF] Belém | Efficient Multiscale Objectbased Superpixel Framework[END_REF], the w root (x, y) = ∥F(R(x)) -F(y)∥ 2 was also evaluated in which uses the immutable features of the seed s, being preferred over w dyn .

Due to oversampling, it is necessary to remove a total of N 0 -N f seeds from S in at most Ω ∈ N >0 > 1 iterations to ensure N f superpixels. For that, a portion of the most irrelevant are removed per iteration i ∈ N >0 < Ω. Or, conversely, the M(i) = max (N 0 ) 1-i/(Ω-1) , N f most relevant seeds are maintained for the subsequent iteration i + 1 while the remaining ones are discarded. As one may note, exactly (Ω -1)(1 -log N0 N f ) + 1 iterations are required for reaching the desired quantity of superpixels, being ideally two.

We can estimate the relevance of a seed s based on certain information (or attributes) extracted from its respective tree T s , like colorbased (e.g., mean color, color histograms, pattern spectra) or shape-based (e.g., size, circularity, compactness) ones. First, in terms of particular features, we can compute its mean features

F(T s ) = v∈Ts F(v)/ |T s |, its mean saliency O(T s ) = v∈Ts O(v)/ |T s |, and its cen- troid c(T s ) = v∈Ts v/ |T s |.
From that, we can compute the color gradient between two trees T s and T t by

∇ F (T s , T t ) = F(T s ) -F(T t ) 2 .
Similarly, we may compute the saliency gradient and the distance between centroids by

∇ O (T s , T t ) = O(T s ) -O(T t ) 1 and ∇ c (T s , T t ) = ∥c(T s ) -c(T t )∥ 2 , respectively.
Also, given a seed s ∈ S, we can establish the (immediate) adjacents of its tree T s from A(T s ) = {T t : ∃ ⟨x, y⟩ ∈ A} given x ∈ T s , y ∈ T t , and s ̸ = t. Consequently, we can compute the maximum and minimum color contrast of T s amongst its neighbors through ∇ + F (T s ) = max Tt∈A(Ts) ∇ F (T s , T t ) and ∇ - F (T s ) = min Tt∈A(Ts) ∇ F (T s , T t ) , respectively. Furthermore, we can indirectly estimate the proximity of the tree T s to an object through its maximum saliency contrast ∇ + O (T s ) = max Tt∈A(Ts) ∇ O (T s , T t ) . Lastly, the shortest adjacent distance of T s , with respect to c, can be calculated by ∇ - c (T s ) = min Tt∈A(Ts) {∇ c (T s , T t )}. Therefore, given the previous definitions, the relevance V * (s) ∈ R ≥0 of any seed s ∈ S can be computed through a criterion v * (s) ∈ R ≥0 based on the characteristics of its tree T s . For instance, in IFT, trees grow by minimizing the path-costs to vertices, making the size-based criterion v size (s) = |T s | / |V| a straightforward option. However, considering that it may favor background seeds [START_REF] Belém | Efficient Multiscale Objectbased Superpixel Framework[END_REF], contrast is vital in accurate relevance estimation. For improving delineation, one could exploit either the minimum contrast or maximum contrast as indicatives of object borders through a size-and minimumcontrast-based v minsc (s) = v size (s)∇ - F (T s ) and a size-and maximum-contrast-based v maxsc (s) = v size (s)∇ + F (T s ) criteria, respectively. Still, any criterion v * is subjected to the existence of an object of interest [START_REF] Belém | Towards a simple and efficient object-based superpixel delineation framework[END_REF][START_REF] Belém | Efficient Multiscale Objectbased Superpixel Framework[END_REF]. For example, a seed that lies far from the object is intrinsically irrelevant for delineation. Consequently, the relevance computation of a seed s ∈ S can be then defined as V * (s) = v * (s)p * (s), where p * (s) ∈ [0, 1] denotes an object-based penalization. Whenever an object saliency map is absent, p none (s) = 1 is applied ∀s ∈ S. For concentrating seeds within and nearby the objects, one may exploit the mean saliency O(T s ) and the maximum saliency gradient ∇ O (T s ) of the tree T s , as considered in the object penalization p obj (s) = max O(T s ), ∇ + O (T s ) .

Compact and Accurate Superpixel Generation

In this section, we present our proposals for producing compact and accurate superpixels in the SICLE framework. Thus, our contributions are related to the Superpixel Generation and Seed Removal steps (Figure 2) for creating more efficient and effective SICLE variants. In Section 4.1, we define two novel saliency-based arc-cost functions and compare them with previous approaches, saliency-based or not, for superpixel segmentation. Similarly, Section 4.2 presents one novel seed relevance criterion and three new seed penalizations, contrasting with those proposed in SICLE.

