Rahat Khan
email: rahat.khan@univ-st-etienne.fr

Cecile Barat
email: cecile.barat@univ-st-etienne.fr

Damien Muselet
email: damien.muselet@univ-st-etienne.fr

Christophe Ducottet
email: ducottet@univ-st-etienne.fr

Spatial histograms of soft pairwise similar patches to improve the bag-of-visual-words model

Keywords: spatial information, object classification, angle and distance histograms, Bag-of-Words

In the context of category level scene classification, the bag-of-visual-words model (BoVW) is widely used for image representation. This model is appearance based and does not contain any information regarding the arrangement of the visual words in the 2D image space. To overcome this problem, recent approaches try to capture information about either the absolute or the relative spatial location of visual words. In the first category, the so-called Spatial Pyramid Representation (SPR) is very popular thanks to its simplicity and good results. Alternatively, adding information about occurrences of relative spatial configurations of visual words was proven to be effective but at the cost of higher computational complexity, specifically when relative distance and angles are taken into account. In this paper, we introduce a novel way to incorporate both distance and angle information in the BoVW representation. The novelty is first to provide a computationally efficient representation adding relative spatial information between visual words and second to use a soft pairwise voting scheme based on the distance in the descriptor space. Experiments on challenging data sets MSRC-2, 15Scene, Caltech101, Caltech256 and Pascal VOC 2007 demonstrate that our method outperforms or is competitive with concurrent ones. We also show that it provides important complementary information to the spatial pyramid matching and can improve the overall performance.

Introduction

In category level and scene classification, the bag-of-visual-words (BoVW) method, first introduced by [START_REF] Sivic | Video google: A text retrieval approach to object matching in videos[END_REF][START_REF] Csurka | Visual categorization with bags of keypoints[END_REF], has shown excellent results in recent years [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF][START_REF] Li | A bayesian hierarchical model for learning natural scene categories[END_REF]. In this method, an image is represented as a histogram of quantized local features called visual words. However, being orderless, histogram representations do not provide any spatial information. This is considered to be one of the major drawbacks of this very successful method.

Different methods have been proposed to incorporate spatial information into the BoVW representation [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF][START_REF] Kim | Disiclass: discriminative frequent pattern-based image classification[END_REF][START_REF] Liu | Integrated feature selection and higher-order spatial feature extraction for object categorization[END_REF][START_REF] Savarese | Discriminative object class models of appearance and shape by correlatons[END_REF][START_REF] Wu | Visual language modeling for image classification[END_REF]. Some of these approaches use spatial context before the vocabulary construction step to incorporate spatial information [START_REF] Qin | Scene categorization via contextual visual words[END_REF][START_REF] Zhou | Spatial context for visual vocabulary construction[END_REF][START_REF] Morioka | Building compact local pairwise codebook with joint feature space clustering[END_REF]. The spatial context is defined at the local feature level as a set of descriptors extracted form neighboring regions so as to enrich the description around a feature point. For example in [START_REF] Morioka | Building compact local pairwise codebook with joint feature space clustering[END_REF], the authors create pairs of local neighbor SIFT descriptors, concatenate them and construct the vocabulary from these 256-D combined features instead of from the classical individual 128-D SIFT. Alternatively, most of the approaches work at the visual word level. They model the spatial arrangements of visual words on the 2D image space as an additional step [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF][START_REF] Kim | Disiclass: discriminative frequent pattern-based image classification[END_REF][START_REF] Wu | Visual language modeling for image classification[END_REF][START_REF] Krapac | Modeling spatial layout with fisher vectors for image categorization[END_REF][START_REF] Tirilly | Language modeling for bag-of-visual words image categorization[END_REF][START_REF] Yuan | Discovery of collocation patterns: from visual words to visual phrases[END_REF][START_REF] Yuan | From frequent itemsets to semantically meaningful visual patterns[END_REF][START_REF] Zhang | Improving bag-of-words model with spatial information[END_REF][START_REF] Zheng | Incorporating spatial correlogram into bag-of-features model for scene categorization[END_REF]. These later methods are more popular as they obtain superior classification accuracies. It is due to the fact that they are able to capture both local and global relationships among the visual words.

In this context, the Spatial Pyramid Representation (SPR) [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF] is probably the most notable work. Its principle relies on the division of an image into sub-windows and the computation of a local BoVW histogram in each. Two images are then compared by using an intersection kernel computed between the two corresponding sets of histograms.

Although SPR performs very well, it only captures the information about approximate absolute locations of the visual words in images. An alternative consists in extracting relative spatial interactions between the visual words [START_REF] Savarese | Discriminative object class models of appearance and shape by correlatons[END_REF][START_REF] Zheng | Incorporating spatial correlogram into bag-of-features model for scene categorization[END_REF][START_REF] Yuan | Discovery of collocation patterns: from visual words to visual phrases[END_REF][START_REF] Zheng | Visual synset: Towards a higher-level visual representation[END_REF][START_REF] Sivic | Discovering objects and their localization in images[END_REF][START_REF] Zhang | Descriptive visual words and visual phrases for image applications[END_REF]. In these representations, the absolute location of each visual word is lost but, for example, information about visual words that are frequently co-occurring together at a certain distance is taken into account. However, the abundance of visual words in an image makes it computationally expensive to explicitly model relative spatial relationships among visual words. Thus, methods like [START_REF] Liu | Integrated feature selection and higher-order spatial feature extraction for object categorization[END_REF][START_REF] Savarese | Discriminative object class models of appearance and shape by correlatons[END_REF] employ vocabulary compression or feature selection and model only local or semi-local spatial information to speed up the computation. Nevertheless, Elfiky et al. [START_REF] Elfiky | Discriminative compact pyramids for object and scene recognition[END_REF] have shown that vocabulary compression before spatial information extraction results into declined classification performance. In another work, Parikh [START_REF] Parikh | Recognizing jumbled images: The role of local and global information in image classification[END_REF] examines the human vs machine performance on jumbled images and concludes that existing machine vision techniques are already effective in modeling local information from images, thus future research efforts should be focused on more advanced modeling of global information.

Based on these observations, in this work, we propose a way to model the global and local relative spatial distribution of visual words over an image. To do so, we introduce the concept of soft pairwise spatial angle-distance histograms to capture the distribution of similar descriptors. The term "soft" means that we apply a soft weighting strategy in a similar way as soft assignment techniques [START_REF] Van Gemert | Kernel codebooks for scene categorization[END_REF][START_REF] Zhou | Hierarchical gaussianization for image classification[END_REF].

The novelty compared with previous approaches is threefold: i) enabling infusion of pairwise relative spatial information (modeling both distances and angles between visual words while most of the approaches consider only distances), ii) adopting a simple word selection scheme that avoid combinatorial explosion, iii) combining hard assignment for visual word selection and soft assignment to weight the contribution of descriptor pairs in spatial histograms.

We experimentally evaluate our new representation on classification tasks with various challenging data sets. The aim is both to study the influence of various parameters of the method and to compare the performances over state of the art concurrent approaches.

The rest of the paper is organized in the following way: the next section describes a review of the related works. Section 3 presents our approach to incorporate spatial in-2 formation into the BoVW representation. Section 4 describes the implementation details and section 5 presents the results on different benchmarks and comparisons with several other methods. Section 6 concludes the article pointing towards our future works.

Related works

There exist two main approaches to incorporate spatial information in the BoVW representation, the difference being the information added over the classical BoVW, i.e. either the approximate absolute positions of the visual words or the relative positions between these visual words.

The SPR [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF] is the most widely used method among the ones representing absolute spatial coordinates. Its principle relies on the division of an image into a sequence of increasingly coarser grids (eg. 4 by 4, 2 by 2 and 1 by 1) and on the computation of a local BoVW in each cell. Two images are then compared using an intersection kernel computed between the two corresponding sets of histograms. Bosch et al. [START_REF] Bosch | Representing shape with a spatial pyramid kernel[END_REF] have generalized the intersection kernel with other quasi-linear kernels like chi-square and learned weights for each pyramid level rather than using fixed weights. Another weight learning method was proposed for SPR in [START_REF] Harada | Discriminative spatial pyramid[END_REF]. More recently, in [START_REF] Cao | Spatial-bag-of-features[END_REF] Cao et al. proposed to extract local BoVW from more sub-windows than in SPR with different shapes (not only rectangular) and to select the best ones with a supervised approach. Finally, in [START_REF] Krapac | Modeling spatial layout with fisher vectors for image categorization[END_REF], Krapac et al. encode the absolute region locations of each visual word using the Fisher kernel.

Among the methods that add spatial relative interactions between the visual words, the most straightforward approaches are those based on correlograms [START_REF] Savarese | Discriminative object class models of appearance and shape by correlatons[END_REF][START_REF] Zheng | Incorporating spatial correlogram into bag-of-features model for scene categorization[END_REF]. The color correlograms have been introduced by Huang et al. in [START_REF] Huang | Image indexing using color correlograms[END_REF]. The idea was to represent the distribution of color pairs as a function of the spatial distance between these colors in the image. Savarese et al. extend this representation to the distribution of pairs of visual words in the image plane. Thus, given K visual words and T distances, the dimension of one correlogram is K × K × T . Then, the authors propose to cluster the T -dimensional vectors representing the evolution of the distribution of each pair across the T distances. The cluster representatives are called correlatons and the descriptor for one image is the histogram of correlatons. Consequently, in this final representation, only the spatial interactions between the visual words are accounted, regardless of the actual identities of these words.

