Distances between frequency features for 3D visual pattern partitioning

1 Distances between Frequency Features for 3D Visual Pattern Partitioning Abstract. In this paper we propose a technique for the decomposition of a 3D image into a set of low level patterns associated to phase congruency, which we call visual patterns. Those patterns have frequency components in a wide range of bands that are aligned in phase. The method involves clustering of the band-pass filtered versions of the image according to a measure of congruence in phase or, what is equivalent, alignment in the filter’s responses energy maxima. This is achieved by defining a distance between the responses of pairs of filters and applying a hierarchical clustering analysis to the resulting distance matrix. To measure the degree of maxima alignment we propose a set of alternative distances and study their suitability. From this study we conclude that a measure of linear dependence between the local energy of filters’ responses is more appropriate than a more general measure of dependence. Keywords: low level feature extraction, visual pattern partitioning, multiresolution analysis, image dissimilarity estimation. 1 Introduction Image analysis involves creating a low level image representation of image contents. In some cases the appropriate selection and detection of features to be the building blocks of higher level tasks is critical for the correct performance of the whole process. It seems to be widely accepted that there is not a unique feature that brings a complete representation of an image, but a set of different features should be used and that these features should correspond to patterns easily identifiable by the human visual system (HVS). There is also agreement in that low level image features representing discontinuities on local properties, like intensity, texture or phase, are detected by the HVS. Therefore, operators that detect these features can yield a meaningful representation of an image. However, HVS can only detect features in a domain of two spatial dimensions plus time. Although 3D spatial features can be inferred from stereo pairs of 2D images, the low level processing mechanisms performed by the cortical areas of brain in mammals are 2D. Despite of this, the algorithms that simulate those mechanisms in 2D can be extended to 3D for their application to the analysis of volumetric data. 1.1 Energy Based Feature Detectors Much effort has been made in the field of image analysis to achieve good detectors for discontinuities on image properties. Canny (1986) stated that a good detector should 2 fulfill the criteria of good detection, good localization and single response. According to Owens et al. (1989) a feature detector should also be a projection. One of the techniques that verify these conditions is the energy based filtering, introduced by Morrone and Owens (1987) and widely studied by several authors (Morrone and Burr, 1988, Owens et al., 1989, Perona and Malik, 1990, Venkatesh and Owens, 1990). In this technique, image features are identified as the local maxima of the local energy of the image, which is calculated as the sum of the squared responses of a pair of filters in quadrature. This kind of operators outperforms previous linear filters (Canny, 1986, Marr and Hildreth, 1980) in many aspects: they do not mark edges in sine-wave signals, detect features that are a mixture of odd and even intensity profiles, do not suffer from multiple responses and are projections. Local energy maxima are also points of maximal phase congruency (PC), which measures the local degree of matching among the phase of Fourier components. Morrone and Owens (1987) sustain that the HVS perceives features at points of high PC. A good measure of PC must have into account the “amount” of frequencies that contribute to a feature to avoid, for instance, high responses to sine-wave signals, which are not perceived as features by the HVS. To this end, some techniques have employed broad bandwidth single-scale analysis (Morrone and Owens, 1987). However, most of them have opted for multiscale approaches (Morrone and Burr, 1988, Malik and Perona, 1990, Venkatesh and Owens, 1990, Kovesi, 1996). Multiresolution analysis is based on the decomposition of the image into a set of feature maps corresponding to different frequency bands and orientations. This is also consistent with the observed behavior of the HVS (Field, 1993). To integrate the information from different frequency bands, most of the approaches combine their responses to produce one single feature map –or “primal sketch” (Marr and Hildreth, 1980) –of the image. Morrone and Burr (1988) developed a method to calculate local energy involving the sum of the responses of a bank of filters. Kovesi (1996) defined a new measure of PC based on statistical measures on