Detecting Anomalous Structures by Convolutional Sparse Models

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1 Detecting Anoalous Structures by Convolutional Sparse Models Diego Carrera, Giacoo Boracchi Dipartiento di Elettronica, Inforazione e Bioingegeria Politecnico di Milano, Italy {giacoo.boracchi, diego.carrera}@polii.it Alessandro Foi Departent of Signal Processing Tapere University of Technology Tapere, Finland alessandro.foi@tut.fi Brendt Wohlberg Theoretical Division Los Alaos National Laboratory Los Alaos, NM, USA brendt@lanl.gov Abstract We address the proble of detecting anoalies in iages, specifically that of detecting regions characterized by structures that do not confor those of noral iages. In the proposed approach we exploit convolutional sparse odels to learn a dictionary of filters fro a training set of noral iages. These filters capture the structure of noral iages and are leveraged to quantitatively assess whether regions of a test iage are noral or anoalous. Each test iage is at first encoded with respect to the learned dictionary, yielding sparse coefficient aps, and then analyzed by coputing indicator vectors that assess the conforance of local iage regions with the learned filters. Anoalies are then detected by identifying outliers in these indicators. Our experients deonstrate that a convolutional sparse odel provides better anoaly-detection perforance than an equivalent ethod based on standard patch-based sparsity. Most iportantly, our results highlight that onitoring the local group sparsity, naely the spread of nonzero coefficients across different aps, is essential for detecting anoalous regions. Keywords Anoaly Detection, Convolutional Sparse Models, Deconvolutional Networks. I. INTRODUCTION We address the proble of detecting anoalous regions in iages, i.e. regions having a structure that does not confor to a reference set of noral iages []. Often, anoalous structures indicate a change or an evolution of the datagenerating process that has to be proptly detected to react accordingly. Consider, for instance, an industrial scenario where the production of fibers is onitored by a scanning electron icroscope (SEM). In noral conditions, naely when the achinery operates properly, iages should depict filaents and structures siilar to those in Figure (a). Anoalous structures, such as those highlighted in Figure (b), ight indicate a alfunction or defects in the raw aterials used, and have to be autoatically detected to activate suitable countereasures. Detecting anoalies in iages is a challenging proble. First of all because, often, no training data for the anoalous regions are provided and it is not feasible to forecast all the possible anoalies that ight appear. Second, anoalies in iages ight cover arbitrarily shaped regions, which can be very sall. Third, anoalies ight affect only the local structures, while leaving acroscopic features such as the average pixel-intensity in the region untouched. Our approach is based on convolutional sparse odels, which in [] were shown to effectively learn id-level features of iages. In convolutional sparse representations, the input iage is approxiated as the su of M convolutions between a sall filter d and a sparse coefficient ap x, i.e. a spatial ap having few non-zero coefficients. A convolutional sparse odel is a synthesis representation [3], where the iage is encoded with respect to a dictionary of filters, yielding sparse coefficient aps. The decoding consists in adding all the outputs of the convolution between the filters and corresponding coefficient aps. Structures fro noral iages are odeled by learning a dictionary of M filters {d } fro a training set of noral iages. Learned filters represent the local structure of training iages, as shown in Figure (a). Each test iage is encoded with respect to the learned filters, coputing the coefficient aps that indicate which filters are activated (i.e. have nonzero coefficients) in each local region of the iage. In noral regions we expect the convolutional sparse odel to describe the iage well, yielding sparse coefficient aps and a good approxiation. This is illustrated in Figure (b), where the green patch in the left half belongs to a noral region and has only few filters activated: the corresponding coefficient aps are sparse. In contrast, in regions characterized by anoalous structures, we expect coefficient aps to be less sparse or to less accurately approxiate the iage. The red patch inside the right (anoalous) half of Figure (b) shows coefficient aps that are not sparse. We detect anoalies by analyzing a test iage and the corresponding coefficient aps in a patch-wise anner. Patches, i.e. sall regions having a predefined shape, are thus the core objects of our analysis. For each patch we copute a low-diensional indicator vector that quantitatively assesses the goodness-of-fit of the convolutional odel, naely the extent to which the patch is consistent with noral ones. Indicators are then used to deterine whether a given patch is noral or anoalous. Given the above considerations, the ost straightforward indicator for a patch would be a vector stacking the reconstruction error and the sparsity of the coefficient aps over each patch. However, in our experients we show that the sparsity of the coefficient aps is too loose a criterion for discriinating anoalous regions, and that it is convenient to consider also the spread of nonzero coefficients across different aps. In particular we observed that, while noral regions can be