Another way to incorporate relative interactions between visual words is to use visual phrases [START_REF] Yuan | Discovery of collocation patterns: from visual words to visual phrases[END_REF][START_REF] Zheng | Visual synset: Towards a higher-level visual representation[END_REF][START_REF] Sivic | Discovering objects and their localization in images[END_REF][START_REF] Zhang | Descriptive visual words and visual phrases for image applications[END_REF] in the final image representation. A visual phrase is a set of frequently locally co-occurring visual words. Most of the times, specific Frequent Itemset Mining (FIM) algorithms are applied in order to discover the most meaningful visual phrases since this selection is the key to these approaches. In [START_REF] Zhang | Descriptive visual words and visual phrases for image applications[END_REF], they consider only pairs of visual words located in a local neighborhood whose radius is related to the scale of the associated SIFT keypoint and use the supervised TF-IDF (Term Frequency -Inverse Document Frequency) information for the selection of the best visual phrases. In [START_REF] Sivic | Discovering objects and their localization in images[END_REF], they first select the 100 most frequent visual words and each visual word is paired with its 5 nearest neighbor in the image space. Unlike the previous approaches, in [START_REF] Yuan | Discovery of collocation patterns: from visual words to visual phrases[END_REF][START_REF] Zheng | Visual synset: Towards a higher-level visual representation[END_REF][START_REF] Zhang | Image retrieval with geometry-preserving visual phrases[END_REF], the visual phrases can contain more than two visual words. Thus, specific FIM algorithms are required to select the best visual phrases among all the local visual word sets. Also, while the visual phrases from [START_REF] Yuan | Discovery of collocation patterns: from visual words to visual phrases[END_REF][START_REF] Zheng | Visual synset: Towards a higher-level visual representation[END_REF] consists of visual words co-occurring in a local neighborhood, the ones from [START_REF] Zhang | Image retrieval with geometry-preserving visual phrases[END_REF] are not constrained by any locality properties. Indeed, in this last paper, the authors consider each image pair (I, I ′ ) and for each pair of the same word in these images, they calculate their offset (∆x, ∆y) which is the location of the word in the image I ′ subtracted from its location in the image I. Then a vote is applied in the offset space in order to detect geometry-preserving visual phrases. In one cell of this offset space are falling all the words whose offset is similar between the two images, but the information about the spatial distribution in the image spaces of the visual words belonging to the same visual phrase is lost. In order to cope with this problem, the authors propose to store in memory the locations of each visual word occurrence within each image. This information is used during the image comparison step. Using a similar storage approach, Wu et al. [START_REF] Wu | Bundling features for large scale partial-duplicate web image search[END_REF] bundle visual words falling in the same local neighborhood, i.e. the same MSER regions [START_REF] Matas | Robust wide baseline stereo from[END_REF]. Then, the orders of the visual words along the x and y axis within each local group is saved in memory and this information is used during the comparison step in order to check the approximate relative positions of the words within each detected MSER region. These two last approaches save more information in memory than the single histogram-like structure and this requires a particular comparison step during the test phase. Even if the approach of [START_REF] Agarwal | Hyperfeatures – multilevel local coding for visual recognition[END_REF] is a bit different from the previous papers, the ideas are similar in the sense that, in this paper, the authors also exploit frequently locally co-occurring visual words to describe images. Indeed, they propose to extract several local BoVW on a grid (as done in the SPR approach) from the images of the dataset and to apply vector quantization on these local BoVW. The cluster representatives are "second order" visual words from which second order BoVW can be computed. And iteratively, the authors extend the classical "first order BoVW" to "n th order BoVW". Obviously, these BoVW contain information about the local spatial interactions between the original visual words.

All the previous papers dealing with relative visual words locations, only consider the distances between the visual words as supplementary information over the classical BoVW. In [START_REF] Khan | Spatial orientations of visual word pairs to improve bag-of-visual-words model[END_REF], we have rather proposed to incorporate angle information in the final representation. Hence, we have shown that global spatial orientations of patch pairs whose corresponding visual words are the same (called intra-type visual word pairs) are discriminative and significantly improve the classification accuracy. The approach proposed by Liu et al. in [START_REF] Liu | Integrated feature selection and higher-order spatial feature extraction for object categorization[END_REF] has the advantage that it considers both distances and angles between the visual words and by this way represents more accurately the relative positions of the visual words in the image space. Nevertheless, this whole information requires much more memory space and computational time during both training and testing steps. Thus, the authors introduce an integrated feature selection and spatial information extraction technique to reduce the number of features and also to speed up the process. The process of feature selection and second order feature (e.g. spatial information) extraction are run alternatively at each iteration of the algorithm. At each round, feature selection selects one feature, and feature extraction pairs this feature with each of the previously selected features. The final image representation is constructed by the concatenation of the first and second order features.

Finally, these works show that both absolute and relative spatial information improve the classification accuracy over the classical BoVW. Consequently, the current trend consists in combining these two information into the final representation [START_REF] Zhang | Improving bag-of-words model with spatial information[END_REF][START_REF] Yang | Spatial pyramid co-occurence for image classification[END_REF][START_REF] Harada | Improving local descriptors by embedding global and local spatial information[END_REF] in order to boost the results. For example, in [START_REF] Yang | Spatial pyramid co-occurence for image classification[END_REF], spatial predicates are introduced and, given a spatial predicate, a co-occurrence matrix is computed over all the visual word pairs. In their approach, an image is represented as a set of co-occurrence matrices of size K × K (if K is the number of visual words) and thus they must limit the size of the vocabulary and the number of considered spatial configurations. Moreover, spatial distances are defined in an absolute way inside a region of a pyramid. In [START_REF] Harada | Improving local descriptors by embedding global and local spatial information[END_REF], Harada et al. incorporate local spatial information by evaluating local auto-correlations between the extracted descriptors and global information by weighting differently the contribution of each cell of a regular grid superimposed to the image (as done by SPR). These weights are learned so that they are related to the discriminative power of the considered cells, given a classification task. Thus, in most of the works combining absolute and relative spatial information, the relative spatial information is combined with the reference SPR approach. Consequently, in this paper, we propose to design a new descriptor that represents the relative spatial information and that is complementary to the SPR descriptor. Our original idea is to account both the distances and the angles between the visual words by using the notion a soft-similarity, which allows us to avoid a complex selection step as required in [START_REF] Liu | Integrated feature selection and higher-order spatial feature extraction for object categorization[END_REF].

Encoding distance-orientations information of similar patches

The principle of our method is to use pairwise spatial histograms to encode spatial cooccurrence of similar word pairs. In this section we first present original pairwise spatial histograms introduced by Liu et al. [START_REF] Liu | Integrated feature selection and higher-order spatial feature extraction for object categorization[END_REF], then we introduce pairwise spatial histograms of similar patches and finally our image representation.

Pairwise spatial histograms

In the standard BoVW model, an image is represented as a histogram of occurrences of its visual words [START_REF] Sivic | Video google: A text retrieval approach to object matching in videos[END_REF]. More formally, a set of local descriptors {d 1 , d 2 , d 3 , d 4 , . . . d N } is first extracted from a sampled set of image patches [START_REF] Lowe | Object recognition from local scale-invariant features[END_REF]. Each descriptor d i is a vector giving the description of the image patch i and N is the total number of patches in the image. The visual vocabulary W = {w 1 , w 2 , w 3 , w 4 . . . w K } is obtained by typically applying an unsupervised K-means clustering on a large set of descriptors from training images. Each visual word w k represents a cluster center, K being the vocabulary size, ie. the number of predefined clusters. Each patch i of the image is assigned to a visual word which corresponds to the nearest center in the descriptor space. If soft assignment or sparse coding is used [START_REF] Van Gemert | Kernel codebooks for scene categorization[END_REF][START_REF] Zhou | Hierarchical gaussianization for image classification[END_REF][START_REF] Yang | Linear spatial pyramid matching using sparse coding for image classification[END_REF], the patch is assigned to a weighted set of visual words. The final histogram is obtained by pooling the contribution of each visual word over the whole image or over spatial regions.

In Liu et al. [START_REF] Liu | Integrated feature selection and higher-order spatial feature extraction for object categorization[END_REF], a pairwise spatial histogram is defined according to a discretization of the spatial neighborhood into several bins encoding the relative spatial position (distance and angle) of two visual words (Figure 1). Given a pair of visual words (w k , w l ), all the pairs of patches (P i , P j ) such that P i is of type w k and P j is of type w l are considered. Then, a pairwise spatial histogram associated to (w k , w l ) is defined as the count of all occurrences of P j falling into a specific spatial bin relatively to P i , the count being averaged for all instances of P i (Figure 1).

Motivation of considering similar cues

The number of possible pairs of visual words is potentially very large and thus, Liu on Discrete AdaBoost with decision stumps as in [START_REF] Viola | Rapid object detection using a boosted cascade of simple features[END_REF]. In [START_REF] Khan | Spatial orientations of visual word pairs to improve bag-of-visual-words model[END_REF], we have proposed another alternative which is to consider only pairs of identical visual words. The motivation came from the previous works [START_REF] Deselaers | Global and efficient self-similarity for object classification and detection[END_REF][START_REF] Shechtman | Matching local self-similarities across images and videos[END_REF] where the authors have argued that modeling the distribution of similar cues across an image can give discriminative information about the content of that image. Figure 2 shows an example which gives an intuition to better understand the idea. In this illustration, we consider patches associated with the same visual word as similar cues.

Pairwise spatial histograms of similar patches

The notions of similar cues and similar words are not equivalent. If we consider clusters delimiting visual words in the descriptor space, two cues at the cluster borders could be very similar being in different clusters. Similar cues in this context are more related to a small inter-patch distance in the descriptor space.

This problem is similar to the visual word ambiguity problem which has been discussed in many works on the BoVW model. Indeed, the vector quantization applied at the visual word assignment step introduces ambiguity if the considered descriptor is at an intermediate position between several cluster centers [START_REF] Van Gemert | Visual word ambiguity[END_REF]. To address this issue, soft assignment approaches have been introduced. Their principle is to use the distance in the descriptor space to compute a weight or a probability estimate associated to a given visual word [START_REF] Van Gemert | Kernel codebooks for scene categorization[END_REF][START_REF] Zhou | Hierarchical gaussianization for image classification[END_REF][START_REF] Liu | In defense of soft-assignment coding[END_REF].

