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**Research Methods Lecturer: Steve Maybank**

Department of Computer Science and Information Systems Autumn 2013 Data Research Methods in Computer Vision 13 November 2013 Birkbeck College, U. London

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**Birkbeck College, U. London**

Digital Images 95 110 40 34 125 108 25 91 158 116 59 112 166 132 101 124 A digital image is a rectangular array of pixels. Each pixel has a position and a value. Original colour image from the Efficient Content Based Retrieval Group, University of Washington 13 November 2013 Birkbeck College, U. London

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**Birkbeck College, U. London**

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! 13 November 2013 Birkbeck College, U. London

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**Birkbeck College, U. London**

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. 13 November 2013 Birkbeck College, U. London

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**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. 13 November 2013 Birkbeck College, U. London

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**Birkbeck College, U. London**

Measurement Space Vectors from 8x8 blocks R64 Each 8x8 block yields a vector in R64. The vectors from natural images tend to lie in a low dimensional subspace of R64. 13 November 2013 Birkbeck College, U. London

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**Strategy for Compression**

vectors from 8x8 blocks Choose a basis for R64 in which the low dimensional subspace is spanned by the first few coordinate vectors. Retain these coordinates and discard the rest. 13 November 2013 Birkbeck College, U. London

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**Discrete Cosine Transform**

13 November 2013 Birkbeck College, U. London

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**Basis Images for the DCT**

UTe(1) UTe(2) UTe(3) UTe(4) 13 November 2013 Birkbeck College, U. London

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**Example of Compression using DCT**

Image constructed from 3 DCT coefficients in each 8x8 block. Original image 13 November 2013 Birkbeck College, U. London

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**Linear Classification**

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

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**Fisher Linear Discriminant**

13 November 2013 Birkbeck College, U. London

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**Birkbeck College, U. London**

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

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**Edge Regions and Regions Without a Peak**

3x3 blocks such that the grey level of the central pixel equals the mean grey level of the 9 pixels 3x3 blocks with large Sobel gradients 13 November 2013 Birkbeck College, U. London

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**Projections Onto a Random Plane**

Sobel edges Regions without a peak Superposed plots 13 November 2013 Birkbeck College, U. London

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**Projections Onto a 1-Dimensional FLD**

Sobel edges Regions without a peak Combined histograms 13 November 2013 Birkbeck College, U. London

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**Histogram of a DCT Coefficient**

The pdf is leptokurtic, i.e. it has a peak at 0 and “fat tails”. 13 November 2013 Birkbeck College, U. London

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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 2013 Birkbeck College, U. London

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**Birkbeck College, U. London**

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

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**Independence and Correlation**

13 November 2013 Birkbeck College, U. London

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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 2013 Birkbeck College, U. London

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**Birkbeck College, U. London**

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 2013 Birkbeck College, U. London

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**Birkbeck College, U. London**

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 2013 Birkbeck College, U. London

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**Birkbeck College, U. London**

Stereo Pair of Images UC Berkeley looking west 13 November 2013 Birkbeck College, U. London

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**Birkbeck College, U. London**

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 2013 Birkbeck College, U. London

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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 2013 Birkbeck College, U. London

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**Illustration of the Kullback-Leibler Divergence**

KL divergence = 0.125 KL divergence = 0.659 Preprint on salience available from SJM 13 November 2013 Birkbeck College, U. London

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