the local phase of the responses of a set of log Gabor filters. Additionally, he incorporated factors related to the spread of filter responses to enlarge the PC of features with contributions from broad ranges of frequencies. An alternative solution is to determine what the spectral bands contributing to each image feature are. Separating the bands corresponding to each feature leads to a partition of the image into its most relevant low level features. This is the kind of analysis performed by Rodríguez-Sánchez et al. (1999) in the so-called RGFF model and extended in Chamorro-Martínez et al. (2003) and Dosil et al. (2004). In the RGFF, visual patterns or integral features are defined as patterns with alignment of frequency components in a set of local properties, not only energy but also entropy, contrast, etc. To isolate visual patterns the RGFF first uses a filter bank to decompose the image into its elementary frequency components. From now on we will call frequency features to the responses of the oriented band-pass filters in the bank. The information from different frequency bands is combined by grouping of similar frequency features together, using cluster analysis. To perform cluster analysis a suitable measure of dissimilarity between frequency features is necessary. Such distance must be small among those features belonging to the same visual pattern, i.e., it must depend on the degree of phase congruency or, equivalently, the degree of alignment among their local energy maxima. 3 1.2 Frequency Feature Dissimilarity Measurement The dissimilarity employed by Rodríguez-Sánchez et al. (1999) and Chamorro-Martínez et al. (2003) to estimate the degree of alignment between two frequency features was inspired by biological processes, combining attention mechanisms and pooling of sensors’ outputs. It involves the measurement of a set of local statistics in a neighborhood of the attention points, located at energy maxima. The distance is obtained by applying a beta-norm function over the weighted sum of the differences between the statistics of each filter’s response. On their part, Dosil et al. (2004) have chosen a distance based on the normalized mutual information of the local energy of filters’ responses, which is less computationally expensive, less parameterized and less dependent on the performance of low level processes like non-maxima suppression and scale estimation. Mutual information is widely employed as a measure of image dissimilarity in different fields of application, with great popularity in medical image registration (Viola, 1995, Studholme et al., 1999). However, we have observed that the behavior of this measure is not completely satisfactory. In some cases it produces the grouping of very dissimilar features together, like in the examples from Fig. 7 and Fig. 11. This is explained by the fact that mutual information treats intensity values in a qualitative fashion. It only considers their joint and marginal probabilities, not the difference between their values. As a result, it might take large values when the concurrence of weak and strong maxima takes place. To put it in another way, mutual information is an underconstrained measure of dependence because it makes no assumptions about the kind of functional dependence between the two images –see Roche et al. (2000) for a detailed explanation. Thus, a measure that allows a general dependence between the images may not be the most appropriate in all applications. A measure of similarity that constrains the type of dependences between two images to those reflecting their true underlying relations should produce better results. 1.3 Our Approach In this work we present a method for the decomposition of a 3D image into a set of low level features that brings a useful description of image contents for further use in high level applications. To this end, we employ a 3D filter bank, where frequency channels are represented by 3D log Gabor filters with rotational symmetry. Frequency features resulting from the application of this filter bank are classified into separate visual patterns by hierarchical cluster analysis. Secondly, we study several image dissimilarity measures to find out which of them is the most appropriate to represent phase congruency. The set of distances has been extracted from the field of image registration, given that the task of registering two images involves the alignment of similar features together, which is a problem analogous to ours. In addition, we have considered the combination of the previous distances with attention mechanisms. In section 2 the method for visual pattern partitioning is described. In section 3 we go into depth in the subject of comparing frequency features. The set of dissimilarity measures between pairs of filtered images is presented in subsection 3.1 and their 4 computational cost is analyzed in subsection 3.2. Section 4 presents an experimental study on the performance of these measures in the task of visual pattern partitioning. The results are discussed in section 5. Section 6 concludes the paper. 