Hence, we propose to analyze the spatial positions of the patches which are situated in proximity in the descriptor space. To avoid the use of a threshold to identify similar patches (hard similarity), we consider all the pairs of patches and we weight the contribution of each pair as a decreasing function g(x) of their distance x in the descriptor space (soft similarity). We propose to use a Gaussian function of standard deviation α defined as:

g(x) = 1 α √ 2π e -x 2 2α 2 (1)
This parameter gives us the control to highly weight patches that are in close proximity in the descriptor space and to ignore the ones which are far. More information about the choice of this parameter can be found in section 4.2. More formally, we consider the set S k of all the pairs of patches where at least one patch in the pair belongs to the visual word w k . A given pair (P i , P j ) ∈ S k is characterized both by a pair of descriptors (d i , d j ) and a pair of positions in the image space denoted (p i , p j ) (Figure 3). Note that both d i and p i are vectors with d i ∈ R D (descriptor space) and p i ∈ R 2 (image space). A pairwise spatial histogram of similar patches is then defined considering a discretization of the image space into M bins denoted {b m , m = 1, . . . , M } with an angle θ ∈ [0, π[ split into M θ equal angle bins and the radius r ∈ [0, R] split into M r radial bins so that M = M θ M r . For example, in the illustration at the bottom of figure 3, the total number of bins M is equal to 15 (M θ =5 and M r =3). The values of M θ and M r will be determined in the experimental section and the maximum radius R is chosen to be the diagonal of the image, in order to reduce scale sensitivity. Note that by using such maximum value for the radius, the representation may indirectly capture the absolute spatial location since patches located close from the image borders have a more limited number of possible angles and distances for pairing. In all spatial methods, such border effects exist and are difficult to quantify.

The count H Sof t-Inter k (m) of bin b m of the spatial histogram of similar pairs H Sof t-Inter k corresponding to the visual word w k is then given by:

H Sof t-Inter k (m) = (Pi,Pj )∈S k g (|d i -d j | 2 ) 1 bm (p j -p i ) (2)
where |d i -d j | 2 is the ℓ 2 distance in the descriptor space and 1 bm is the indicator function of bin b m such that:

1 bm : R 2 → {0, 1} v → 1 if v ∈ b m 0 otherwise (3)
Note that, due to symmetric considerations, angle bins are discretized in the [0, π[ interval. Indeed, as the two members of a pair are related to the same visual word w k (i.e. inside the corresponding cluster or not far from its border), only the absolute orientation modulo π is registered. Thus, given a pair of positions (p i , p j ) the corresponding histogram bin is determined by taking either p i or p j as reference point which is equivalent to consider either (p j -p i ) or (p i -p j ) vectors (see figure 3).

To independently evaluate the benefits of using either the soft-weighting (over hard) or the inter-word pairs (over intra), we also introduce two other spatial histograms. The first one, called H Hard-Inter k , is using only binary weights t defined as: To encode spatial information, we use the distance and orientation information between pairs of patches in the image space (left) as well as their distance in the descriptor space (top-right). We consider inter and intra type word pairs based on their proximity in the descriptor space. At the left-bottom, the spatial discretization is illustrated.

t(x) = 1 if x < 3α 0 otherwise (4
Translating the reference patch P i (resp. P j ) at the center, the position of patch P j (resp P i ) gives the bin number (b 9 here) that would be affected in the histogram (bottom-right). The contribution of the patch P j to the histogram is related to the distance D between the two descriptors d i and d j by using the equation [START_REF] Sivic | Video google: A text retrieval approach to object matching in videos[END_REF].

where α is still the standard deviation used in equation 1.

Then, the count H Hard-Inter k (m) of the bin b m is given by:

H Hard-Inter k (m) = (Pi,Pj )∈S k t (|d i -d j | 2 ) 1 bm (p j -p i ). (5)
The second one, called H Hard-Intra k , is also using the same binary weight t but does not consider inter-words pairs (only intra). Formally, if S * k denotes the set of all the pairs of patches for which both patches belong to visual word w k , the hard pairwise histogram H Hard-Intra k is defined as:

H Hard-Intra k (m) = (Pi,Pj )∈S * k t (|d i -d j | 2 ) 1 bm (p j -p i ) (6)
For a visual word w k , equations 2, 5 and 6 can be used to compute the entire histograms H Sof t-Inter k , H Hard-Inter k and H Hard-Intra k respectively. In the next section we introduce a way to combine a given set of spatial histograms H k into one image histogram.

Image representation

As explained in the previous section, for each visual word w k , we obtain one spatial histogram. This histogram encodes spatial information (distance and orientation) of pairwise similar patches, where at least one of the patches belongs to w k . This modu-

Soft Pairwise Similarity angle distance histogram SP S ad representation

To obtain the SP S ad representation from the classical BoVW histogram, we use a 'bin replacement' technique. Bin replacement literally means to replace each bin of the BoVW frequency histogram with the spatial histogram H Sof t-Inter k associated to w k . The sum of all the bins of the spatial histogram obtained from one visual word w k is normalized to the number of occurrences of this word in the whole image. The final whole histogram is subsequently ℓ 1 normalized as in the BoVW method. By this way, we keep the frequency information intact and add the spatial information. The dimensionality of our representation S = K × M depends on the vocabulary size (K) and the number of angle-distance bins of the spatial histogram (M ). On the other hand, if we only use hard weighting t (eq. 4) instead of soft weighting g (eq. 1) as done in H Hard-Inter k (eq. 5), we obtain a hard pairwise similarity histogram, denoted as HP S Inter ad . And finally, if we just consider intra-type visual words pairs and hard weighting as done in H Hard-Intra k (eq. 6), the final representation is denoted HP S Intra ad .

Combination of SP S ad with SPR

Since SP S ad represents the relative spatial interactions between the visual words in an image, it is complementary to descriptors that represent absolute locations of the visual words such as SP R. Thus, we propose to combine SP S ad with SP R. This concatenation (without any weights) is called SP S ad +. For a vocabulary size of K the dimensionality of SP S ad + is K ×(1+4+16+M ) because the number of local histograms are respectively 1, 4 and 16 in the 0 th , 1 st and 2 nd levels.

Dimensionality reduction

One of the drawbacks of the SP S ad + compared to SPR is the high dimensionality of the feature vectors. Typically, for a K = 200 words vocabulary and M = 45 angledistance bins, the dimensionality of the feature vectors is 13200. To get a more compact representation, dimension reduction techniques as feature selection or feature clustering can be applied. We propose to use a divisive information theoretic clustering (DITC) approach introduced by Dhillon et al. [START_REF] Dhillon | A divisive information-theoretic feature clustering algorithm for text classification[END_REF]. The DITC algorithm minimizes the withincluster Jensen-Shannon divergence while simultaneously maximizing the between-cluster Jensen-Shannon divergence in a clustering like algorithm based on the estimation of the joint probability of a visual word in a class. This method was used by Elfiky et al. [START_REF] Elfiky | Discriminative compact pyramids for object and scene recognition[END_REF] to compress the SPR representation. They found that it provides the size reduction of a high dimensional pyramid representation up to an order of magnitude with little or no loss in accuracy. Alternative methods as the AdaBoost feature selection method integrated in Liu et al. approach [START_REF] Liu | Integrated feature selection and higher-order spatial feature extraction for object categorization[END_REF] could also be used. Compared to this latter method which is based on a local optimization of the classification accuracy, DITC algorithm performs a global minimization of the loss of information due to feature clustering.

Starting from SP S ad + with K = 200 and M = 45 (13200 dimensions), we compress our feature vectors by DITC algorithm down to 1800 dimensions. We denote this representation as SP S 1800 ad +.

Comparison with related approaches

Our representation differs on many points compared to original spatial histograms introduced by Savarese et al. [START_REF] Savarese | Discriminative object class models of appearance and shape by correlatons[END_REF] and refined by Liu et al. [START_REF] Liu | Integrated feature selection and higher-order spatial feature extraction for object categorization[END_REF]. We present here the differences in the methodology and in the computational point of view.

On the methodological point of view all the three versions are based on pairwise spatial histograms (called correlogram elements in [START_REF] Savarese | Discriminative object class models of appearance and shape by correlatons[END_REF]). Each spatial histogram is associated to a specific pair of visual words (w k , w l ). In the original versions, all the combinations of visual words are considered leading to K(K + 1)/2 possible histograms (K still denotes the number of visual words). In our version, only pairs of identical (or similar) visual words are considered leading to only K possible histograms. Another difference lies in the discretization of the spatial neighborhood. In [START_REF] Savarese | Discriminative object class models of appearance and shape by correlatons[END_REF], only distance divisions are considered whereas in [START_REF] Liu | Integrated feature selection and higher-order spatial feature extraction for object categorization[END_REF] both distances and angle divisions are considered but in both cases, the neighborhood size is limited to a certain distance in pixels or in function of the patch size. In our case, the neighborhood size is relative to the image size and thus covers the whole image. Moreover, compared to [START_REF] Liu | Integrated feature selection and higher-order spatial feature extraction for object categorization[END_REF], our distance divisions are linear and our angles covers only the [0, π[ due to symmetric considerations. Also note that angles are measured absolutely which does not provide invariance of the representation to image rotations but prevent the decrease of discriminative power in object categorization [START_REF] Zhang | Local features and kernels for classification of texture and object categories: A comprehensive study[END_REF].