Arc-cost Estimation

The arc-cost function [START_REF] Belém | Towards a simple and efficient object-based superpixel delineation framework[END_REF][START_REF] Belém | Efficient Multiscale Objectbased Superpixel Framework[END_REF], led to effective superpixel delineation whenever a saliency map is absent. Moreover, its performance is equivalent to a dynamic arc-cost function [START_REF] Bragantini | Graph-based image segmentation using dynamic trees[END_REF], whose results surpass state-of-the-art methods whenever such map is absent. Also, since it is based on immutable features (i.e., the seed's feature vector), w root exhibits more stability than the aforementioned function [START_REF] Belém | Efficient Multiscale Objectbased Superpixel Framework[END_REF]. Finally, the use of w root allows differential computation of the IFT [START_REF] Falcão | Interactive volume segmentation with differential image foresting transforms[END_REF] throughout the iterations. However, both functions are unable to improve their delineation for superior saliency maps.

w root = ∥F(R(x)) -F(y)∥ 2 , considered in
A key idea of our proposals is to exploit a function o(x, y) ∈ R ≥0 that indicates, for a given vertex y ∈ V, if it is likely to be concatenated to an optimal path ρ x ∈ P ending at a vertex v ∈ V, with respect to the object saliency O. This likelihood value o(x, y) is weighted by a constant parameter α ∈ R ≥0 that intuitively indicates the degree of trust we have towards the saliency variation between the root of x and y. More precisely, we define such function as o(x, y) = α ∥O(R(x)) -O(y)∥ 1 . Given that, we propose a generalization of w root , as shown in Equation 2:

w α root (x, y) = (w root (x, y)) 1+o(x,y) (2)
Note that, by disregarding any saliency variation, thus w α=0 root = w root , our proposal does not incorporate incorrect estimations and, therefore, maintains the effective delineation provided by w root (i.e., robustness). Conversely, by setting α > 0, the arc costs that cross a saliency border increase significantly, promoting a higher adherence to it and preventing superpixel leaking.

However, even with w α root , such SICLE configuration is still unable to produce compact superpixels. For that, one may suggest using the f sum and the w sum (Equation 3) functions as proposed in [START_REF] Vargas-Muñoz | An iterative spanning forest framework for superpixel segmentation[END_REF]:

w sum (x, y) = (ι ∥F(R(x)) -F(y)∥ 2 ) β + ∥x -y∥ 2 (3)
in which ι ∈ R ≥0 and β ∈ R ≥0 control the superpixels' irregularity and boundary adherence, respectively. Clearly, this function is unaware of any object information, and consequently, one could recur to the generalization proposed in [START_REF] Belém | Superpixel segmentation by object-based iterative spanning forest[END_REF][START_REF] Belém | Superpixel generation by the iterative spanning forest using object information[END_REF]. Nevertheless, although it presents high effectiveness in delineation, especially with few superpixels, incorrect saliency estimations may deteriorate its performance significantly [START_REF] Belém | Towards a simple and efficient object-based superpixel delineation framework[END_REF]. Therefore, we aim to develop an arc-cost function that maximizes object delineation, but with compact superpixels, while minimizing errors derived from inaccurate saliency estimation.

For that, we propose a novel generalization (Equation 4) in which the saliency information only impacts the irregularity factor:

w α sum (x, y) = ((ι + o(x, y)) ∥F(R(x)) -F(y)∥ 2 ) β + ∥x -y∥ 2 (4
) As one can see, similarly with w α root , w α=0 sum = w sum neglects the saliency information while, for α > 0, the irregularity factor changes substancially whenever the arc crosses the saliency border. That is, for the former, w α sum promotes incorrect saliency robustness whereas, for the latter, promotes adherence to saliency borders and prevents leakings.

Seed Criteria and Penalization

When regularity is not desired, the criteria v size , v maxsc , and v minsc promote accurate seed relevance estimation. On the other hand, they often lead to a higher superpixel concentration in different regions of the image (e.g., background and near object borders), which may favor irregularity. Therefore, in this work, we introduce a novel criterion v spread (s) = v size (s)∇ - c (T s ) for spreading superpixels across the image (i.e., favoring regularity) by exploiting the shortest adjacent distance ∇ - c (T s ) of each tree T s . In terms of seed penalization, p obj , as expected, promotes a high superpixel concentration within and near the objects. Still, as one may note, it favors internal superpixels which exert limited influence on delineation. Thus, when irregularity and delineation are of utmost importance, we propose favoring only trees T s which are near a saliency border, increasing competition near the objects. More formally, the object border criterion is based on the tree's maximum saliency contrast such that p bord (s) = ∇ + O (T s ). Clearly, by focusing on delineation, the previous penalizations p obj and p bord favor irregularity (e.g., near object borders), which compromises compactness when it is desired. Conversely, if saliency is provided, it is expected to favor the object delineation to the detriment of the others. Thus, it is essential to provide a penalization that minimizes the impacts on compactness whilst improving the object delineation.