2 (a) (b) Fig. : Exaples of SEM iages depicting a nanofibrous aterial produced by an electrospinning process: Fig. (a) does not contain anoalies, and is characterized by specific structures also at local-level. Fig. (b) highlights anoalies that are clearly visible aong the thin fibers. typically approxiated by few filters, the representation of anoalous ones often involves any filters. Therefore, we include in the indicator vector a ter easuring the a local group-sparsity of the coefficient aps, and show that this is inforation is effective at discriinating anoalous regions. Suarizing, the contribution of the paper is two-fold. First, we develop a novel approach for detecting anoalies by eans of convolutional sparse odels describing the structures of noral iages. Second, we show that the local group sparsity of the coefficient aps is a relevant prior for discriinating anoalous regions. The reainder of the paper is organized as follows: Section II provides an overview of related works, while Section III forulates the anoaly-detection proble. The proposed approach is outlined in Section IV while details about convolutional sparse representations and the indicator vectors are provided in Sections IV-A and IV-B, respectively. Experients are presented and discussed in Section V. II. RELATED WORKS Anoaly detection [4], refers to the general proble of detecting unexpected patterns both in supervised scenarios (where training saples are labeled as noral, or either noral and anoalous) and in unsupervised scenarios (where training saples are provided without labels). Anoaly detection is also referred to as novelty detection [], [5], [6], in particular when anoalies are intended as patterns that do not confor to a training set of noral data. In the achine learning literature, novelty detection is forulated as a one-class classification proble [7]. In this paper, we shall refer to the patterns being detected as anoalies, despite the novelty detection context. An overview of novelty-detection ethods for iages is reported in []. Convolutional sparse odels were originally introduced in [] to build architectures of ultiple encoder layers, the so-called deconvolutional networks. These networks have been shown to outperfor architectures relying on standard patchbased sparsity when learning id-level features for object recognition. More sophisticated architectures involving both decoder and encoder layers showed to be effective in visual recognition tasks, such as supervised pedestrian detection [8] and unsupervised learning of object parts [9]. Deconvolutional networks are strictly related to the well-known convolutional networks [0], which in contrast are analysis representations, where the iage is subject to ultiple layers where it is directly convolved against filters. Convolutional sparse odels have not previously been used for the anoaly-detection proble, such as that considered in this work. For siplicity, we presently develop and describe a single-layer architecture, although ore layers ay also be used. Convolutional sparse odels can be seen as extensions of patch-based sparse odels [], the forer providing a representation of the entire iage while the latter independently represent each iage patch. Patch-based sparse odels have been recently used for anoaly detection purposes [], where an unconstrained optiization proble is solved to obtain the sparse representation of each patch in a test iage. Then, the reconstruction error and the sparsity of the coputed representation are jointly onitored to detect the anoalous structures. In [3] anoalies are detected by eans of a specific sparse-coding procedure, which isolates anoalies as data that do not adit a sparse representation with respect to a learned dictionary. Sparse representations have been used for detecting unusual events in video sequences [4], [5] by onitoring the functional iniized during the sparse coding stage. The detection of structural changes a proble closely related to anoaly detection was addressed in [6], where sparse representations were used to sequentially onitor a strea of signals. Convolutional sparse odels offer two ain advantages copared to patch-based ones: first of all, they directly support the use of ultiscale dictionaries [8], whereas this is not straightforward for standard patch-based sparse representations. Second, the size of the patch that has to be analyzed in the anoaly detection can be arbitrarily increased at a negligible coputational overhead when convolutional sparse odels are exploited. In contrast, this requires additional training and also increases the coputational burden [] in the case of patch-based sparsity.

