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Empirical Evaluation of Dissimilarity Measures for Color and Texture Presented by: Dave Kauchak Department of Computer Science University of California, San Diego Jan Puzicha, Joachim M. Buhmann, Yossi Rubner & Carlo Tomasi

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The Problem: Image Dissimilarity D( ) = ?,

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Where does this problem arise in computer vision? Image Classification Image Retrieval Image Segmentation

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Classification ? ? ?

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Retrieval Jeremy S. De Bonet, Paul Viola (1997). Structure Driven Image Database Retrieval. Neural Information Processing 10 (1997).

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Segmentation

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Histograms for image dissimilarity Examine the distribution of features, rather than the features themselves General purpose (i.e. any distribution of features) Resilient to variations (shadowing, changes in illumination, shading, etc.) Can use previous work in statistics, etc.

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Histogram Example

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Histogramming Image Features Color Texture Shape Others… Create histogram through binning or some procedure to get a distribution

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Color Which is more similar? L*a*b* was designed to be uniform in that perceptual “closeness” corresponds to Euclidean distance in the space.

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L*a*b* L – lightness (white to black) a – red-greeness b – yellowness-blueness

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Texture Texture is not pointwise like color Texture involves a local neighborhood Gabor Filters are commonly used to identify texture features

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Gabor Filters Gabor filters are Gaussians modulated by sinusoids They can be tuned in both the scale (size) and the orientation A filter is applied to a region and is characterized by some feature of the energy distribution (often mean and standard deviation)

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Examples of Gabor Filters Scale: 4 at 108°Scale: 5 at 144°Scale: 3 at 72°

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Creating Histograms from Features Regular Binning – Simple – Choosing bins important. Bins may be too large or too small Adaptive Binning – Bins are adapted to the distribution (usually using some form of K-means)

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Marginal Histograms Marginal histograms only deal with a single feature Normal Binning Marginal binning resulting in 2 histograms

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Cumulative Histogram Normal Histogram Cumulative Histogram

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Dissimilarity Measure Using the Histograms Heuristic Histogram Distances Non-parametric Test Statistics Information-Theoretic diverges Ground distance measures

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Notation D(I,J) is the dissimilarity of images I and J f(i;J) is histogram entry i in histogram of image J f r (i;J) is marginal histogram entry i of image J F r (i;J) is the cumulative histogram

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Heuristic Histogram Distances Minkowski-form distance L p Special cases: – L 1 : absolute, cityblock, or Manhattan distance – L 2 : Euclidian distance – L : Maximum value distance

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More heuristic distances Weighted-Mean-Variance (WMV) – Only includes minimal information about distribution

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Non-parametric Test Statistics Kolmogorov-Smirnov distance (K-S) Cramer/von Mises type (CvM)

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Cumulative Difference Example Histogram 1Histogram 2Difference - = K-S = CvM =

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Non-parametric Test Statistics (cont.) 2 -statistic (chi-square) – Simple statistical measure to decide if two samples came from the same underlying distribution

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Information-Theoretic diverges How well can one distribution be coded using the other as a codebook? Kullback-Leibler divergence (KL) Jeffrey-divergence (JD)

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Ground Distance Measure Based on some metric of distance between individual features Earth Movers Distance (EMD) – Minimal cost to transform one distribution to the other Only measure that works on distributions with a different number of bins

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EMD One distribution can be seen as a mass of earth properly spread in space, the other as a collection of holes in that same space Distributions are represented as a set of clusters and an associated weight Computing the dissimilarity then becomes the transportation problem

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Transportation Problem Some number of suppliers with goods Some other number of consumers wanting goods Each consumer-supplier pair has an associated cost to deliver one unit of the goods Find least expensive flow of goods from supplier to consumer

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Various properties of the metrics K-S, CvM and WMV are only defined for marginal distributions L p, WMV, K-S, CvM and, under constraints, EMD all obey the triangle inequality WMV is particularly quick because the calculation is quick and the values can be pre- computed offline EMD is the most computationally expensive

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Key Components for Good Comparison Meaningful quality measure – Subdivision into various tasks/applications (classification, retrieval and segmentation) Wide range of parameters should be measured An uncontroversial “ground truth” should be established

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Data Set: Color Randomly chose 94 images from set of 2000 – 94 images represent separate classes Randomly select disjoint set of pixels from the images – Set size of 4, 8, 16, 32, 64 pixels – 16 disjoint samples per set per image

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Data Set: Texture Brodatz album – Collection of wide range of texture (e.g. cork, lawn, straw, pebbles, sand, etc.) Each image is considered a class (as in color) Extract sets of 16 non-overlapping blocks – sizes 8x8, 16x16,…, 256x256

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Setup: Classification k-Nearest Neighbor classifier is used – Nearest Neighbor classification: given a collection of labeled points S and a query point q, what point belonging to S is closest to q? – k nearest is a majority vote of the k closest points – k = 1, 3, 5 and 7 Average misclassification rate percentage using leave-one-out

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Setup: Classification (cont.) Bins: {4, 8, 16, 32, 64, 128, 256} Texture case, three sets of filters were used of sizes 12, 24 and 40 filters 1000 CPU hours of computation

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Results: Classification, color data set

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Results: Classification, texture data set

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Results: Classification For small sample sizes, the WMV measure performs best in the texture case. – WMV only estimates means and variances – Less sensitive to sampling noise EMD also performs well for small sample sizes – Local binning provides additional information For large sample sizes 2 test performs best

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Results: Classification (cont.) For texture classification, marginal distributions do better than multidimensional distributions except for very large sample sizes (256x256) – Binning is not well adapted to the data since it is fixed for all the 94 classes – EMD, which uses local adaption does much better For multidimensional histograms, the more bins the better the performance For texture, usually 12 filters is enough

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Setup: Image Retrieval Vary sample size Vary number of images retrieved Performance measured based on precision (i.e. percent correct of the images retrieved) vs. the number of images retrieved

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Results: Image Retrieval

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Results: Image Retrieval (cont.) Similar to classification – EMD, WMV, CvM and K-S performed well for small sample sizes – JD, 2 and KL perform better for larger sizes

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Setup: Segmentation 100 images Each image consists of 5 different Brodatz textures For multivariate, the bins are adapted to specific image

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Setup: Segmentation (cont.) Image is divided into sites (128 x 128 grid) A histogram is calculate for each site Each site histogram is then compared with 80 randomly selected sites Image sites with high average similarity are then grouped

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Results: Segmentation

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Results: Segmentation (cont.) Median20% Quantile L 1 marginal8.2%12% 2 marginal 8.1%13% JD marginal8.1%12% KS marginal10.8%20% CvM marginal10.9%22% L 1 full6.8%9% 2 full 6.6%10% JD full6.8%10%

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Results: Segmentation (cont.) Binning can be adapted to image – Increased accuracy in representing multidimensional distributions – Adaptive multivariate outperforms marginal Best results were obtained by 2 EMD suffers from high computational complexity

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Conclusions No measure is overall best Marginal histograms and aggregate measures are best for large feature spaces Multivariate histograms perform well with large sample sizes EMD performs generally well for both classification and retrieval, but with a high computational cost 2 is a good overall metric (particularly for larger sample sizes

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