One possible approach is to exploit the quantity of seeds in each region. That is, for those majorly in the object, a higher concentration and, thus, a higher competition can suppress eventual leakings, which produce irregularity. Conversely, for those mainly in the background, we argue in favor of spreading instead of selecting near the object. That is, we propose favoring those trees located within the object, indicated by their mean saliency, and spreading those in the background, with respect to their shortest adjacent distance. Simply put, such penalization is defined by

p osb (s) = max O(T s ), (1 -O(T s ))∇ - c (T s )/d , considering d ∈ R >0
as the maximum distance between any two vertices.

By the same assumption presented for p bord , we may infer that superpixels within the object, due to their size, also present limited influence on the overall compactness, such that prioritizing them may not favor delineation nor compactness. Therefore, we may focus on those near the object borders to improve delineation while, for those at the background, spreading for promoting regularity and compactness. We argue that such selection tends to impact the compactness slightly since the major influence lies on the superpixels in the background. Finally, by using the tree's T s maximum saliency contrast ∇ + O (T s ) instead of its mean saliency O(T s ), the new penalization differs from p osb and can be formally defined as p bobs (s

) = max ∇ + O (T s ), (1 -O(T s ))∇ - c (T s )/d .

Experimental Results

In this section, we describe the experimental framework for evaluating SICLE with our proposals against state-of-the-art approaches. In Section 5.1, we present the experimental setup, including datasets, metrics, and baselines. Then, in Sections 5.2 and 5.3, we provide a quantitative and qualitative analysis whenever an object saliency map is absent or not, respectively. Finally, we discuss, analyze and exemplify some limitations of our proposals in Section 5.4.

Experimental Setup

In order to provide a wide-ranging analysis of the methods' performance in generating superpixels, we have selected five distinct medical and natural image datasets. First, we considered the Berkeley Segmentation Dataset (BSDS500) [START_REF] Arbelaez | Contour detection and hierarchical image segmentation[END_REF], a popular segmentation dataset containing 500 natural images. However, since it is contour-driven, such dataset is not applicable whenever a single object is desired. To cope with this drawback, we chose the well-known Extended Complex Scene Saliency (ECSSD) [START_REF] Shi | Hierarchical image saliency detection on extended cssd[END_REF] dataset, which contains 1000 natural images with complex objects and backgrounds. Third, we selected the Insects [START_REF] Mansilla | Oriented image foresting transform segmentation: Connectivity constraints with adjustable width[END_REF] dataset (130 images), given the challenge of delineating the thin legs of various bugs. In terms of medical images, the Liver [START_REF] Vargas-Muñoz | An iterative spanning forest framework for superpixel segmentation[END_REF] dataset (40 images) imposes a major difficulty in segmenting CT slices of the human liver, given its monochromaticity and smooth borders. Finally, the fifth dataset, named Parasites [START_REF] Belém | Superpixel segmentation by object-based iterative spanning forest[END_REF] is composed of 72 images of human intestinal parasite eggs which, although colored, are often attached to impurities and also exhibit smooth borders. We randomly selected 30% for optimization and 70% for testing for each dataset, except for BSDS500, which already establishes such division. For producing the saliency maps, we opted for the U 2 -Net estimator [START_REF] Qin | U2-net: Going deeper with nested u-structure for salient object detection[END_REF], finetuned using the same training set and its default parameters.

For measuring the performance of all methods, we selected popular evaluation metrics. First, let the two partitions R = {R 1 , . . . , R n } ∈ P(V) and O = {O 1 , . . . , O k } ∈ P(V) denote, respectively, the set of all superpixels and the set of all objects, considering n, k ∈ N ≥1 and P to be the power set. The Boundary Recall (BR) [START_REF] Stutz | Superpixels: An evaluation of the state-of-the-art[END_REF] evaluates the ratio between object boundaries and superpixel borders and it is defined by Equation 5:

BR(R, O) = |tp r (R, O)| |tp r (R, O)| + |fn r (R, O)| (5) in which tp r (R, O) ∈ N (resp. fn r (R, O) ∈ N)
computes the number of true positive (resp. false negatives) boundary pixels of the segmentation R with respect to O, considering a tolerance radius of r ∈ N ≥1 pixels. Similarly to [START_REF] Stutz | Superpixels: An evaluation of the state-of-the-art[END_REF], we set r = 0.0025d √ 2, being d the image's diagonal size. Under-segmentation Error (UE) [START_REF] Neubert | Superpixel benchmark and comparison[END_REF] measures the error (Equation 6) from object overlap with either internal or external superpixel (i.e."leaking"): [START_REF] Stutz | Superpixels: An evaluation of the state-of-the-art[END_REF] is also a famous metric, but it differs from both previous ones by evaluating whether the superpixels present a compact shape (being maximum when it is a circle). Thus, it is defined by Equation 7:

UE(R, O) = 1 |V| Oj ∈O Ri∈R min{|R i ∩O j |, |R i \O j |} (6) considering that R i ∩ O j ̸ = ∅. Compactness (CO)
CO(R) = 1 |V| Ri∈R 4π|R i | ℘(R) 2 (7)
in which ℘(R) ∈ N >0 denotes the perimeter of R i . Finally, we estimated the elapsed time of each method in a 64-bit Intel(R) Core(TM) i7-4790S PC with a CPU Speed of 3.20GHz. In this work, our goal is to provide SICLE variants for generating effective delineation with irregular and compact superpixels (hereafter named SICLE-IRREG and SICLE-COMP, respectively). Moreover, differently from [START_REF] Belém | Efficient Multiscale Objectbased Superpixel Framework[END_REF][START_REF] Belém | Fast and effective superpixel segmentation using accurate saliency estimation[END_REF], these novel versions of SICLE exploit differential computations of the IFT (DIFTs) [START_REF] Falcão | Interactive volume segmentation with differential image foresting transforms[END_REF] for improving speed. For comparison, we selected the following state-of-theart algorithms as baselines based on their reported performance: (i) ERS [START_REF] Liu | Entropy rate superpixel segmentation[END_REF] 1 ; (ii) SLIC [START_REF] Achanta | SLIC superpixels compared to state-of-the-art superpixel methods[END_REF] 2 ; (iii) SH [START_REF] Wei | Superpixel hierarchy[END_REF] 3 ; (iv) IBIS [START_REF] Bobbia | Iterative boundaries implicit identification for superpixels segmentation: A real-time approach[END_REF] 4 ; (v) LSC [START_REF] Li | Superpixel segmentation using linear spectral clustering[END_REF] 5 ; (vi) GMM [START_REF] Ban | Superpixel segmentation using gaussian mixture model[END_REF] 6 ; (vii) OISF [START_REF] Belém | Superpixel segmentation by object-based iterative spanning forest[END_REF][START_REF] Belém | Superpixel generation by the iterative spanning forest using object information[END_REF] 7 ; and (viii) DAL-HERS [START_REF] Peng | Hers superpixels: Deep affinity learning for hierarchical entropy rate segmentation[END_REF] 8 . For all of the aforementioned methods, we used their default parameter setting. Regarding CO, SLIC, IBIS, and GMM offer highly compact superpixels. Conversely, SH, LSC, OISF, and ERS present highly irregular ones but with accurate object delineation (i.e., BR and UE). Finally, we included the recent DAL-HERS algorithm for comparison.

Non-object-based Analysis

We first optimized SICLE-IRREG and SICLE-COMP through grid-search under the situation of not having any object information (i.e., α = 0 and p none ). For all variants, we optimized considering N 0 ∈ {2000, 3000, 5000, 8000}, Ω ∈ {2, 3, 5, 7, 10}, f * = {f max , f sum }, w * = {w root , w sum }, ι ∈ {0.05, 0.10, 0.12, 0.15, 0.20, 0.25, 0.50, 0.75, 1.00}, and v * ∈ {v size , v minsc , v maxsc , v spread }. For SICLE-IRREG, we found the following values: (i) N 0 = 3000; (ii) Ω = 5; (iii) f * = f max and w * = w α root ; and (iv) v * = v minsc . These values converge to those found in [START_REF] Belém | Superpixel segmentation using dynamic and iterative spanning forest[END_REF][START_REF] Belém | Towards a simple and efficient object-based superpixel delineation framework[END_REF][START_REF] Belém | Efficient Multiscale Objectbased Superpixel Framework[END_REF]. For the novel compact variant SICLE-COMP, we fixed β = 12 as suggested in [START_REF] Vargas-Muñoz | An iterative spanning forest framework for superpixel segmentation[END_REF], and we obtained the following optimized configuration: (i) N 0 = 3000; (ii) Ω = 7; (iii) f * = f sum and w * = w α sum ; (iv) ι = 0.12; and (v) v * = v maxsc . The optimization of ι prioritized delineation, crucial in segmentation, at the expense of on-par compactness performance. From Ω, we can infer that compactness is sensible to the quantity of seeds removed per iteration, requiring slightly more than SICLE-IRREG. Also, from v maxsc , we may assume that seeds positioned near high contrast boundaries favor the generation of compact superpixels without compromising delineation. We believe that, due to the proximity, such border is assured at first by competition while possible leakings, often in low contrast regions, are prevented by the spatial limitation imposed by w sum .