3 Learned Filters Training Iage Test Iage Feature aps anoalous patch Feature aps noral patch a) b) Fig. : (a) Learned dictionary (8 filters are of size 8 8 and 8 are of size 6 6) reports the proinent local structures of the training iage. (b) A test iage used in our experients: the left half represents the noral region (filters were learned fro the other half of the sae texture iage), while the right half represents the anoalous region. The ideal anoaly detector should ark all pixels within the right half as anoalous, and all pixels within the left half as noral. The coefficient aps corresponding to the two highlighted regions (red and green squares) have approxiately the sae l nors. However, the right-ost figures show that there is a substantially different spread of nonzero eleents across coefficient aps. Thus, in this exaple the local group sparsity of the coefficient aps is ore inforative than sparsity for discriinating anoalous regions. Feature aps have been rescaled for better visualization. III. PROBLEM FORMULATION Let us denote by s : X R + a grayscale iage, where X Z is the regular pixel grid representing the iage doain having size N N. Our goal is to detect those regions in a test iage s where the structures do not confor to those of a reference training set of noral iages T. To this purpose, we analyze the iage locally, and forulate the detection proble in ters of iage patches. In particular, we denote s c := Π c s = {s(c + u), u U}, c X () the patch centered at a specific pixel c X, where U is a neighborhood of the origin defining the patch shape, and Π c denotes the linear operator extracting the patch centered at pixel c. We consider U a square neighborhood of P P pixels (indicating by P the cardinality of U), even though patches s c can be defined over arbitrary shapes. We assue that patches in anoaly-free iages are drawn fro a stationary, stochastic process P N and we refer to these as noral patches. In contrast, anoalous patches are generated by a different process P A, which yields unusual structures that do not confor to those generated by P N. The training set T is used to learn a suitable odel specifically for approxiating noral iages. Instead, no training saples of anoalous iages are provided, thus it is not possible to learn a odel approxiating anoalous regions. In this sense, P A reains copletely unknown. Anoalous structures are detected at the patch level: each patch s c is tested to deterine whether it does or does not confor to the odel learned to approxiate iages generated by P N. This will result in a ap locating anoalous regions in a test iage. IV. METHOD We treat the low and high-frequency coponents of iages separately; we perfor a preprocessing to express s as s = s l + s h, () where s l and s h denote the low-frequency and high-frequency coponents of s, respectively. Typically, s l is first coputed by low-pass filtering s and then s h = s s l. In particular, we copute the convolutional sparse representation of s h with respect to a dictionary of filters learned to describe the high-frequency coponents of noral iages. Restricting the convolutional sparse representation to the highfrequency coponents allows the use of few sall filters in the dictionary. For each patch s c we copute a vector g h (c) that siultaneously assesses the accuracy and the sparsity of the convolutional representation around c. The low-frequency content of s is instead onitored by coputing the saple oents of s l ( ) over patches, yielding vectors g l ( ). Then, for each patch, we define the indicator vector g(c) as the concatenation of g h (c) and g l (c). Anoalies are detected as patches yielding outliers in these indicator vectors. Let us briefly suarize the proposed solution, presenting the high-level schee in Algoriths and ; details about convolutional sparse representations and indicators are then provided in Section IV-A and IV-B, respectively. Training: Anoalies are detected by learning a odel describing noral patches fro the training set T. In particular, we learn a dictionary of filters {d } yielding convolutional sparse representation for high-frequency coponents of noral iages (Algorith, line and Section IV-A). The indicator vectors are then coputed fro all the noral patches, as described in Section IV-B, and a suitable confidence region