Concerning computation, the main point is that we avoid the combinatorial explosion by considering only similar word pairs. First, it limits the size of the final histograms (which becomes linear on K) and second it makes unnecessary to apply any additional clustering technique (as in [START_REF] Savarese | Discriminative object class models of appearance and shape by correlatons[END_REF]) or additional feature selection step (as in [START_REF] Liu | Integrated feature selection and higher-order spatial feature extraction for object categorization[END_REF]). Note that we still need to consider all the patch pairs of an image (complexity O(N 2 )), but in practice, we can fix an upper limit on the number of pairs and choose them randomly to cut down the complexity.

Experimental protocol

In this section, we present the data sets used and the implementation details. Then, we evaluate different aspects of the SP S ad representation for image classification.

Image data sets

For this work, we use MSRC-v2, 15Scene, Caltech101, Caltech256 and Pascal VOC 2007 data sets for experiments.

This subsection provides short descriptions of these image data sets. MSRC-v2: In this data set, there are 591 images that accommodate 23 different categories. All the categories in the images are manually segmented. Different subsets of these categories have been used by several authors to derive a classification problem in which each region or image is assigned to a class label [START_REF] Savarese | Discriminative object class models of appearance and shape by correlatons[END_REF][START_REF] Su | Visual word disambiguation by semantic contexts[END_REF]. 15Scene: This data set [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF][START_REF] Li | A bayesian hierarchical model for learning natural scene categories[END_REF][START_REF] Oliva | Modeling the shape of the scene: A holistic representation of the spatial envelope[END_REF] comprises indoor (i.e. office, kitchen, bedroom, etc.) and outdoor (i.e. beach, mountain, tall building, etc.) scenes. Images were collected from different sources, predominantly from Internet, COREL collection and personal photographs. Each category has 200 to 400 images. Caltech101: The Caltech101 data set [START_REF] Fei-Fei | Learning generative visual models from few training examples: an incremental Bayesian approach tested on 101 object categories[END_REF] has 102 object classes. It has been widely used for evaluation purpose but has some limitations. Namely, most images feature relatively little clutter and possess homogeneous backgrounds. In addition, there are very less variations among the objects of the same category. Despite the limitations, this data set is quite a good resource containing a lot of interclass variability. Caltech256: This is a challenging set of 257 object categories containing a total of 30,607 images [START_REF] Griffin | Caltech-256 object category dataset[END_REF]. The minimum number of images in any category is 80. This dataset is collected the same way as Caltech101 while coping with its limitations. Pascal VOC 2007: The challenging Pascal VOC 2007 database1 is constituted of images downloaded from internet, containing 9,963 images split into train, val and test sets. Each of the images contains at least one occurrence of the 20 object classes. In many images there are objects of several classes present. Altogether the images contain 24,640 objects. The most common object class ("person") is present in 40% of the images, the rarest ("sheep") in 1.9%.

Implementation Details

For MSRC-v2 data set we selected the 15 classes: building, grass, tree, cow, sheep, sky, aeroplane, face, car, bike, flower, sign, bird, book, and chair as in [START_REF] Liu | Integrated feature selection and higher-order spatial feature extraction for object categorization[END_REF][START_REF] Savarese | Discriminative object class models of appearance and shape by correlatons[END_REF]. We used filter-bank responses for feature extraction as in [START_REF] Liu | Integrated feature selection and higher-order spatial feature extraction for object categorization[END_REF][START_REF] Savarese | Discriminative object class models of appearance and shape by correlatons[END_REF]. The training and testing sets are also chosen in accordance with those works for the sake of comparison. For Caltech101 and 15Scene, we follow the experimental setup consistent with [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF]. Thus, we use single scale dense detector (8 pixels period) and SIFT descriptor (16 pixels size) for feature extraction. To be able to compare our results with other spatial representations, we use the standard BoVW representation (hard assignment). Finally, for Caltech256 and Pascal VOC 2007, we use the experimental setup of [START_REF] Chatfield | The devil is in the details: an evaluation of recent feature encoding methods[END_REF] ie. a dense SIFT detector sampled every three pixels at four scales (16, 24, 32 and 40 pixels respectively). For all data sets, we apply K-means on the descriptors to construct the vocabularies. Each descriptor is then mapped to the nearest visual word based on euclidean distance. One versus all multi-class Support Vector Machine (SVM) is used to perform the classification tasks. We use the intersection kernel [START_REF] Swain | Color indexing[END_REF] for the first three data sets and the χ 2 kernel for Pascal VOC 2007 and Caltech256. The cost parameter C was optimized for all the experiments using a 10-fold method on the training set or using the validation set (Pascal VOC 2007). Note that, the new representation does not require any quantization for 2nd order descriptors as opposed to [START_REF] Savarese | Discriminative object class models of appearance and shape by correlatons[END_REF]. So, the output of our algorithm is directly fed into the classification algorithm.

Parameter tuning

In our approach, three parameters (M θ , M R and α) have to be set to compute classification results. We study their influence in this section. In figure 4, on the left, we plot the effect of the number of angle bins (M θ ) and distance bins (M R ) on classification accuracy on 15-scene and Caltech101 data sets for a vocabulary size of 200. The evaluation was done using 10-fold cross-validation on the training set. A 45 bins (9×5) spatial histogram appears to be a good compromise for both datasets. Considering finer quantization does not improve the accuracy significantly, but highly increases feature dimension. On the bottom of figure 4, we show the effect of the weighting parameter α on accuracy. For a very low α, not all similar patches are taken into account and for a higher α, there are patches which may not be similar and could be regarded as noise. For both data sets, the value α=0.3 gives the best results. This value seems related to the descriptor in use (SIFT in this case). We have also experimented that the previous values for the three parameters (M θ , M R and α) are also relevant for the filter bank descriptor used with MSRC-v2 dataset. They thus will be used in the following sections.

Experimental results

In this section, we present our results organized into two main parts. In the first part, we propose to analyze the performance improvements obtained with our concept of soft pairwise spatial angle-distance histograms representation, i.e. the SP S ad representation. We first evaluate the performance increase brought by each of the three following contributions, namely i) infusing spatial information in the BoVW ii) considering inter type visual word pairs and iii) adding a soft weighting when inserting a new pair in the final representation. Then, we compare the SP S ad representation to previous works that also deal with relative visual words positions encoding. In the second part, we focus on our key SP S ad + representation that exploits SP S ad advantages and also models local and global information. We first compare it with SP R to highlight the two methods complementarity and with similar systems that also enrich the SP R representation either by introducing relative spatial information or by using advanced coding and spatial pooling methods based on soft assignment. Second, for a complete overview of the performance of SP S ad +, we compare it with state-of-the-art methods for image classification, that are built upon the bag-of-words pipeline or not.

(%) (%) (%)

The contributions of our SP S ad representation

The purpose of this section is to assess the improvements provided by the proposed concept of soft pairwise spatial angle-distance histograms representation, studying first the impact of the different factors combined in this concept, and second some computational advantages over close methods.

Independent contributions of each factor to the SP S ad model

Three types of information are combined into the SP S ad representation: spatial information, soft weighting and inter-type visual words pairs consideration. The aim, here, is to evaluate the contribution of each of these factors using the three versions introduced in section 3.4.1: pairwise spatial information with HP S Intra ad , inter type visual word pairs with HP S Inter ad and pairwise similarity weighting with SP S ad . For this study, we chose to report some results obtained on two types of datasets, Caltech101 and [START_REF] Yuan | From frequent itemsets to semantically meaningful visual patterns[END_REF] Scenes, that enable a good understanding of the analysed factors on both object and scene classification tasks.

Table 1 shows the results for four different vocabulary sizes. From these results, it is clear that for each data set the HP S Intra ad representation improves the results over BoVW. This means that the spatial information is very useful for image classification. For larger dictionaries, spatial information does not seem to be as effective as in the smaller ones. This was also observed in some of the previous works [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF][START_REF] Zheng | Incorporating spatial correlogram into bag-of-features model for scene categorization[END_REF][START_REF] Yang | Evaluating bag-of-visual-words representations in scene classification[END_REF].

Second, by comparing HP S Intra ad and HP S Inter ad , we understand how it is important to account both intra type visual word pairs and inter type visual word pairs, since the second one outperforms the first one in most of the cases. This is explained by the new spatial information brought by pairs located near the inter visual word boundary as discussed in section 3.3. Finally, since the contribution of each pair should be different depending on the distance between the two considered descriptors, we have introduced a weighting scheme when inserting a pair in the final representation. This soft insertion is the difference between HP S Inter ad and our proposed representation SP S ad . We notice that this final representation outperforms HP S Inter ad overall, specially when the number of visual words is increasing. Indeed, it is interesting to note that with the increase of vocabulary size BoVW, HP S Inter ad and SP S ad representation improve their results whereas HP S Intra ad reaches an optimal and decreases with the increasing vocabulary size. The reason is that for larger dictionaries intra type words pairs become scarce (one cluster is divided into multiple clusters) and thus HP S Intra ad cannot provide important spatial information. On the other hand, SP S ad should always be able to add spatial information into the BoVW representation regardless of the state of the vocabulary since the soft weighting contribution is not varying with the number of visual words.