The quantitative results of all methods are shown in Figure 3. It is possible to see that SICLE-IRREG achieves superior delineation in all datasets for BR, except for Insects. It is important to notice that, although ERS excels on such dataset, its performance is fair for both medical image datasets. For BSDS500 and Parasites, DAL-HERS achieves on-par performance with our proposal but presents a poor delineation for Liver and Insects, which shares similarities with BSDS500. In fact, apart from SICLE-IRREG and SH, the remaining irregular methods do not present a consistent performance throughout the baselines, indicating a tendency to be best suited for specific conditions rather than a general solution for superpixel segmentation. Regarding UE, we can see that SICLE is continuously on pair with the best methods on each dataset.

We can also notice such effective performance and stability in SICLE-COMP for all metrics considered. For BR and UE, our proposal achieves onpar performance with GMM and LSC in BSDS500 and Insects while surpassing them in Liver and Parasites by a significant margin. For CO, we can see that SICLE-COMP consistently achieves a similar absolute performance irrespective of the dataset. Under the same perspective, IBIS consistently produces the most compact superpixels in all datasets considered. On the other hand, similarly to the irregular baselines, the remaining methods, such as SLIC, GMM, and LSC, are unable to exhibit the same behavior seen on BSDS500 and Insects, considering all metrics on Liver and Parasites.

Regarding speed, Table 1 compares the performance of the fastest methods in the BSDS500 dataset. It is important to notice that, although SICLE is O(|V| log |V|), bounded by the IFT, and the other referred methods are O(|V|), our proposal executes fewer iterations as N f increases. As a result, it becomes faster for usual superpixel quantities, whereas SLIC and LSC, for instance, perform a strict number of 10 iterations independently of N f . Also, in a single execution, SICLE generates the exact number of scales in a multiscale segmentation, while SH may unnecessarily exceed such quantity by computing the whole hierarchy [START_REF] Belém | Efficient Multiscale Objectbased Superpixel Framework[END_REF].

From Figure 4, we can perceive the effective performance of the novel SICLE variants by their delineation of the parasite egg. Compared with irregular methods, SICLE-IRREG best approximates the object borders and presents minimal leaking errors. Although SH also manages to present accurate object delineation, it still presents leakings around the object, like the segmentations from ERS and LSC. For those approaches which aim for compactness, it is possible to see that the result from SICLE-COMP achieves an accurate delineation of the object with compact superpixels, while the remaining ones, although generated highly compact and regular superpixels, exhibit major segmentation errors.

Object-based Analysis

In this section, we further optimized the configuration found in the previous section for considering prior object information. First, we analyzed the impact of the saliency quality only into delineation (i. state-of-the-art [START_REF] Qin | U2-net: Going deeper with nested u-structure for salient object detection[END_REF] (U 2 ); and (iii) ideal, represented by the object's ground-truth (GT). Concurrently, we optimized the saliency importance α ∈ {0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0} with respect to each case. We emphasize that the GT is only used in this experiment. Coincidently, the best values of α for each case, for both SICLE-IRREG and SICLE-COMP, were 0, 2, and 3, respectively, and they are represented in Figure 5. As expected, we note that the trustiness of the information correlates with its quality. For BB, α = 0 avoids considering saliency inaccuracies and maintains an effective delineation performance. On the other hand, by increasing α, our method can significantly improve its performance for better quality estimations, as seen for U 2 and GT. Moreover, we found that such improvement does not impair the compactness achieved by SICLE-COMP. Thus, our approach can maximize compactness and delineation, whereas other approaches only offer a compromise. It is worth noticing the discrepancy between the performances when considering U 2

and GT, indicating that, although accurate, the state-of-the-art estimation is not ideal. Subsequently, we evaluated the best penalization p * ∈ {p none , p obj , p osb , p bobs } for each variant. As expected, p bord is more suited for SICLE-IRREG by increasing the number of seeds around the object, thus increasing competition in crucial regions. But, for SICLE-COMP, p bobs presented the best results without compromising compactness, indicating the importance of spreading background superpixels to suppress the irregularity within the object. Finally, in the last experiment, we verified whether such final relevant seed estimation is accurate by varying values for Ω ∈ {2, 3, 5, 7, 10}. That is, the more accurate it is, the fewer iterations are required to achieve effective delineation (i.e., ideally two). For both SICLE-IRREG and SICLE-COMP, Ω = 2 suffice for achieving top performance with no significant variation on BR, UE, and CO.