4 R γ that encopasses ost of these indicators is defined (Algorith, lines and 3). Testing: During operation, each test iage s is preprocessed to separate the high frequency content s h fro the low frequency content s l (Algorith, line ). The convolutional sparse representation of s h with respect to the dictionary {d } is obtained by the sparse coding procedure described in Section IV-A (Algorith, line ). Then, for each pixel c, g h (c) is coputed by analyzing the convolutional representation of s h in the vicinity of c, and g l (c) is coputed by analyzing s l in the vicinity of c (Algorith, lines 4 and 5). The indicator vector g(c) is then obtained by stacking g h (c) and g l (c), naely g(c) = [g h (c), g l (c)]. Any indicator g(c) that falls outside a confidence region R γ estiated fro noral iages is considered an outlier and the corresponding patch anoalous (Section IV-B3, Algorith, line 7). Input: training set of noral iages T:. Learn filters {d } solving (4). Copute g h ( ) (9) and g l ( ) (0) for all the noral patches, define g as in () 3. Define R γ in () setting the threshold γ > 0 Algorith : Training the anoaly detector using convolutional sparse odels. Input: test iage s:. Preprocess the iage s = s l + s h. Copute the coefficient aps {x } solving (7) 3. foreach pixel c of s do 4. Copute g h (c) as (9) 5. Copute g l (c) as (0) 6. Define g(c) = [g l (c), g l (c)] as () 7. if g(c) / R γ then 8. c belongs to an anoalous region else 9. c belongs to a noral region end end Algorith : Detecting anoalous regions using convolutional sparse odels. A. Convolutional Sparse Representations Convolutional sparse representations [] express the highfrequency content of an iage s R N N as the su of M convolutions between filters d and coefficient aps x, i.e. M s h d x, (3) = where denotes the two diensional convolution, and the coefficient aps x R N N have the sae size of the iage s. Filters {d } ight have different sizes, but are typically uch saller than the iage. Coefficient aps are assued to be sparse, naely only few eleents of each x are nonzero, thus x 0 (the nuber of nonzero eleents) is sall. Sparsity regularizes the odel and prevents trivial solutions of (3). For notational siplicity we oit {,..., M} fro the collection of filters and coefficient aps. ) Dictionary learning: To detect anoalous regions we need a collection of filters {d } that specifically approxiate the local structures of noral iages. These filters represent the dictionary of the convolutional sparse odel, and can be learned fro a noral iage s provided for training. Dictionary learning is forulated as the following optiization proble arg in {d },{x } M M d x s h + λ x, = = subject to d =, {,..., M}, where {d } and {x } denote the collections of M filters and coefficient aps, respectively. To siplify the notation, (4) presents dictionary learning on a single training iage s, however, extending (4) to ultiple training iages is straightforward []. The first ter in (4) denotes the reconstruction error, i.e. the squared l nor of the residuals, naely d x s h = ( (d x )(c) s h (c)), c X (5) while the l nor of the coefficient aps is defined as x = x (c). (6) c X In practice, the penalization ter in (4) prootes the sparsity of the solution [7], naely the nuber of nonzero coefficients in the feature aps. Thus, the l nor is often used as replaceent of l 0 nor to ake (4) coputationally tractable. The constraint d = is necessary to resolve the scaling abiguity between d and x (i.e. x can be ade arbitrarily sall if the corresponding d is ade correspondingly large). We solve the dictionary learning proble using an efficient algorith [8] that operates in Fourier doain. Learned filters typically report the proinent local structures of training iages, as shown in Figure (a). We observe that filters {d } learned in (4) ay have different sizes [8], which is a useful feature for dealing with iage structures at different scales. Figure (a) provides an exaple where 8 filters of size 8 8 and 8 filters of size 6 6 were siultaneously learned fro a training iage. ) Sparse Coding: The coputation of coefficient aps {x } of an input iage s h with respect to a dictionary {d } is referred to as sparse coding, and consists in solving the following optiization proble []: arg in d {x } x s h + λ x, (7) where filters {d } were previously learned fro (4). The sparse coding proble (7) can be solved via the Alternating Direction Method of Multipliers (ADMM) algorith [9], exploiting an efficient forulation [] in the Fourier doain. The dictionary-learning proble is typically solved by alternating the solution of (4) with respect to the coefficient aps {x } when filters are fixed (sparse coding) and then with respect to the filters {d } when coefficient aps are fixed. (4)

