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13 November 2013Birkbeck College, U. London1 Research Methods Lecturer: Steve Maybank Department of Computer Science and Information Systems sjmaybank@dcs.bbk.ac.uk Autumn 2013 Data Research Methods in Computer Vision

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13 November 2013Birkbeck College, U. London2 Digital Images A digital image is a rectangular array of pixels. Each pixel has a position and a value. 951104034 1251082591 15811659112 166132101124 Original colour image from the Efficient Content Based Retrieval Group, University of Washington

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13 November 2013Birkbeck College, U. London3 Size of Images Digital camera, 5,000x5,000 pixels, 3 bytes/pixel -> 75 MB. Surveillance camera at 25 f/s -> 1875 MB/s. 1000 surveillance cameras -> ~1.9 TB/s. Not all of these images are useful!

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13 November 2013Birkbeck College, U. London4 Image Compression Divide the image into blocks, and compress each block separately, e.g. JPEG uses 8x8 blocks. Lossfree compression: the original image can be recovered exactly from the compressed image. Lossy compression: the original image cannot be recovered.

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13 November 2013Birkbeck College, U. London5 Why is Compression Possible? Natural image: values of neighbouring pixels are strongly correlated. White noise image: values of neighbouring pixels are not correlated. Compression discards information.

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13 November 2013Birkbeck College, U. London6 Measurement Space Each 8x8 block yields a vector in R 64. The vectors from natural images tend to lie in a low dimensional subspace of R 64. R 64 Vectors from 8x8 blocks

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13 November 2013Birkbeck College, U. London7 Strategy for Compression Choose a basis for R 64 in which the low dimensional subspace is spanned by the first few coordinate vectors. Retain these coordinates and discard the rest. R 64 vectors from 8x8 blocks

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13 November 2013Birkbeck College, U. London8 Discrete Cosine Transform

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13 November 2013Birkbeck College, U. London9 Basis Images for the DCT U T e(1) U T e(2)U T e(3)U T e(4)

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13 November 2013Birkbeck College, U. London10 Example of Compression using DCT Original image Image constructed from 3 DCT coefficients in each 8x8 block.

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13 November 2013Birkbeck College, U. London11 Linear Classification x x xx x x y y yy Given two sets X, Y of measurement vectors from different classes, find a hyperplane that separates X and Y. A new vector is assigned to the class of X or to the class of Y, depending on its position relative to the hyperplane.

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13 November 2013Birkbeck College, U. London12 Fisher Linear Discriminant

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13 November 2013Birkbeck College, U. London13 Maximisation of J(v)

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13 November 2013Birkbeck College, U. London14 Edge Regions and Regions Without a Peak 3x3 blocks with large Sobel gradients 3x3 blocks such that the grey level of the central pixel equals the mean grey level of the 9 pixels

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13 November 2013Birkbeck College, U. London15 Projections Onto a Random Plane Sobel edgesRegions without a peak Superposed plots

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13 November 2013Birkbeck College, U. London16 Projections Onto a 1-Dimensional FLD Sobel edges Regions without a peak Combined histograms

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Histogram of a DCT Coefficient 13 November 2013Birkbeck College, U. London17 The pdf is leptokurtic, i.e. it has a peak at 0 and fat tails.

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Sparseness of the DCT Coefficients For a given 8×8 block, only a few DCT coefficients are significantly different from 0. Given a DCT coefficient of low to moderate frequency, there exist some blocks for which it is large. 13 November 2013Birkbeck College, U. London18

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Whitening the Data 13 November 2013Birkbeck College, U. London19

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Independence and Correlation 13 November 2013Birkbeck College, U. London20

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Independent Components Analysis Let z be a rv with values in R^n such that cov(z)=I. Find an orthogonal matrix V such that the components of s =Vz are as independent as possible. 13 November 2013Birkbeck College, U. London21

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Method Assume a particular leptokurtic pdf p for the components si, e.g. a two sided Laplace density. Find the orthogonal matrix V with rows v1,…,vn that maximises the product p(s1)…p(sn) = p(v1.z)…p(vn.z) 13 November 2013Birkbeck College, U. London22

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Example Natural Image Statistics: a probabilistic approach to early computational vision, by A. Hyvarinen, J. Hurri and P.O. Hoyer. Page 161. ICA applied to image patches yields Gabor type filters similar to those found in the human visual system. 13 November 2013Birkbeck College, U. London23

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Stereo Pair of Images 13 November 2013Birkbeck College, U. London24 UC Berkeley looking west http://arch.ced.berkeley.edu/kap/wind/?p=32

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Salience A left hand image region is salient if the correct right hand matching region can be found with a high probability. H: hypothesis that (v(1),v(2)) is a matching pair. B: hypothesis that (v(1),v(2)) is a background pair 13 November 2013Birkbeck College, U. London25

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Formulation using PDFs The region yielding v(1) is salient if p(v(2)|v(1), H) differs significantly from p(v(2)|v(1), B) Measurement of difference: the Kullback-Leibler divergence: 13 November 2013Birkbeck College, U. London26

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Illustration of the Kullback-Leibler Divergence KL divergence = 0.125 13 November 2013Birkbeck College, U. London27 KL divergence = 0.659 Preprint on salience available from SJM

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