Comparison with closely related works

Here, we compare our method with Savarese et al. [START_REF] Savarese | Discriminative object class models of appearance and shape by correlatons[END_REF] and Liu et al. [START_REF] Liu | Integrated feature selection and higher-order spatial feature extraction for object categorization[END_REF]. These two works are the most notable among those which concern modeling spatial relationships among the visual words. They rely on the use of new features composed of pairs (or higher number) of words having a specific relative position in order to build spatial histograms. Note that, contrary to our method, the previous approaches do not directly incorporate the spatial information of pair of similar words. We focus on several criteria to compare our work with the mentioned ones. Table 2 shows the details of the comparisons on MSRC-v2 dataset for 400 visual words. For this dataset, SP S ad representation provides the best classification results. Our method also holds different other advantages 14 over the existing methods. For example, Liu et al. [START_REF] Liu | Integrated feature selection and higher-order spatial feature extraction for object categorization[END_REF] integrates feature selection and spatial information extraction to boost recognition rate. However, as the spatial feature extraction becomes a part of the learning step, the modification in the training set would lead to recomputation of features and thus making it difficult to generalize. Let's also note that, SP S ad models only global association and unlike Savarese et al. [START_REF] Savarese | Discriminative object class models of appearance and shape by correlatons[END_REF], does not require a 2 nd -order feature quantization. On the other hand, our method has the highest feature dimensionality (the representation size of Savarese et al. method is not available in their article but it seems to be lower than ours) because, unlike the two other methods, it does not include any feature selection nor additional quantization step. Note that this dimension is still compatible with a fast SVM based classification.

SP S ad+ for image classification

As discussed previously, SP S ad+ is our novel representation, based on SP S ad , that enriches the SP R model by encoding both the global and local relative distribution of visual words over an image. In this section, we first propose to study the gain of SP S ad+ over SP R-based methods. Then, we enlarge our comparison to a large panel of state-ofthe-art methods for image classification, from BoVW-based ones to completely different ones to discuss the interest of our approach.

SP S ad+ vs SP R-based methods

The current trend in BoVW-based method is over the use of SPR combined with other spatial methods or advanced coding methods. The goal of this section is twofold. First, we study the complementarity of the relative spatial information introduced by SP S ad over the absolute one provided by original SPR. For that purpose, we compare SP S ad + over SPR and over concurrent methods that also combine relative spatial information with SPR. Second, we study the performance of our combined transform over recent advanced coding methods. As previously mentioned, the word ambiguity due to hard assignment coding in the BoVW model prevents the use of large visual vocabularies and thus limits the performance of methods. Two main improvements have been proposed to overcome this problem: 1) to use a soft assignment to account for the proximity of several visual words through Kernel CodeBooks (KCB) [START_REF] Van Gemert | Kernel codebooks for scene categorization[END_REF] or Sparse Coding (SC) [START_REF] Yang | Linear spatial pyramid matching using sparse coding for image classification[END_REF] methods, 2)to use a locality constraint to enable stability in the set of visual words used to represent similar descriptors through Locality-constrained Linear enCoding (LLC) [START_REF] Wang | Locality-constrained linear coding for image classification[END_REF] or Localized Soft-assignment Coding (LSC) [START_REF] Liu | In defense of soft-assignment coding[END_REF]. These advanced coding methods also use several spatial pooling strategies to integrate SPR model.

Experiments reported on Table 3 were made using two different pipelines to focus on a fair comparison that enhance the benefits of our approach rather than fine-tuning and datasets Pascal VOC2007 and Caltech256. Let us also note that all results reported for Pascal VOC2007 are taken from [START_REF] Chatfield | The devil is in the details: an evaluation of recent feature encoding methods[END_REF] to be strictly comparable to SP S ad + computed with the same VLFeat library.

Comparison to state-of-the-art

In Table 4, for completeness, we give some top results for the four different datasets used so far. The aim of this table is to show where the state-of-the-art methods are today, compared with our results.

From Table 4, we can see that the most performing approaches are mainly the ones based on convolutional neural networks (CNN) and on Fisher vectors (FV). Indeed, recently, the CNN have shown to provide outstanding results on most of the datasets. However, it is worth mentioning that the CNN require huge amounts of data and time to learn the features. Nowadays, most of the CNN-based approaches are learning their features from the big ImageNet dataset because it provides sufficient data to learn and then try to adapt these pre-learned features to other smallest datasets. Consequently, the CNN provide very good results on datasets that contain classes which are not far (semantically) from the ImageNet classes (chairs, dogs, faces, ...). For specific problems with relative small datasets, such as flower or bird classification, they still not compete with handcraft features. In this case, our approach can help, since it does not require any supervised learning step.

BossaNova and FV enrich BoW representation with extra knowledge from the set of local descriptors but FV use parametric models that lead to very high dimensional image representation. For example, in Table 4, FV require more than 1,376,000-D in order to get that results while the other approaches such as BossaNova, SLC or our solution require less than 300,000-D. Furthermore, there is no guarantee that the local descriptors follow a gaussian distribution around each visual words for all the datasets. If this gaussian assumption does not hold, the representation may be unrepresentative of the local descriptor statistics.

The solutions based on FV such as [START_REF] Seidenari | Local pyramidal descriptors for image recognition[END_REF] or [START_REF] Krapac | Modeling spatial layout with fisher vectors for image categorization[END_REF] are adding information to the classical FV, increasing this way the huge dimension of this descriptor. And sometimes, this dimension increase does not lead to representative improvements. For example, it appears that simple FV have a mean average precision of 59.5% for Pascal VOC20007, while the spatial-FV from [START_REF] Krapac | Modeling spatial layout with fisher vectors for image categorization[END_REF] have 56.7%. Likewise, on 15-scenes the spatial-FV reaches 88.2% while the classical FV are at 88.1%.

BossaNova is a concatenation of a classical BoW and a histogram of distances (in the local feature space) between each cluster center (visual word) and all the descriptors. The first problem of this approach is that the range of distances (related to the number of bins in the distance histograms) has to be adapted to the dataset and the used local features. In practice, they set up differently the range bounds for each visual words. Second, the 2 histograms (BoW and distance histograms) are weighted before concatenation and the weights are learned via cross validation on a training/validation subset. Finally, the results provided in Table 4 are the ones obtained by the concatenation between BossaNova and FV. Without concatenation, BossaNova get 85.3% on the 15 Scene dataset while we get 83.7%, without supervised learning step.

Finally, the spatially local coding [START_REF] Mccann | Spatially local coding for object recognition[END_REF] is inserting the spatial 2D positions (in the image) of the pixels in the local feature vector before creating the dictionary. Consequently, the visual words contain information about their positions in the image space. This kind of representation is helpful only for datasets where the objects have stable positions in the images such as Caltech101 and Caltech256 ones.

Conclusion

In this paper, we proposed a new computationally efficient method to model global spatial distribution of visual words and improved the standard BoVW representation. This method exploits spatial orientations and distances of all pairs of similar descriptors in the image. The evaluation was made on an image classification task, using an extensive set of standard data sets.

Experiments demonstrate that: i) our approach succeeds in adding relative spatial information into the BoVW model, ii) it outperforms all other concurrent local histogram based methods, iii) it provides competitive results compared with recent systems that enrich SPR representation by using advanced coding and spatial pooling methods based on soft assignment. Moreover, it also has significant advantages over very recent state of the art approaches.

A direct extension of this work could be to try to combine advanced BoVW encoding techniques as Fisher kernel [START_REF] Perronnin | Improving the fisher kernel for large-scale image classification[END_REF] to our pairwise spatial histograms. Another interesting future direction could be to include some new encoding techniques directly into our pairwise spatial histograms. For example, local soft assignment [START_REF] Liu | In defense of soft-assignment coding[END_REF] could be combined with soft similarity weighting. Finally, spatial information provided by multiple cues e.g. color and shape, is also promising as a future direction.

Figure 1 :

 1 Figure1: Spatial discretization of the image space. p i and p j are the 2D positions of two patches P i and P j respectively, P i being of type w k and P j of type w l . Concentric circles and angles are drawn centered at position p i . The sector where pixel p j is falling informs about the relative positions between the two patches[START_REF] Liu | Integrated feature selection and higher-order spatial feature extraction for object categorization[END_REF].

Figure 2 :

 2 Figure 2: Discriminative power of spatial distribution of intra type visual words pairs. Four images from Caltech101 dataset are shown. The red squares refer to identical visual words across all the images (for a dictionary of 200 words created from SIFT descriptors). For the two motorbikes in the left, the global distribution of the identical visual words is more similar than the ones in Helicopter or Bugle image.

Figure 3 :

 3 Figure3: Pairwise spatial histogram using soft-similarity weighting. To encode spatial information, we use the distance and orientation information between pairs of patches in the image space (left) as well as their distance in the descriptor space (top-right). We consider inter and intra type word pairs based on their proximity in the descriptor space. At the left-bottom, the spatial discretization is illustrated. Translating the reference patch P i (resp. P j ) at the center, the position of patch P j (resp P i ) gives the bin number (b 9 here) that would be affected in the histogram (bottom-right). The contribution of the patch P j to the histogram is related to the distance D between the two descriptors d i and d j by using the equation (1).

  larity facilitates simple way to assemble the spatial histograms and to obtain the final representation. Starting from the proposed word specific histogram H Sof t-Inter k , we define three different representations: the soft pairwise similarity angle-distance histogram SP S ad derived from the classical BoVW histogram, SP S ad + its combination with the SPR and SP S 1800 ad + a compact version of SP S ad +. On the other hand, word specific histograms H Hard-Inter k and H Hard-Intra k are used to create hard pairwise similarity angle histograms HP S Inter ad and HP S Intra ad respectively.

Figure 4 :

 4 Figure 4: Parameter tuning for SP S ad representation. On the top, the influence of the number of bins for Caltech101 (left) and 15Scene (right) data sets and at the bottom, the influence of α for the same data sets.