5 B. Indicators To deterine whether each patch s c is noral or anoalous we copute an indicator vector g(c) that quantitatively assesses the extent to which s c is consistent and with noral patches. Indicators g(c) are coputed fro the decoposition of s in () to assess the extent to which the dictionary {d } atches the structures of s h around c (Section IV-B), as well as the siilarity between low frequency content of Π c s l and noral patches (Section IV-B). ) High-Frequency Coponents: In anoalous regions filters are less likely to atch the local structures of s h, thus it is reasonable to expect the sparse coding (7) to be less successful, and that either the coefficient aps would be less sparse or that (3) would yield a poorer approxiation. This iplies that we should onitor the reconstruction error (5) and the sparsity of coefficient aps (6), locally, around c. However, we observed that onitoring the sparsity ter (6) is too loose a criterion for discriinating anoalous regions. To iprove the detection perforance, we take into consideration also the distribution of nonzero eleents across different coefficient aps. This choice was otivated by the epirical observation that often, within noral regions where filters are well atched with iage structures only few coefficient aps are siultaneously active; in contrast, within regions where filters and iage structures do not atch, ore filters are typically active. Figure (b) copares two coefficient aps within noral and anoalous regions, and shows that in the latter case ore filters are active. For this reason, we also include a ter in g(h) to onitor the local group sparsity of the coefficient aps, naely: M Π c x = (x (c + u)). (8) = u U The indicator based on the high frequency coponents of the iage is defined as g h (c) = Π c (s h d x ) Π cx, (9) Π cx where first and second coponents represent the reconstruction error of s c and the sparsity of the coefficient aps in the vicinity of c, respectively: these directly refer to the ters in (7), thus inherently indicate how successful the sparse coding was. The third eleent in () represents the group sparsity, and indicates the spread of nonzero eleents across different coefficient aps in the vicinity of pixel c. ) Low-Frequency Coponents: Anoalies affecting the low frequency coponents of s are in principle easy to detect as these affect, for instance, the average value of each patch. We analyze the first two saple oents over patches Π c s l, extracted fro s l. More precisely, for each pixel c of s l we define [ ] µc g l (c) =, (0) where µ c = u U s l(u + c)/p denotes the saple ean and σc = u U (s l(u + c) µ c ) /(P ) the saple variance coputed over the patch of s l centered in c. σ c Both g h and g l can be stacked in a single vector to be jointly analyzed when testing patches, naely [ ] gl (c) g(c) =. () g h (c) This is the indicator we use to deterine whether a patch is anoalous or not, analyzing both the high and the low frequency content in the vicinity of pixel c. 3) Detecting Anoalous Patches: We treat indicators as rando vectors and detect as anoalous all the patches yielding indicators that can be considered outliers. Therefore, we build a confidence region R γ around the ean vector [0] for noral patches, naely: R γ = { φ R : (φ g) T Σ (φ g) γ }, () where g and Σ denote the saple ean and saple covariance atrix of indicators extracted fro noral iages in T, and γ > 0 is a suitably chosen threshold. R γ represents an highdensity regions for indicators extracted fro noral patches, since the ultivariate Chebyshev s inequality ensures that, for a noral patch s c, holds Pr({g(s c ) / R γ }) γ, (3) where Pr({g(s c ) / R γ }) denotes the probability for a noral patch s c to lie outside the confidence region (false-positive detection). Therefore, outliers can be siply detected as vectors falling outside a confidence region R γ, i.e. (g(c) µ) T Σ (g(c) µ) > γ, (4) and any patch s c yielding an indicator g(c) satisfying (4) is considered anoalous. V. EXPERIMENTS We design two experients to assess the anoaly-detection perforance of convolutional sparse representations and, in particular, the advantages of onitoring the local group sparsity of the coefficient aps. In the first experient we detect anoalies by onitoring both the low and high frequency coponents of an input iage, while in the second experient we exclusively analyze the high-frequency coponents. The latter experient is to assess the perforance of detectors based exclusively on sparse odels. Considered Algoriths: We copare the following four algoriths built on the fraework of Algorith and : Convolutional Group: a convolutional sparse odel is used to approxiate s h. Anoalies are detected by analyzing the indicator g in () which includes also the local group sparsity of the coefficient aps (8). The indicator has five diensions. Convolutional: the sae dictionary of filters for the Convolutional Group solution is used to approxiate s h, however, the local group sparsity ter of the coefficient aps (8) is not considered in g. Thus, the indicator has four diensions.