Figure 6 shows the quantitative results of all methods. It is possible to see that, contrasting Quantitative results on the test set of each dataset using the U 2 -Net map [START_REF] Qin | U2-net: Going deeper with nested u-structure for salient object detection[END_REF].

with Figure 3, SICLE variants improved their performance significantly, up to surpassing ERS in Insects. Although the saliency quality plays an essential role in such improvement, we argue that our proposals intelligently incorporate such information. For instance, OISF, which uses the same information, is unable to achieve similar performance in all datasets, especially in Insects. Similarly to SICLE-IRREG being the best amongst all baselines in BR and UE, SICLE-COMP presents the best performance in both metrics for all compact methods, being on par with state-of-the-art methods in delineation, such as SH. Finally, by comparing with the non-object-based scenario, this upgrade did not deteriorate SICLE's performance in CO, as previously stated. We further argue that our object-based approaches offer a more suitable methodology for object incorporation, especially when compared to the performance instability of DAL-HERS in both BR and UE, for producing either irregular or compact superpixels.

The speed performance of the fastest methods in ECSSD is presented in Table 2. It is important to state that the performance of SICLE-IRREG differs from the results shown in [START_REF] Belém | Fast and effective superpixel segmentation using accurate saliency estimation[END_REF]. We argue that the speedup achieved from exploiting DIFTs is limited when a significantly high quantity of seeds is removed between iterations, which is a direct result when Ω = 2. Intuitively, removing superpixels can generate inconsistencies during the IFT execution, which are absent in the sequential version and must be handled in the differential one. Thus, the higher the number to be removed, the higher the quantity of inconsistencies to be solved, impacting the overall performance. We may note such behavior on the performance discrepancy for N f = 1000 and N f = 500. Still, differently from [START_REF] Belém | Fast and effective superpixel segmentation using accurate saliency estimation[END_REF], our implementation benefits from multiscale segmentation. As the number of scales k ∈ N ≥1 ≤ Ω increase, more iterations are required and, consequently, M estimates higher quantities of relevant seeds to be maintained per iteration. Therefore, by removing fewer superpixels at each iteration, fewer inconsistencies are generated and need to be handled by DIFT. Lastly, when comparing with the baselines, both SICLE-IRREG and SICLE-COMP still exhibit the fastest segmentation performances irrespective of N f . As an example of how SICLE variants profit from our proposals and use object information appropriately, we may note, in Figure 7, that OISF replicates the errors of the saliency map in its segmentation, whereas our proposals overcame them and provided an accurate object delineation. Compared to non-object-based baselines, the proper incorporation of object information assisted SICLE-IRREG and SICLE-COMP in avoiding severe leaks. Finally, it is interesting to notice that SICLE-COMP favored the superpixels within the object less than SICLE-IRREG but more than GMM and LSC, as represented by the larger background superpixels.

Limitations

Although the performance of SICLE was improved after considering our proposed modifications, it still presents several limitations. From Figure 8, one can see that, even though the saliency map partially detects the insect's thin parts (e.g., antenna and legs), SICLE cannot, at least, delineate the same regions with similar accuracy. We argue that this behavior is a drawback of seedbased methods since far away seeds have more difficulty to offer strongly connected paths to pixels in thin parts of the object, even with help of a saliency map. That is, for small quantities of superpixels, seeds tend to be farther, and consequently, it is more challenging to conquer the pixels associated with its classification (i.e., object or background seed). In such case, one may propose a more meticulous path-cost or arccost function for improving the delineation of thin object parts or, conversely, conceive different seed strategies (i.e., seed sampling or removal) for increasing the seed resolution in such regions and, consequently, increasing competition in such crucial region. Either way, knowing the peculiarities and particularities of the object is utterly vital for solving this difficulty.

Multiobject Handling

In this section, we will debate briefly over the challenges of segmenting multiple objects of interest in an image using SICLE. First, let O = {O 1 , . . . , O k } ∈ P(I) as the set of all k ∈ N >0 objects within the image I and, equivalently, in the digraph G. As one may see, k = 1 consists of the usual object saliency task.