2 Visual Pattern Detection As aforementioned, the method for visual pattern extraction consists of the decomposition of an image into a set of frequency channels and their grouping according to some dissimilarity measure. The process can be described as a sequence of steps: 1. Selection of active filters in the 3D filter bank, i.e., channels with high information content, by analyzing their spectral energy. 2. Calculation of the energy maps corresponding to the active filters’ responses. 3. Measurement of dissimilarity between frequency features. 4. Hierarchical clustering of the frequency features based on the dissimilarity matrix. 5. Visual pattern reconstruction by linear summation of the energy of the filters in each cluster. In the next subsections these procedures are detailed. 2.1 3D Filter Bank Visual patterns are represented as a combination of the responses of a set of band-pass filters tuned in to different scales and orientations. The filters’ transfer function T is designed as the product of separable factors R and S in the radial and angular components respectively, such that T = R·S. The radial term R is given by the log Gabor function (Field, 1987) () ()() ()() ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −= 2 2 log2 log exp; i i i R ρσ ρρ ρρ ρ , (1) where σ ρ is the standard deviation and ρ i the central radial frequency. To achieve orientation selectivity, the angular component is usually defined as a scattering function centered at the filter’s direction. Here, the scattering function is a Gaussian on the angular distance α between the directions of the filter and the position vector of a given point f in the spectral domain (Faas and Van Vliet, 2003) () fvf ⋅= arccos),( ii θφα , (2) where v = (cos φ i cos θ i , cos φ i sin θ i , sin φ i) is a unit vector in the filter’s direction and f is expressed in Cartesian coordinates. Then, for a given angular standard deviation σ α ()() ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛−== 2 2 2 ),( exp),(,;, α σ θφα θφαθφθφ ii iiii SS . (3) 5 The shape of S from equation (3) is depicted in Fig. 1. It can be seen that S from expression (3) has rotation symmetry. The complete 3D bank is composed of a number of the above described filters to tile the frequency domain, selecting a number of wavelengths and orientations and tuning the bandwidths to cover the spectrum properly. In our configuration, elevation is sampled uniformly while the number of azimuth values decreases with elevation in order to keep the “density” of filters constant. This is achieved by maintaining equal arclength between adjacent azimuth values over the unit radius sphere instead of taking uniform angular distances. Following this criterion, the filter bank has been designed using a number of orientations N = 6 in half equator, this is, the region with { θ i = 0; φ ∈ [0, π]}, producing 23 3D orientations. The angular bandwidth is 25º. In the radial axis, four values have been taken, with wavelengths 4, 8, 16 and 32 pixels, and 2 octaves bandwidth. These settings yield a highly redundant bank. Small images must be given an especial treatment. It may happen that some of the bands in the bank have wavelengths larger than half the image size. In these cases, their responses approximately represent the average intensity level. Therefore, these bands are discarded and only the bands with highest frequencies are considered. In the case of images with different sizes in each image axis direction, the projections of the wavelength in each direction are studied for each filter in a band. Fig 1. 2.2 Selection of Active Bands To decrease the computational cost, the number of filters is reduced by discarding filters with wavelengths greater than half the image size, roughly representing the average intensity, and with low information content, named non-active. The measure of information density is E = log(|F| + 1), where F is the Fourier transform of the image. A band is active when it comprises any value of E over the maximum spectral noise. The maximum noise level is estimated as m + x σ , where m is the mean noise energy, σ is its standard deviation and x ≥ 0. Here, m and σ have been measured in the band of frequencies greater than twice the largest of the bank’s central frequencies. Assuming that the spectral noise is uniform and that it fits a Gaussian distribution, most of the spectral noise energy is eliminated by taking x = 3. To eliminate remaining spurious noise “spots” an especial median operator is applied. Standard median filters produce the elimination of thin lineal structures. To avoid this, we have