6 Fig. 3: Ten of the textures selected fro Brodatz dataset (textures 0, 7, 36, 54, 66, 83, 87, 03, 09, ). For visualization purposes, we show only regions cropped fro the original iages, which have size Patch-based: a standard sparse odel [] rather than a convolutional sparse odel is used to describe patches extracted fro sh, as in []. The indicator includes the reconstruction error and the sparsity of the representation. To enable a fair coparison against convolutional sparse odels, the indicator includes the saple ean over each patch fro sh and sl, as well as the saple variance over each patch fro sl. The indicator has five diensions. Saple Moents: no odel is used to approxiate the iage and we copute the saple ean and variance over patches fro sl and sh. The indicator has four diensions. In the second experient, where only the high-frequency content of iages is processed, the sae algoriths are used. In particular, the Convolutional Group coputes threediensional indicator (gh ), the Convolutional coputes a twodiensional indicator (only the first two coponents of gh are used), the Patch-based operates only on sh coputing a threediensional indicator and the Saple Moents also operates on sh coputing a two-diensional indicator. We learned dictionaries of 6 filters (8 filters of size 8 8 and 8 of size 6 6), solving (4) with λ = 0.. The sae value λ was used also in the sparse coding (7). The dictionaries in the patch-based approach are.5 ties redundant, and learned by []. In all the considered algoriths, indicators are coputed fro 5 5 patches. Preprocessing: We perfor a low-pass filtering of each test iage s to extract the low frequency coponents. More precisely, sl corresponds to the solution of the following optiization proble arg in sl α ksl sk + k sl k, (5) where sl denotes the iage gradient, α > 0 regulates the aount of high frequency coponents in sl (in our tests α = 0). The proble (5) can be solved as a linear syste and adits a closed-for solution. Dataset: To test the considered algoriths we have selected 5 textures fro the Brodatz dataset [3] having a structure that can be properly captured by 5 5 filters and Fig. 4: Exaple of anoaly-detection perforance for the Convolutional Group algorith. Any detection (red pixels) on the left half represents a false positive, while any detection on the right half a true positive. The ideal anoaly detector would here detect all the points in the left half and none on the right half. Patches across the vertical boundary are not considered in the anoaly detection to avoid artifacts. As shown in the highlighted regions, ost of false positives in this exaple are due to structure that do not confor to the noral iage in Figure (a). patches (0 of the selected textures are shown in Figure 3). A dataset of test iages has been prepared by splitting each texture iage in two halves: the left half was exclusively used for training, the right half for testing. The right halves of the 5 textures are pair-wise cobined, creating 600 test iages by horizontally concatenating two different right halves. An exaple of test iage is reported in Figure (b). We consider the left half of each test iage as noral and the right half as anoalous, therefore we perfor anoaly detection using the odel learned fro the texture on the left half. Note that, since anoalies are detected in a patch-wise anner, having anoalies covering half test iage does not ease the anoaly detection task with respect to having localized anoalies like those in Figure (a). Figures of Merit: The anoaly detection perforance are assessed by the following figures of erit: FPR, false positive rate, i.e. the percentage of noral patches detected as anoalous. TPR, true positive rate, i.e. the percentage of correctly detected patches. Figure 4 provides an exaple of false positives and true positives over a test iage (the left half being noral, the right half anoalous). Both FPR and TPR depend on the specific value of γ used when defining Rγ. To enable a fair coparison aong the considered algoriths we consider a wide range of values for γ and plot the receiver operating characteristic (ROC) curve for each ethod. In practice, each point of a ROC curve corresponds to the pair (FPR, TPR) achieved for a specific value of γ. When coputing FPR and TPR, we exclude patches overlapping the vertical boundary between different textures.