Difficulties arise when k > 1, especially on manipulating different object informations in SICLE. A straightforward approach could be incorporating multiple saliencies, one dimension for each object, such that O(v) = ⟨O 1 (v), . . . , O k (v)⟩ ∈ [0, 1] k given v ∈ V. However, we argue that storing k saliencies is memory inefficient since background vertices are redundantly marked as such ∀O j , given 1 ≤ j ≤ k. Instead, we claim that, for SICLE, a single saliency O(v) = max 1≤j≤k {O j (v)}, for v ∈ V, is sufficient for handling multiple objects. Aside being more memory efficient, conceptually speaking, our proposed saliency still depicts the probable likelihood of a vertex belonging to an object. For such case, and without loss of generality and abuse of notation, we analyze three situations, as exemplified in Figure 9, considering two distinct objects O x , O y ∈ O and ideal object estimation: (i) O y ̸ ∈ A(O x ) ; (ii) O y ∈ A(O x ) ; and (iii) O x ∩ O y ̸ = ∅. Thus, Figure 9(a) depicts (i), while Figure 9(c) relates to (iii). Both Figures 9(b,d) are associated with (ii) in which, for the former, the distinction of both triangles of equal shape is facilitated by the colors, and highly difficulted for the latter due to the color gradient. Intuitively, (i) presents the case of two non-adjacent objects and, consequently, every path ρ u⇝v in which u ∈ O x and v ∈ O y is ensured to cross a saliency border, being properly penalized by o. However, for (ii) and (iii), ρ u⇝v can be composed of nonbackground vertices of distinct objects, resulting in no penalization. Nevertheless, we can verify that w α root and w α sum tend to estimate a distinction between v, u, being v ∈ O x and u ∈ O y , even for o(v, u) = 0, if they exhibit sufficient color divergence. Finally, for case (iii), we argue that it is an ill-posed problem due to the impossibility to define and, thus, to detect the overlapped object's boundaries, either in F or O, in a bi-dimensional representation.

Conclusion and Future Work

This work proposes novel arc-cost functions, a new seed relevance criterion, and three new seed relevance penalizations for improving the effectiveness and efficiency of the state-of-the-art superpixel segmentation method Superpixels through Iterative CLEarcutting (SICLE). Due to our proposals, SICLE can now compute compact superpixels with high adherence to object boundaries and can exploit the accurate saliency information during delineation. Moreover, our proposals maintain the properties of previous SICLE variants, such as saliency inaccuracy robustness and saliencybased multiscale segmentation. Results show that, with our modifications, SICLE produces accurate object delineation with irregular and compact superpixels whenever a saliency map is absent, achieving a faster performance than several stateof-the-art methods analyzed in this work. When object information is provided, the novel proposals led SICLE variants to achieve superior performance in delineation and speed for distinct objects of interest, requiring only two iterations for segmentation. Finally, we propose a brief discussion on the challenges and a possible solution for multiobject handling in SICLE. For future work, we intend to extend SICLE for video supervoxel segmentation and evaluate its applicability as an interactive segmentation tool.

Original Saliency map

SICLE-IRREG SICLE-COMP

Fig. 8 Examples of the limitations of our approaches, even when assisted by a saliency map from the U 2 -Net estimator [START_REF] Qin | U2-net: Going deeper with nested u-structure for salient object detection[END_REF], requiring 50 superpixels. The object boundaries and the superpixel borders are depicted in red and cyan, respectively. Yellow circles indicate regions where SICLE could not properly approximate to the object borders.

88887.191730/2018-00, 88887.712530/2022-00) for the financial support.

Fig. 1

 1 Fig. 1 Segmentation requiring 100 superpixels. The object boundaries and the superpixel borders are depicted in red and cyan, respectively. Yellow circles indicate regions where SICLE best approximates the object borders in contrast with the baselines.

  Fig.2The top image shows the SICLE framework for generating 100 superpixels using a deep-learning saliency estimator[START_REF] Qin | U2-net: Going deeper with nested u-structure for salient object detection[END_REF]. Superpixel borders and seeds are depicted by cyan lines and dots, respectively. Whereas the bottom image explicit the inputs and outputs of each process in the pipeline.

Fig. 3

 3 Fig.3Quantitative results on the test set of each dataset when no object information is provided.

Fig. 4

 4 Fig. 4 Qualitative results with an image from Parasites requiring 100 superpixels. The object boundaries and the superpixel borders are depicted in magenta and cyan, respectively. The images were cropped for visualization purposes. Yellow circles indicate regions where SICLE best approximates the object borders in contrast with the baselines.

Fig. 5

 5 Fig. 5 Quantitative results on the train set of each dataset using the ground-truth (GT), U 2 -Net map [43] (U 2 ) and the object's minimum bounding box (BB) as saliency maps.

Fig. 6

 6 Fig.6Quantitative results on the test set of each dataset using the U 2 -Net map[START_REF] Qin | U2-net: Going deeper with nested u-structure for salient object detection[END_REF].

Fig. 7

 7 Fig.7Qualitative results with an image from ECSSD requiring 100 superpixels and using the U 2 -Net estimator[START_REF] Qin | U2-net: Going deeper with nested u-structure for salient object detection[END_REF]. The object boundaries and the superpixel borders are depicted in red and cyan, respectively. Yellow circles indicate regions where SICLE best approximates the object borders in contrast with the baselines.

Table 1

 1 Speed performance, in seconds, of the fastest methods on the BSDS500 dataset when no object information is provided. The values depicted in bold, blue, and red indicate, respectively, the first, second, and third faster performances for each N f .