designed a radial median operator. The difference with an ordinary median filter is that, for a given pixel in the spectral domain, it only considers neighbors that are anterior or posterior in the radial direction. This eliminates isolated peaks but preserving the continuity of structures along scales. The expression of the radial median filter mask M of size L × L × L is as follows () ⎩ ⎨ ⎧=−− = otherwise0 ]||||/[]||||/)[(if1 ,M pppqpq pq , (4) where p and q are points in the image and mask domains respectively, [·] represents rounding to the nearest integer and the origin is placed in the image center. We choose a 6 mask size L = 3. Larger masks bring little improvement and make the filtering much slower, as the mask coefficient must be calculated for each pixel location. The behavior of this filtering is illustrated in Fig. 2. Fig 2. 2.3 Feature Clustering Our approach generates visual patterns by clustering of active bands. Dissimilarities between each pair of frequency features are computed to build a dissimilarity matrix. To determine the clusters from the dissimilarity matrix, a hierarchical clustering method has been chosen. Other clustering techniques, like k-means, are not adequate due to the nature of our data. Divisive clustering methods deal with the input data themselves treating them as coordinate vectors in a multidimensional space. In the this case, the input data are the energy maps, giving place to a space of as much as N 3 coordinates. Agglomerative methods are better suited, since they work directly with the dissimilarity matrix, not having into account data coordinates. Hierarchical clustering has been applied using a complete-link algorithm. The distance between clusters is defined as the maximum of all pairwise distances between features in the two clusters, thus producing compact clusters. The number of clusters Nc that a hierarchical technique generates is an input parameter of the algorithm. The usual strategy to determine the Nc is to run the algorithm for each possible Nc and evaluate the quality of each resulting configuration according to a given validity index. A modification of the Davies-Boulding index proposed by Pal and Biswas (1997) has proved to produce good results for our application. It is a graph-theory based index that measures the compactness of the clusters in relation to their separation. 3 Measuring Feature Phase Alignment Phase alignment imposes a relationship among the energy of the responses of a subset of frequency channels, so that one frequency feature is, to some point, predictable from another frequency feature belonging to the same visual pattern. Given that statistical dependence is defined as the predictability of a random variable given knowledge of another one, then there exists some degree of statistical dependence among frequency features if they are aligned in phase. Consequently, it seems reasonable to expect statistical measures of dependence to produce good results when applied to frequency feature clustering. The question now is what kind of measure is best suited to reflect phase alignment. The use of a measure that is invariant to unexpected or undesired types of dependence will prevent from grouping of features belonging to different patterns. On the other hand, an overconstrained measure could discard allowed relations. To determine which is the most appropriate measure one has to make some assumptions regarding the nature of the relations among the components of a same visual pattern. Here, the following a priori assumptions are adopted: the energy values of two frequency features belonging to the same cluster are linked; the link increases with energy maxima alignment, but total dependence is not possible, i.e., one response cannot 7 be perfectly predicted by the other, given that they are versions of the same image viewed through different sensors, this is, the filters’ parameters are different. The assumptions adopted for the present application are quite similar to those imposed in the field of image registration, like in multimodal medical image registration. The difference is that in image registration the alignment might not be produced among maxima, but among the intensity values associated to the same tissue in the various imaging modalities. Despite this difference, it is likely that the image distances employed in such application are also suitable for frequency feature comparison. One of the most popular measures in image registration is mutual information, I, (Viola, 1995) and its normalized versions (Studholme et al. 1999). I is a measure of arbitrary statistical functional dependence. This means that this measure does not make any assumptions regarding the kind of functional relation between the intensity values in the two images. Another measure of this kind is the correlation ratio, η (Roche et al., 