Table 1 :

 1 13 Classification accuracy comparison among BoVW, HP S Intra

	Dataset	Voc. Size	BoVW µ	σ	HP S Intra ad µ σ	HP S Inter ad µ σ	SP S ad µ	σ
		100	39.83% 1.32 53.01%	1.1 55.15% 0.77 53.91% 1.23
	Caltech101	200	41.12% 1.06	55.3%	0.9 55.86% 0.82 57.47% 1.00
		400	45.56 % 1.54 52.11 % 1.38 57.13% 1.32 57.62 % 1.38
		1000	48.08 % 1.42 50.82 % 0.92 58.12% 0.79 58.66 % 0.77
		100	70.83%	0.6	76.11% 0.46 78.02% 0.43 77.96% 0.46
	15 Scene Dataset	200	72.2%	0.6	77.52% 0.59 79.41% 0.57 79.38% 0.67
		400	75.7 % 0.33 78.11 % 0.5 79.65% 0.56 79.58 % 0.8
		1000	76.82 % 0.61 76.52 % 0.52 80.00% 0.58 80.38 % 0.44
	ad sentations. Mean (µ) and Standard Deviation (σ) over 10 individual runs are presented. , HP S Inter and SP S ad repre-ad

Table 2 :

 2 Comparison among existing methods on a 15 class problem derived from MSRC-V2 dataset. The '-' means that the dimensionality is not mentioned in the corresponding work.

	Criteria of Comparison	SP S ad Savarese et al. [7] Liu et al. [6]
	Accuracy	83.5%	81.1%	83.1%
	Feature dimensionality	18000	-	1200
	Global Spatial Association	Y	N	N
	2nd Order Feature Quantization	N	Y	N
	Pre-processing/Feature Selection Step	N	Y	Y

Table 4 :

 4 Classification accuracy (%) (or mean average precision (%) for Pascal VOC2007) provided by state-of-the-art methods.

http://pascallin.ecs.soton.ac.uk/challenges/VOC/voc2007/index.html

Methods

Caltech101 15 Scenes Pascal VOC2007 Caltech256 Dimension * 30 train 100 train 30 train SPR (L=2) 64.6 [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF] 81.1 [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF] 53.42 [START_REF] Chatfield | The devil is in the details: an evaluation of recent feature encoding methods[END_REF] 34.1 [START_REF] Griffin | Caltech-256 object category dataset[END_REF] 4200 P IW AH+ [START_REF] Khan | Spatial orientations of visual word pairs to improve bag-of-visual-words model[END_REF] 67.1 82.5 --5000 Zhang et al. [START_REF] Zhang | Improving bag-of-words model with spatial information[END_REF] 65.93 81.5 --13200 Yang et al. [START_REF] Yang | Spatial pyramid co-occurence for image classification[END_REF] -82.5 --5565 KCB [START_REF] Van Gemert | Kernel codebooks for scene categorization[END_REF] 64.14 76.67 54.60 [START_REF] Chatfield | The devil is in the details: an evaluation of recent feature encoding methods[END_REF] 27.17 4200 SC [START_REF] Yang | Linear spatial pyramid matching using sparse coding for image classification[END_REF] 73.20 searching maximal performance for each dataset. The first pipeline, used for Caltech101 and 15 Scenes datasets, is consistent with [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF] with a single scale SIFT sampled every 8 pixels and a small vocabulary of 200 visual words. The second one, used for Pascal VOC2007 and Caltech256, is consistent with [START_REF] Chatfield | The devil is in the details: an evaluation of recent feature encoding methods[END_REF] with a multiscale SIFT sampled every 3 pixels and a large vocabulary of 4000 words. Even the VLFeat open library [START_REF] Vedaldi | VLFeat: An open and portable library of computer vision algorithms[END_REF] proposed by the authors is used to compute our results. The motivation is to use a relatively basic pipeline for the relatively simple datasets Caltech101 and 15 Scene and a more refined one for the more challenging Pascal VOC2007 and Caltech256. However, for advanced coding methods (rows 5-8) the results were not always available with the right parameters in related publications (particularly for Caltech101 and 15 Scenes, they are only available for 1000 visual words). In these cases, we chose publications related to the closest parameters. We can first see that for the four datasets, SP S ad + outperforms the SPR baseline (row 1). It clearly demonstrates the complementarity of the additional relative spatial information provided by our approach. Moreover, compared to other concurrent methods which combine relative and absolute spatial information (rows 2-4), the new method is also the best performing one. It is worth mentioning that the improvement from P IW AH+ to SP S ad + on 15 Scenes is representative since the standard deviations are 0.55 and 0.73 respectively.

Concerning the comparison over advanced coding methods, except for Caltech101, SP S ad + also outperforms other coding methods for all datasets. The worse results for Caltech101 are due to the small vocabulary used (200 words) compared to other coding methods which use a 1000 words vocabulary and thus have a higher dimensionality (21000 vs 13200). For the 15 Scenes dataset, SP S ad + is still the best performing even with the same low vocabulary dimensionality. The trade-off between accuracy and dimensionality is even better when using our reduced-dimension feature vector SP S 1800 ad + (introduced in section 3.4.3) whose dimension is 1800 and which is providing 67.5% and 83.0% for Caltech101 and 15 Scenes respectively. These results are still very competitive with the other methods from Table 3, despite the low feature dimensionality.

On the other hand, if high accuracy is desired, the use of a refined pipeline (4000 words and denser descriptor) outperforms all improved coding methods on the two challenging


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Intrapixel measurement techniques on large focal plane arrays for astronomical applications : a comparative study

Intrapixel measurement techniques on large focal plane

arrays for astronomical applications : a comparative study

INTRODUCTION

The intrapixel response is the signal detected by a single pixel illuminated by a Dirac distribution as a function of the position of this Dirac inside this pixel. It is also known as the pixel response function (PRF). This function measures the sensitivity variation at the subpixel scale and gives a spatial map of the sensitivity across a pixel. The variation of the pixel sensitivity finds its origin in the physical properties of the pixel. The measurement of this function gives information on effects which occur after the absorption of the incoming photons by the material. Therefore the knowledge of this function is of great interest, on the one hand for the technologist, who could check the quality of the pixels and study the phenomena related to the detection; and on the other hand, for the instrument scientist, who could evaluate the impact of the spatial filtering of the pixel on image quality. The knowledge of the intra-pixel sensitivity is also important to control and reduce the systematic errors in under-sampled instruments. Indeed, in these instruments, the variation of pixel sensitivity limits the performances of the system and contributes to the instrument error budget [START_REF] Ingalls | Intra-pixel gain variations and high-precision photometry with the Infrared Array Camera (IRAC)[END_REF][START_REF] Anderson | Toward high-precision astrometry with WFPC2. I. Deriving an accurate PSF[END_REF]. All these reasons brings ESA to require the measurement of the intra-pixel sensitivity with a resolution close to the tenth of the pixel in detector technology development activities. The techniques used to measure the intra-pixel response are based on the projection of optical probes on the detector. These techniques can be divided in two main approaches: the direct approach and the indirect approach.

In the direct approach, we can enumerate the spot scan techniques, which consist in projecting a single spot and scanning it across the detector under test (DUT). These techniques require high-quality optics coupled with high-precision scanning systems. They are effective to understand the subpixel properties but they are time-consuming. It is the reason why the published results in literature are commonly limited to only a few pixels [START_REF] Barron | Sub-pixel response measurement of near-infrared sensors[END_REF]. To overcome this limitation, Biesiadzinski et al [START_REF] Biesiadzinski | A method for the characterization of subpixel response of near-infrared detectors[END_REF] designed a method based on a multi-spot projector, the spots-O-Matic. The spots-O-Matic enables a two-dimensional scan of every pixel in a detector by projecting 160000 spots (on a 400x400 grid) onto the device. In the indirect approach, the principle consists in estimating the Modulation Transfer Function (MTF) in the spatial domain and computing the detector Point Spread Function, PSF (or its Line Spread Function, LSF) by inverse Fourier transform. The MTF is estimated by projecting onto the detector a pattern with a known spatial frequency content. Guérineau et al [START_REF] Guérineau | New techniques of characterisation[END_REF] have developed an original technique based on this Fourier-transform approach. It is based on the projection of high-spatial resolution periodic patterns onto the whole sensor without classic optics such as lens or other focusing components but using the self-imaging property (known as the Talbot effect) of a particular class of diffracting gratings illuminated by a plane wave. The main advantage of this approach is that the projection of patterns is optics-less. However this technique allows one to obtain a MTF (then PSF or LSF) of an average pixel corresponding to an area of the detector. Ketchazo et al [START_REF] Ketchazo | A new technique of characterization of the intrapixel response of astronomical detectors[END_REF] presented an adaptation of this technique for the characterization of the intrapixel response of each pixel of a detector. The idea is to couple the projection of the interferograms made of sub-pixel details with a precise micro-scanning process. Viale et al [START_REF] Viale | High accuracy measurements of the intrapixel sensitivity of VIS to LWIR astronomical detectors: experimental demonstration[END_REF] presented the experimental demonstration and the preliminary results. In the following paragraph, we will discuss the physical origins of the sensitivity variations. This section will be followed by the comparison of the main measurement techniques allowing one to evaluate the intrapixel sensitivity. We will conclude with a discussion on the potentialities of these techniques to fulfill the requirements of astronomical applications.

II. THE ORIGIN OF THE VARIATION OF THE INTRAPIXEL SENSIVITY

The variation of the pixel sensitivity finds its origin in the physical processes that occur with charges created by the interaction of the photons with the material. A photon with sufficient energy generates at least an electron-hole (e-h) pair in the semiconductor material. This e-h pair is free to move and can diffuse in the structure. In the ideal case, the signal carriers are collected in the pixel where they are created without spreading into neighbouring pixels. However, lateral charge diffusion into other pixels can occur. In this scenario, the charge carriers are collected by the adjacent pixels. This effect is known as the diffusion pixel

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Biarritz, France International Conference on Space Optics 18 -21 October 2016 crosstalk and widens the pixel response function. The consequence is the degradation of the image quality.