	N f	SLIC	SH	IBIS	LSC	SICLE-IRREG SICLE-COMP
	50	0.610±0.027 0.856±0.026 0.762±0.027 0.774±0.026	0.808±0.065	1.031±0.047
	250 0.612±0.027 0.857±0.028	0.753±0.025	0.789±0.027	0.678±0.030	0.746±0.035
	500 0.615±0.027 0.853±0.026	0.752±0.025	0.787±0.029	0.567±0.024	0.718±0.030
	1000 0.620±0.028 0.857±0.026	0.748±0.025	0.803±0.030	0.538±0.021	0.566±0.023

Table 2

 2 Speed performance, in seconds, of the fastest methods on ECSSD dataset using the U 2 -Net map[START_REF] Qin | U2-net: Going deeper with nested u-structure for salient object detection[END_REF]. The values depicted in bold, blue, and red indicate, respectively, the first, second, and third faster performances for each N f .

	N f	SLIC	SH	IBIS	LSC	SICLE-IRREG SICLE-COMP
	50	0.576±0.029 0.813±0.028 0.710±0.027 0.702±0.036 0.501±0.055	0.514±0.046
	250 0.581±0.027 0.816±0.028 0.700±0.028 0.713±0.036 0.503±0.040	0.511±0.043
	500 0.583±0.028 0.813±0.028 0.694±0.026 0.714±0.039 0.507±0.039	0.517±0.042
	1000 0.583±0.028 0.812±0.027 0.694±0.028 0.718±0.036 0.420±0.090	0.482±0.060

https://github.com/mingyuliutw/EntropyRateSuperpixel

https://www.epfl.ch/labs/ivrl/research/slic-superpixels/

https://github.com/semiquark1/boruvka-superpixel

https://github.com/xapha/IBIS

https://jschenthu.weebly.com/projects.html

https://github.com/ahban/GMMSP-superpixel

https://github.com/LIDS-UNICAMP/OISF

https://github.com/hankuipeng/DAL-HERS

Acknowledgements

The authors thank the Conselho Nacional de Desenvolvimento Científico e Tecnológico -CNPq -(Universal 407242/2021-0, PQ 303808/2018-7, 310075/2019-0), the Fundação de Amparo a Pesquisa do Estado de Minas Gerais -FAPEMIG -(PPM-00006-18), the Fundação de Amparo a Pesquisa do Estado de São Paulo -FAPESP -(2014/12236-1) and the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -CAPES -Finance code 001 (COFECUB


Jérémy Monteiro

Antoine Lefèvre

Adeline Oury

Diane Dufour-Rainfray

Lydie Nadal-Desbarats

Patrick Emond

Diane Dufour

3-NPH derivatization of SCFAs in human biofluids for longitudinal and multi-matrix exploration

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction Material & Methods

Results

Conclusion

In this work, we propose an absolute quantification method of SCFAs suitable for human biological samples such as feces, saliva, serum, DBS, CSF and urine using 3-NPH derivation and LC-MS/MS. We have adapted the pre-analytical parts for each sample with the aim of using a common calibration curve, facilitating multi-array quantification studies. A chromatographic gradient has been developed in order to separate, after derivatization, isomeric and isobaric SCFAs and to be free from interferences, whatever the matrix studied.

Quantitative Data

  fatty acids (SCFAs) play an important role in human biochemistry. They originate primarily from the digestive system through the microbial fermentation of carbohydrates and bind to the GPR41/43 receptor on the epithelial barrier or with cellular receptors to regulate glucose storage, fatty acid oxidation or fuel the TCA cycle. SCFAs are also involved in regulating the levels of several neurotransmitters such as GABA, glutamate/glutamine, dopamine, norepinephrine. Their role is also highlighted in many inflammatory and neurodegenerative diseases such as Alzheimer's disease or Parkinson's disease.

  Comparison of different SCFA concentrations between biological matrices, from healthy patients. The results from intra-day and inter-day repeatabilities on 10 aliquots from 2 different pools. The concentrations found in saliva are in µM (left axis) whereas they are expressed in nM for the other matrices. Determination of residual SCFA concentrations from blotting papers. Three types of blotter are used in clinical routine: neonatal screening (NeoNat), patient monitoring (Adult Tracking) and internal quality control (IQC) samples. The data was produced with 30 aliquots from a blotter, with spots made randomly over the entire surface. The results show that it is necessary to take into account the residual SCFA concentrations contained in blotters for any assays carried out on DBS. SCFAs have been extensively and intensively studied by 3-nitrophenylhydrazine-LC-MS² derivatization due to their involvement in metabolisms and diseases. We propose a unique methodology to quantify SCFAs in 6 different biological matrices to generate FAIR data. Reference values are proposed for CSF and DBS. We also propose a new methodology for sampling patient faeces that could be widely applicable. These validated approaches are now used to characterize metabolism and better investigate the inflammatory component of several diseases.