1998, 2000). Unlike I, η assumes that the joint and marginal probability density distributions of the intensity levels in the two images can be properly modeled by means of Gaussian functions. These measures can be considered underconstrained, because they allow any kind of functional dependence. However, as posed before, the alignment should be produced only among maxima in the two images. A more restrictive measure is the correlation coefficient ρ . Nevertheless, this measure constrains the possible functional relation to be linear, so it could be too restrictive for our application. Alternative measures can be obtained by modifying the previous measures so as to directly apply them over energy maxima maps instead of raw energy maps. In this way, it is ensured that the correspondence is produced between energy maxima pairs and not between other concurrent intensity values. This can be easily accomplished by applying non-maxima suppression to the energy of the frequency features before distance estimation. The weakness of this approach is that both the number and the location of energy maxima are strongly influenced by scale, so that there is not perfect match among maxima locations in different features of the same pattern. For this reasong, this measure is expected to disjoint the bands composing a visual pattern. This problem could be reduced considering also the regions of influence of each energy maxima. Even when the maxima of two energy maps do not match, the energy profiles in the maxima surroundings should have a similar trend. Furthermore, if the neighborhood of the maxima is not taken into account the notion of size of the spatial features is lost. The previous approaches can be considered to introduce attention mechanisms, like those that take place in the HVS. Nevertheless, this is the only aspect in which the distance estimation emulates the biological process. On the other hand, the RGFF model presented in Rodríguez-Sánchez (1999) does combine the information from the attention points, simulating the pooling of the responses that is supposed to occur in the HVS (Quick, 1974, Graham, 1989). The main drawbacks of attention based measures are their dependence on the selection of threshold parameters and on the performance of local maxima detectors. To study the suitability of the different alternatives, an experimental study on their performance has been realized. Next subsection presents a formal definition of the diverse distances used in the test. Section 4 presents the tests and its results and in 8 section 4.2 these results are discussed and the conclusions obtained from them are presented. 3.1 Dissimilarity between Energy Maps 3.1.1 Statistical Measures of Dependence All of the dissimilarity measures presented here are derived from a similarity measure δ by applying to it a transformation to enhance intercluster distances, invert its range and limit it to the interval [0, 1]. What follows is the list of similarity measures δ followed by the distance functions Dδ derived from them. In our notation, X and Y represent two energy maps and M is the number of bins in histogram calculations. a) Normalized Mutual Information The mutual information I of two random variables is the reduction of uncertainty of the first variable brought by the knowledge of the second variable when Shannon's entropy H is used as the measure of uncertainty. It can also be considered the Kullback's divergence between the joint probability distribution of the two variables PX,Y and the predicted distribution in case of total statistical independence PX×Y = PX × PY. ),()()(),( YXHYHXHYXI −+= , where () () ( ) ( ) ∑ ∑== ji yxyx i xx jiPjiPYXHiPiPXH , ,, ,log,),( and log)( , Studholme proposed a normalized measure of image distance for medical image registration (Studholme et al., 1999). This measure is independent of the marginal entropies of the two images under comparison, so that it measures uncertainty reduction regardless of the uncertainty amount itself. The normalized similarity measure employed here is derived from Studholme's distance and has the following expression )()( ),()()( 2 )()( ),( 2),( YHXH YXHYHXH YHXH YXI YXNI + −+ ⋅= + ⋅= , This is a similarity measure with values in the interval [0, 1]. To transform it into a distance, its range must be inverted. To this end, the following transformation is applied, which also increases high distances and reduces small ones, enhancing the compactness of the clusters. () () () 2 ,1, YXNIYXDNI −= (5) This measure has a strong dependence on the bin size chosen in the histogram calculations needed to estimate the probability densities. It has been observed that a small number of bins, of O(16), produce the best results and incidentally speed up the calculations. 