The last process which can occur after the generation of signal carriers is the recombination of the e-h pairs without being collected. This leads to a loss of information. We should note that the pixel sensitivity changes with the wavelength. Finally, we should mention that other effects can degrade the pixel sensitivity. We enumerate among others: the pixels defects, the reflection losses of the incoming light and gates absorption (for front-illuminated CCD for example). Therefore, to improve the collection charge efficiency and the sensitivity in the case of CCDs for example, the backside devices are preferred, the substrate is thinned enough to reduce the epitaxial region and to minimize the field free regions. In this case, charges are created close to the depletion region and are easily collected by the target pixel. However, to increase the detector quantum efficiency (QE) and improve the absorption of shorter and longer wavelengths, the photosensitive thickness is often made larger. For example, the thick substrates detectors are developed to achieve high QE in red and near-infrared wavelength. To overcome the problem of charge diffusion, the CCDs are deep or fully depleted and an appropriated bias is applied to the substrate to minimize the diffusion [START_REF] Janesick | Scientific charge-coupled devices[END_REF]. Hence there is a trade-off between various aspects of the detector that have impact on its efficiency. This discussion demonstrates that the pixel response function is far to be a top-hat function with a constant value within the boundaries of a pixel (i.e. uniform pixel sensitivity) and zero outside (i.e, zero crosstalk). The discussion also raises the question of the homogeneity of the intra-pixel responses among the pixels of the detector. One can expect that there is no difference among the pixels (i.e pixel A has the same intrapixel response as pixel B) because the detectors are fabricated by photolithography processes using different masks on the same wafer. However these processes can suffer from potential alignment defects of masks, and the masks can also present geometric defects. These can introduce dispersion in pixel sizes and shapes. Therefore, it is important to measure the pixel response function and check its variation at the scale of a detector. The techniques developed to measure the pixel response function are derived from the commonly used techniques to evaluate the spatial properties of electro-optics systems (cameras, imagers, …). They are based into the projection of the optical probes or input images on the DUT. The detectors have a periodic and pixelated structure then any input image is sampled at a fixed spatial frequency. The process of pixel spatial sampling can create aliasing and "fold" high frequencies of the modulation transfer function. To overcome these limitations, super-resolution techniques are often implemented and the results are corrected from the pixel sampling effect.

In the next section, we will review the classical techniques used to characterize the spatial properties of the detectors and we will highlight those which are well adapted for the study of the intra-pixel response.

III. OVERVIEW OF MAIN MEASUREMENT TECHNIQUES

In the following paragraph, we first define the criteria on which the comparative study will be carried. This definition is followed by a review on the measurement techniques.

A. Measurement figures of Merit

The Pixel Response Function (PRF) is the parameter to measure. In practice, the PRF is sampled and determined on a subpixel grid of pitch p ech /M where p ech is the sampling pitch of the focal plane array and M is an oversampling factor. The Fourier transform of the PRF is the pixel Optical Transfer Function (OTF) of which the modulus gives the pixel MTF. The pixel MTF describes the ability of the pixel to reproduce the contrast modulation present in the scene (here the pixel field of view) at any frequency. The knowledge of the PRF at p ech /M corresponds to exploring the pixel MTF up to M times the Nyquist frequency (f Ny ) where f Ny =1/2p ech .

In the following paragraphs, the techniques comparison we make is, as far as possible, in respect to the PRF oversampling factor and the accuracy of PRF estimation.

B. Overview of the measurement techniques

As we have already mentioned in section I, the measurement techniques can be divided in two approaches. Hereinafter, the first three techniques we present belong to the direct approach. We conclude this section with a discussion on the indirect techniques.

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Biarritz, France International Conference on Space Optics 18 -21 October 2016

Part 1: the direct techniques 1. The spot scan: it consists in projecting a point source (an infinitesimal pulse i.e., a Dirac distribution in an ideal case) onto the detector. The projection is often achieved by a microscope objective lens [START_REF] Kavaldjiev | Subpixel sensitivity map for a charge-coupled device sensor[END_REF][START_REF] Swindells | MTF and PSF measurements of the CCD273-84 detector for the Euclid visible channel[END_REF][START_REF] Niemi | Measuring a charge-coupled device point spread function[END_REF].

The microscope allows the de-magnification and focusing of the pinhole illumination onto the DUT. The size of the spot depends on the wavelength and the aperture of the optics (~l/NA). The scanning of the spot across the DUT is usually achieved by the displacement of the detector with two motorized stages.

The spot scan provides the entire-two dimensional PRF with only one measurement. Niemi et al [START_REF] Niemi | Measuring a charge-coupled device point spread function[END_REF] measured, with uncertainties of 3s, the PRF of one version of the CCD-273 using the spot projection. The CCD-273 is the device that will populate the visible channel of the Euclid payload [START_REF] Endicott | Charge-Coupled Devices for the ESA Euclid M-class mission[END_REF]. Niemi et al estimated the CCD PRF by fitting the data to a model using a Bayesian approach. They showed that the PRF size changes with the wavelength and narrows towards longer wavelengths. They also showed that the PSF size grows as a function of intensity. One of the major drawback of this technique is that pointsource objects often provide too little flux, which, coupled with the multiple noise sources (source fluctuations, sensor noise…), could drive to a poor signal-to-noise ratio (SNR). 3. The knife-edge technique: the technique uses an image of a sharp, high-contrast edge. This image is the convolution of the PSF with the step function and it is called the Edge Spread Function (ESF). The LRF is deduced by calculating the derivative of the ESF. And the one-dimensional MTF can be deduced as the 1-D Fourier transform of the LRF. The 2D MTF estimation requires several one-dimensional slices at different angles. To overcome the aliasing effect, the knife-edge is tilted slightly and the construction of the oversampled ESF is realised at the subpixel spacing [START_REF] Reichenbach | Characterizing digital image acquisition devices[END_REF]. Experimentation showed that a subpixel bin of one-quarter pixel had the best results [START_REF] Nugent | Measuring the modulation transfer function of an imaging spectrometer with rooflines of opportunity[END_REF]. Recently virtual knife edge has been proposed by Karcher et al [START_REF] Karcher | Measurement of lateral charge diffusion in thick, fully depleted, back-illuminated CCDs[END_REF]. Instead of using a real Foucault knife-edge, they selected a grid of pixels and the idea consists in measuring the variation of the total charge in the grid as the beam is scanned across a grid edge [START_REF] Karcher | Measurement of lateral charge diffusion in thick, fully depleted, back-illuminated CCDs[END_REF].

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Biarritz, France International Conference on Space Optics 18 -21 October 2016

Fig. 2. The knife edge

The tilted edge was used to construct a superresolution 1-D ESF, which was resampled at quarter-pixel increments to provide data with four times the resolution of the detector. Image courtesy [START_REF] Nugent | Measuring the modulation transfer function of an imaging spectrometer with rooflines of opportunity[END_REF] We notice that the estimation of the PRF through the ESF must be done carefully to avoid bias in the result as the intensity profile (erf function) or its derivative (Gaussian function) are very sensitive to noise.

Intrapixel analysis compatibility

Yes

No No Among all these techniques, the spot scan is the most effective for the investigation of the intrapixel sensitivity analysis. Indeed, despite the fact that all these techniques allow the exploration of sub-pixel properties, only the spot scan technique is local enough to permit this analysis for only each pixel considered individually. However, considering the spot scan, the time required to create a detector-wide map is prohibitive. To overcome this limitation, Biesiadzinski et al [START_REF] Biesiadzinski | A method for the characterization of subpixel response of near-infrared detectors[END_REF] designed a method based on a multi-spot projector, the spots-O-Matic.

a. Principle of the Spots-O-Matic

The principle of this system is derived from the spot-o-matic firstly designed by the same authors to project and scan a single spot [START_REF] Barron | Sub-pixel response measurement of near-infrared sensors[END_REF]. Instead of the spot-o-matic, the spots-o-matic projects around 100k spots by the mean of a 7 in. x 7 in. photolithography mask containing an array of circular apertures of 17 µm diameter each (a pinhole array) back-illuminated by narrow-band laser diodes. surface is curved. Hence, the authors restrict their analysis to a local region in the image centre. For example, considering the detector H2RG-236 (18 µm pitch), the authors restrict their measurement to a 1024x1024 pixel region in the image centre, a quarter of the total number of pixels. Nevertheless, the authors announced it woud take only a few weeks to characterize all the pixels of the entire device by a proper focus procedure of each zone of the device.

b. The result

The PRF is characterized by scanning the spots vertically and horizontally with a step of 2 µm. The spot diameter is 7 µm with a laser diode centered at 1050 nm. During the scan, each pixel is sampled by different spots. The algorithm based on a matched filter approach permits to estimate the pixel response function. In the indirect approach, the principle consists in estimating the Modulation Transfer Function (MTF) in the Fourier domain and computing the Point Spread Function, PSF by inverse Fourier transform. The MTF is estimated by projecting onto the detector a pattern with a known spatial frequency content. The pattern is usually the fringes generated by a double aperture interferometer [START_REF] Andersen | An interferometric method for measurement of the detector MTF[END_REF] or by a modified Michelson interferometer [START_REF] Willemin | Optical characterization methods for solid-state image sensors[END_REF][START_REF] Tackacs | PSF and MTF measurement methods for thick CCD sensor characterization[END_REF]. These techniques are global as the interferogram can be made large enough to characterize the overall detector sensitive surface. The main drawback is that they often require complex optics to produce these fringes. Guérineau et al. [START_REF] Guérineau | New techniques of characterisation[END_REF] developed an original technique based on the projection of high-spatial resolution patterns onto the whole sensor without classic optics but using the self-imaging property (known as the Talbot effect) of a diffracting element called Continuously Self-Imaging Grating (CSIG). The CSIG is illuminated by a plane wave, the projection of the pattern onto the detector is optic-less, the test bench alignment budget is low. Figure 5 below gives the principle of the test bench.