7 ROC curves fro s = s h + s l ROC curves fro s h TPR Convolutional Group Convolutional Patch-based Saple Moents TPR Convolutional Group Convolutional Patch-based Saple Moents FPR (a) The Area Under the Curve values are: Convolutional Group: 0.950; Convolutional: 0.937; Patch-Based: 0.94; Saple Moents: FPR (b) The Area Under the Curve values are: Convolutional Group: 0.998; Convolutional: ; Patch-Based: ; Saple Moents: Fig. 5: ROC curves for the algorith considered in Section V, obtained by varying the paraeter γ in the definition of the confidence region R γ. Figure (a) shows the perforance in detecting anoalies when the whole spectru of the iages is considered, while in Figure (b) are reported the ROC curves obtained by onitoring s h. Figures 5(a) and 5(b) report the average ROC curves when processing the whole iage spectru and high frequency coponents only, respectively. The Area Under the Curve (AUC) is the typical scalar etric to copare the detection perforance: AUC values are indicated in the captions of the Figures and in Figure 6, which displays the AUC averaged over all the iages having a specific texture as noral. Discussions: These experients indicate that convolutional sparsity provides an effective odel for describing local iage structures and detecting anoalous regions. In both the ROC curves of Figure 5, the best perforance is achieved when considering the group sparsity of the coefficient aps, suggesting that the spread of nonzero coefficients across different aps is relevant inforation for detecting anoalies. This clearly eerges fro Figure 5(a), where the Convolutional Group algorith substantially outperfors the others, and fro Figure 5(b) which shows an increased perforance gap when anoalies can be only perceived fro the high-frequency coponents. VI. CONCLUSIONS We have presented a novel approach for detecting anoalous structures in iages by learning a convolutional sparse odel that describes the local structures of noral iages. Convolutional sparse odels are shown to be effective at detecting anoalies in the high frequency coponents of iages, and this is essential in applications where the anoalies are The curves were averaged over the whole dataset setting the sae FPR values. not very apparent fro the low-frequencies of the iage, or where anoalies affecting the low frequency content have not to be detected. Our experients also show that the local group sparsity of the coefficient aps is an essential inforation for assessing whether regions in a test iage confor or not the learned convolutional sparse odel. In our ongoing work we will investigate whether the local group sparsity represents a general regularization prior for convolutional sparse odels, and design a specific sparse-coding algorith that leverages a penalization ter to proote this for of regularity. REFERENCES [] M. A. Pientel, D. A. Clifton, L. Clifton, and L. Tarassenko, A review of novelty detection, Signal Processing, vol. 99, pp. 5 49, 04. [] M. D. Zeiler, D. Krishnan, G. W. Taylor, and R. Fergus, Deconvolutional networks, in Proceedings of IEEE Conference on Coputer Vision and Pattern Recognition (CVPR), 00, pp [3] M. Elad, Sparse and redundant representations: fro theory to applications in signal and iage processing. Springer Science & Business Media, 00. [4] V. Chandola, A. Banerjee, and V. Kuar, Anoaly detection: A survey, ACM Coputing Surveys (CSUR), vol. 4, no. 3, p. 5, 009. [5] M. Markou and S. Singh, Novelty detection: a review - part : statistical approaches, Signal processing, vol. 83, no., pp , 003. [6], Novelty detection: a review - part : neural network based approaches, Signal processing, vol. 83, no., pp , 003. [7] B. Schölkopf, R. C. Williason, A. J. Sola, J. Shawe-Taylor, and J. C. Platt, Support vector ethod for novelty detection. in Advances in Neural Inforation Processing Systes (NIPS), 999, pp

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