9 b) Correlation Coefficient The correlation coefficient is a measure of the degree of linear functional dependence. It is defined as follows. )Var()Var(),Cov(),( YXYXYX = ρ The values of ρ are in the range [−1, 1]. When ρ = 1 the data set perfectly fits a linear function and its sign reflects the sign of the slope of the function. The closer ρ is to zero, the lesser the linear tendency. In the present application the sign of the coefficient turns out to be very useful to constrain the possible relations to positive slope linear functions, thus preventing from the clustering of features with concurrence of maxima and minima. () () 2 2),(11),( YXYXD ρ ρ +−= (6) This distance function takes values in the range [0, 1]. The minimum value corresponds to perfect linear dependence with positive slope and the maximum corresponds to the case of perfect fit with negative slope, like, for example, an image and its inverse. This measure does not depend on the selection of any parameter. This is an advantage over any divergence measure, which involves the discrete estimation of joint and/or marginal probabilities. c) Correlation Ratio The correlation ratio was proposed by Roche et al. (1998) as an image dissimilarity estimation for multimodal medical image registration. Like mutual information, it is a measure of functional dependence, but it is restricted to the case when the conditional probability densities of the image intensities can be modeled as Gaussians (Roche et al., 2000). The correlation ratio of X conditioned to Y is defined as ()()() XYXXYX Var|Var1)|( 2Ε−−= η It represents the amount of uncertainty of X that can be predicted by Y in relation to the total uncertainty of X. η 2 can be considered as a measure of information gain when the variance is taken as a measure of uncertainty. The correlation ratio is not a symmetric measure. One signal may tell us much about another one but the opposite may not be true −let's think of one image and a degraded version of it corrupted by noise. As η (X|Y) ≠ η (Y|X), the symmetric measure is obtained by taking the maximum of both. Again, it is a similarity measure, so that its range is inverted and transformed to accentuate inter-cluster distances. () )|(),|(max1),( 22 XYYXYXD ηη η −= (7) d) Toussaint’s Distance 16 Field, D.J., 1993. Scale–Invariance and self-similar “wavelet” Transforms: An Analysis of Natural Scenes and Mammalian Visual Systems, in: Farge, M., Hunt, J.C.R., Vassilicos, J.C., (Eds.), Wavelets, fractals and Fourier Transforms. Clarendon Press, Oxford, pp. 151-193. Graham, N.V.S., 1989. Visual Pattern Analyzers. Oxford University Press. Kovesi, P.D., 1996. Invariant Measures of Image Features from Phase Information, The University or Western Australia. http://www.cs.uwa.edu.au/pub/robvis/theses/PeterKovesi/ Malik, J., Perona., P., 1990. Preattentive Texture Discrimination with Early Vision Mechanisms. J. Opt. Soc. Am. A. Vol. 7(5), pp. 923-932. Marr, D., Hildreth, E., 1980. Theory of Edge Detection. Proc. R. Soc. Lond. B. Vol. 207, pp. 187-217. Morrone, M.C., Burr, D.C., 1988. Feature Detection in Human Vision: a PhaseDependent Energy Model. Proc. R. Soc. Lond. B. Vol. 235, pp. 221-245. Morrone, M.C., Owens, R.A., 1987. Feature Detection from Local Energy. Pattern Recognition Letters. Vol. 6, pp. 303-313. Owens, R., Venkatesh, S., Ross, J., 1989. Edge Detection is a Projection. Pattern Recognition Letters. Vol. 9, pp. 233-244. Pal, N.R., Biswas, J., 1997. Cluster Validation Using graph Theoretic Concepts. Pattern Recognition. Vol. 30(6), pp. 847-857. Perona, P., Malik, J., 1990. Detecting and Localizing Edges Composed of Steps, Peaks and Roofs. Third Int. Conf. on Computer Vision, pp. 52-57. Quick, R.F., 1974. A vector magnitude model of contrast detection. Kybernetic. Vol. 16, 65-67. Roche, A., Malandain, G., Ayache, N., 2000. Unifying Maximum Likelihood Approaches in Medical Image Registration. Int. J. of Imaging Systems and Technology. Vol. 11, pp. 71-80. Roche, A., Malandain, G., Pennec, X., Ayache, N., 1998. The Correlation Ratio as a New Similarity Measure for Multimodal Image Registration, in LNCS 1496: MICCAI’98. Springer-Verlag, pp. 115-1124. Rodríguez-Sánchez, R., García, J.A., Fdez-Valdivia, J., Fdez-Vidal, X.R., 1999. The RGFF Representational Model: A System for the Automatically Learned Partition of 17 “Visual Patterns” in Digital Images. IEEE Trans. Pattern Anal. Mach. Intell., Vol. 21(10), pp. 1044-1073. Sarrut, D., and Miguet, S., 1999. Similarity Measures for Image Registration. 1st European Workshop on Content-Based Multimedia Indexing, Toulouse, France, pp. 263-270. Studholme, C., Hill, D.L.G., Hawkes, D.J., 1999. An Overlap Invariant Entropy Measure of 3D Medical Image Alignment. Pattern Recognition. Vol. 32, pp. 71-86. Venkatesh, S., Owens, R., 1990. On the Classification of Image Features. Pattern Recognition Letters. Vol. 11, pp. 339-349. Viola, P.A., 1995. Alignment by Maximization of Mutual Information. Massachusetts Institute of Technology, Artificial Intelligence Laboratory, Technical Report 1548 (1995). 