Fig.5. Schematic representation of the principle of the test bench

The CSIG is a diffractive element, a complete description of its properties can be found in [START_REF] Guérineau | Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings[END_REF]. Depending on the substrate material, different CSIGs can be manufactured to cover the visible, the near-infrared and the long-wave infrared bands. CSIGs allow exploring the spatial response of the detector up to frequencies several times higher than the Nyquist frequency. The highest frequency we can reach depends on the CSIG properties, the pixel pitch and the wavelength. Viale et al [START_REF] Viale | High accuracy measurements of the intrapixel sensitivity of VIS to LWIR astronomical detectors: experimental demonstration[END_REF] presented the results of an estimation of the pixel response function of a Dalsa camera (1024x1024 CCD, 12 µm pitch) using a highly resolved CSIG (cutoff frequency higher than 500 mm Nyquist frequency (i.e the oversampling factor M is equal to 12) and therefore to estimate the PRF with a resolution of pixel/12 (namely 1 µm).

We can consider two measurements procedure: global and local:

1. The global procedure: hypothesis of identical pixels

The procedure is based on the assumptions that all the pixels under study are identical. The mean PRF corresponding to at least an area of one CSIG period can be estimated. To overcome the aliasing effect, the superresolution procedure consists to displace the CSIG in front of the DUT at the step of pixel/M and acquire MxM images. The combination of the MxM images permits the estimation of the over-sampled interferogram.

Viale et al [START_REF] Viale | High accuracy measurements of the intrapixel sensitivity of VIS to LWIR astronomical detectors: experimental demonstration[END_REF] presents in detail the procedure. The deconvolution is based on a Bayesian approach [START_REF] Mugnier | From Data to Object : the Inverse Problem[END_REF] which takes into account the noise in the image and some prior information on the object to be restored, the latter being estimated in an unsupervised way from the interferogram itself [START_REF] Gratadour | Near-infrared adaptive optics dissection of the core of NGC 1068 with NAOS-CONICA[END_REF]. The SNR of the reconstructed interferogram must be high enough to improve the accuracy of the PRF estimation. Indeed, studies showed that with a SNR of 500, the maximum error in the estimation is below 2%. The demonstrator test bench has been developed to validate the measurement technique. The experimental results obtained are given in Figure 6 hereafter: at left, a period (a 0 =380µm) of the oversampled interferogram considered for the estimation of the pixel response function, and at right, the reconstructed pixel response function with an error bar estimated at 8.5%.

Fig. 6. Oversampled inteferogram reconstructed after microscan (left). 2D view of the mean PRF estimated (right) [START_REF] Viale | High accuracy measurements of the intrapixel sensitivity of VIS to LWIR astronomical detectors: experimental demonstration[END_REF] Applying this methodology on the CCD-273 (4096 x 4132 pix, 12 µm pitch) could allow the estimation of 16800 PRFs corresponding to an area of 380 x 380 µm (32x32pix) each. This quite good statistic is get in only 2.4h.

The local procedure: hypothesis of different pixels

The objective is to evaluate the intrapixel response of each pixel considered individually. To get rid of aliasing effect, the superresolution procedure consists to acquire M(a 0 /pixel) x M(a 0 /pixel) images taken as different subpixel positions of the CSIG (e.g. considering a scenario with the H2RG-236, it comes to acquire 253x253 images). The M(a 0 /pixel) x M(a 0 /pixel) images are thereafter combined to get an interferogram over a period a 0 for each pixel of the FPA. With the H2RG, the intrapixel detector-wide map could be obtained in lesser than 1 week. The calculation is made under the hypothesis of 100kpix/s output frequency, 8 outputs and the integration time of 2s.

The implementation of this ultimate measurement requires among others a precise and reliable motion of the CSIG, a stable test bench (i.e vibration-less) and good quality of the optics. Currently ONERA/DOTA and CEA/Irfu work on the development of a test bench named Intrapix to satisfy these requirements [START_REF] Viale | High accuracy measurements of the intrapixel sensitivity of VIS to LWIR astronomical detectors: experimental demonstration[END_REF]. The displacement of the CSIG will be ensured by means of 2 piezo nano-positioners (ANP Series, Attocube) used in closed loop control based on fiber-optic Fabry-Perot interferometers (FPS, Attocube). The optics shall be made in RSA 6061 (RSA stands for Rapid Solidified Aluminium) and aligned to satisfy a wavefront error at the exit of the collimator better than l/10 RMS.

IV. COMPARISON BETWEEN SPOTS-O-MATIC AND INTRAPIX BENCHES

We have presented two experimental test-benches for the measurement of the intrapixel sensitivity. These benches use highly resolved probes and require precise scanning systems. Because of field limitations due to the use of optics in the Spots-o-Matic, the Intrapix test bench appears to be more appropriated to characterize the intrapixel sensitivity of large detectors than the Spots-o-Matic.

V. CONCLUSION

We have reviewed different techniques commonly applied to characterize the spatial properties of the detectors in laboratory conditions. The aim was to evaluate their compatibilities with intrapixel measurement and compare their performances. We have shown that even if a technique permits the exploration of the sub-pixel properties it cannot be compatible with the measurement of the intrapixel response. The Line Response Function and Edge Scan Function techniques are not local enough to allow the investigation of sensitivity of an individual pixel, but the line or the edge are sampled over respectively 4 or 8 pixels. In the scenario where the need is to measure a mean PRF of a detector, the tilted edge technique appears however to be more appropriated (high contrast, high SNR, easy to implement). The spot scan however is well effective in measuring the intrapixel response. The measurement resolution depends mainly on the spot size and the performances of the displacement plates. Therefore the smaller the spots (and its displacement pitch) are, the better the PRF is resolved. However a trade-off has to be made between the size of the spot and the SNR.

In the case of astronomical detectors constituted of millions of pixels, a pure (simple) local technique as the spot scan is not appropriate. Indeed, it is important to check the uniformity of the intrapixel response over the whole detector. We have shown in these cases that global techniques coupled with a micro-scanning process are required. The technique must solicit the entire FPA and also provide a local assessment of every pixel. We have presented and compared two test benches: the spots-o-matic and the Intrapix. The comparison leads to conclude that the Intrapix bench is the one that could really permit the study of the intrapixel variations at the scale of a large detector.

2 .

 2 The line response function scan/tilt: instead of a spot, a line (slit image) can be projected and scanned on the device so to produce a line response function. The line response function (LRF) is the twodimensional convolution of the line source object with the detector PRF. The detector MTF along one direction (perpendicular to the line source) is then deduced from the LRF by a Fourier transform. In contrast with a spot source, a better SNR is obtained using a line source but the information is truncated and needs several slit angles to get a 2D MTF. Instead of displacing successively the line by a small fraction of a pixel pitch, an alternative static technique known as the Vernier technique has been devised. With this technique, the line is tilted slightly allowing to get rid of aliasing effects. Successive pixel rows sample the response function at the pixel pitch but each set of samples is displaced by a fraction of a pixel per row. The technique is used at e2v and the LRF is sampled with a resolution of 1/8 pixel[START_REF]Technical note on the MTF of CCD sensors, A1A-CCDTN[END_REF].

Fig. 1 .

 1 Fig. 1. Vernier sampling technique to measure the Line Response Function [13]. The well focused line image is rotated onto approximately the desired angle when it is displaced by one pixel pitch over an interval of say 8 rows. Image credit: e2v. e2v estimate the MTF of the CCD-273 devices as part of the ESA pre-development phase for the VIS CCD detectors (Euclid) [10]. The slit is focused using a 20x objective lens, the measured LRF points are interpolated to get a regularly sampled The results of MTF obtained in horizontal and vertical orientations of the slit are presented. The MTFs are corrected from the optical MTF and the results at the pixel Nyquist Frequency are given for different wavelength.

Fig. 3 .

 3 Fig.3. Schematic diagram of the Spots-O-MaticThe pinhole array is de-magnified and imaged onto the detector by a camera lens. The pinhole array is mounted at room temperature. The detector is mounted in a cryogenic dewar and operates at 140 K. The lens is affixed in front of the cryostat window on the room temperature side. The setup suffers from spherical aberration and distortion. Therefore the spot array image may not be co-planar with the detector and the focal

Fig. 4 .

 4 Fig.4. 2D view of pixel response function estimated with the H2RG-236 [17] Part 2: indirect techniques

  -1 ) illuminated by white light. The CSIG allows one to explore the detector MTF up to 12 times the detector

Table 1 :

 1 Summary and comparison of the techniques Techniques

Spot Scan Line Response Function (the Vernier technique) Tilted knife edge

	Test bench	Complex	Complex	Simple
	Scanning system	Yes	No	No
	Sub-pixel sampling		Pixel/8 (e2v)	Pixel/4

Table 2 :

 2 Color table.

	White	1	Gren	0.5
	Red	0.9	Cyan	0.3
	Orange 0.8	Blue	0.2
	Yellow 0.7	Violet 0

ICSO 2016 Biarritz, France International Conference on Space Optics 18 -21 October 2016 focus

  of spots array while the Intrapix approach is a projection of patterns using the self-imaging properties of the gratings. The table below summarizes and compares the main characteristics of these test-benches.

The Spots-O-Matic is based on the

Table 3 :

 3 Summary and comparison of the Spots-o-matic and the Intrapix test-benches

	Bench	Approach	Principle	Setup	Detector	Multi-	PRF	PRF	Time
				alignment	Large format	bands	Measurement	estimation	consuming
				budget	compatible	compatible	resolution	accuracy
	Spots-o-	Direct	Focusing	High	No	Yes	7 µm spot size	Not available	Yes
	Matic		an array of		(Optics	(Optics	for 18 um pixel
			spots		aberrations	must be	pitch
					reduce the	changed)
					field of
					measurement)
	Intrapix	Indirect	Projection	Low	Yes	Yes	1 µm	2%	No
			of 2D			(CSIG must		(simulation)
			patterns			be changed)		8.5%
								experimentally
								achieved in the
								demonstrator