18 Tables Filter Pair NI DNI ρ D ρ NI* D*NI {32, 31} 0.3146 0.1752 0.8391 0.0017 0.5239 0.0763 {1, 31} 0.1157 0.5559 –0.0096 0.0878 0.0146 0.7729 {13, 31} 0.2041 0.3621 –0.1630 0.1247 0.0079 0.8302 Table 1. Values of the normalized mutual information, correlation coefficient and their corresponding dissimilarities for each feature pair in the last row of Fig. 10. Asterisk indicates that measures are taken over maxima maps. 19 Figure Captions Fig 1. Left: Representation of S ( φ i = π/3, θ i = π/4) in the ( φ , θ ) plane by isolevel curves. Right: Representation of S ( φ i = π/3, θ i = π/4) over a unit radius sphere by isolevel curves. Fig 2. 2D example of radial median filtering. (a) Input image of a nematode. (b) E = log(|F| + 1), where |F| is the spectral energy. (c) Noise subtraction on E, non null values depicted in white. (d) Radial median filtering, non null values in white. (e) Bands comprising non null values. (f) Associated active filters. Fig 3. Examples of the different types of images used in the text. Cases in first row correspond to 2D images of (a) biomedical 2D data –a nematode in the example– and (b) 2D synthetic images composed from natural textured images –Brodatz textures. The remainder cases are 3D synthetic images represented by two cross-sections of the volume comprising different kinds of visual patterns: (c) regions with different grating patterns, (d) textures, (e) odd symmetric surfaces, (f) even symmetric surfaces, (g) surfaces caused by a phase shift, (h) lines, (i) blobs, (j) junctions –star-shaped junctions in this case–, (k) mixture of various features of different types, (l) superimposed features and (m, n) textures composed by grating patterns of different orientations and scales. Fig 4. The best results obtained for example cases from (a) to (g) in Fig. 3. The second subscript is the index for the set of resulting visual patterns. All results correspond to the dissimilarity measure based on the correlation coefficient, except from (c) and (g), which can be obtained, for example, using DRGFF. Fig 5. The best results obtained for example cases from (h) to (m) in Fig. 3. The second subscript is the index for the set of resulting visual patterns. All results correspond to the dissimilarity measure based on the correlation coefficient, except from (i) and (h), which can be obtained, for example, using DRGFF. Fig 6. Percentage of correct (OK), incorrect (X) and indecisive (?) results for each distance. Fig 7. The two visual patterns obtained for image in Fig. 3.m using DNI. Left: Clusters of filters represented by isosurfaces of level exp(−1/2) of the filters’ transfer function. Right: Patterns associated to the previous clusters. Fig 8. For image in Fig. 3.m, Left: frequency feature with label 13, with {λ13=16, φ 13=2π/3, θ13=0}, associated to the inner region, Middle: frequency feature 11, with {λ11=4, φ 11=0, θ11=0}, associated to the outer region and Right: frequency feature 12, with {λ12=8, φ 12=0, θ12=0}, associated to the outer region. 20 Fig 9. The two visual patterns obtained for image in Fig. 3.m using Dρ Left: Clusters represented by isosurfaces of level exp(−1/2) of the filters’ transfer function. Right: Patterns associated to the previous clusters. Fig 10. (a) 2D test image and some of its frequency features: (b) frequency feature with label 13, with {λ13=4, φ 13=π/3}, (c) frequency feature 31, with {λ31=4, φ 31=2π/3}, (d) frequency feature 32, with {λ32=16, φ 32=2π/3} and (e) frequency feature 1, with {λ1=4, φ 1=0}. (f, g) Frequency features 13 and 31 respectively after non maxima suppression and binarization. (h, i) Frequency features 13 and 31 after thresholding of energy maxima and binarization. The bottom row shows the joint histograms for the energies of the following couples of frequency features from the previous row: (j) {31, 13} (k) {31, 32} and (l) {31, 13} from energy maps after non maxima suppression and thresholding. Fig 11. (a, b) The two filter clusters obtained for image in Fig. 10.a using the dissimilarity DNI, represented as the sets of level exp(−1/2) of the filters’ transfer function. (c, d) Visual patterns associated to previous clusters. (e, f, g) The three filter clusters obtained for image in Fig. 10.a using dissimilarities D ρ and D*NI, represented as the sets of level exp(−1/2) of the filters’ transfer function. (h, i, j) Visual patterns associated to clusters in previous row. 21 Fig. 1. 22 (a) (b) (c) (d) (e) (f) Fig. 2. 23 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) Fig. 3. 24 (a).i (a).ii (a).iii (b).i (b).ii (c).i (c).ii (c).iii (d).i (d).ii (e).i (e).ii (f).i (f).ii (g).i (g).ii Fig. 4. 25 (h).i (h).ii (i).i (i).ii (j).i (j).ii (k).i (k).ii (k).iii (l).i (l).ii (m).i (m).ii Fig. 5.