L8(LinFilt)

L8(LinFilt) - Linear Filtering CS / ECE 181B Convolution,...

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Linear Filtering CS / ECE 181B ± Convolution, Fourier Transforms and Correlation ± Ack: Prof. Matthew Turk for the lecture slides. 2 Additional Pointers • See my ECE 178 class web page http://www.ece.ucsb.edu/~manj/ece178 • A good understanding of linear filtering and convolution is essential in developing computer vision algorithms. • Topics I recommend for additional study (that I will not be able to discuss in detail during lectures)--> sampling of signals, Fourier transform, quantization of signals.
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3 Area operations: Linear filtering • Point, local, and global operations – Each kind has its purposes • Much of computer vision analysis starts with local area operations and then builds from there – Texture, edges, contours, shape, etc. – Perhaps at multiple scales • Linear filtering is an important class of local operators – Convolution – Correlation – Fourier (and other) transforms – Sampling and aliasing issues 4 Convolution • The response of a linear shift-invariant system can be described by the convolution operation R ij = H i ± u , j ± v F uv u , v ² Input image Convolution filter kernel Output image F H F H R ± = = * Convolution notations n j m i M m N n n m ij F H R ± ± ± = ± = ²² = , 1 0 1 0
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5 Convolution • Think of 2D convolution as the following procedure • For every pixel (i,j): – Line up the image at (i,j) with the filter kernel – Flip the kernel in both directions (vertical and horizontal) – Multiply and sum (dot product) to get output value R ( i,j ) (i,j) 6 Convolution • For every ( i,j ) location in the output image R , there is a summation over the local area F H R 4,4 = H 0,0 F 4,4 + H 0,1 F 4,3 + H 0,2 F 4,2 + H 1,0 F 3,4 + H 1,1 F 3,3 + H 1,2 F 3,2 + H 2,0 F 2,4 + H 2,1 F 2,3 + H 2,2 F 2,2 n j m i M m N n n m ij F H R ± ± ± = ± = ²² = , 1 0 1 0 = -1 *222 +0 *170 +1 *149+ -2 *173+ 0 *147+ 2 *205+ -1 *149+ 0 *198+ 1 *221 = 63
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7 Convolution: example 1 1 4 1 0 2 5 3 0 1 2 x(m,n) 1 1 1 0 1 -1 0 1 h(m,n) m n m n -1 1 1 1 h(-m, -n) -1 1 1 1 h(1-m, n) y(1,0) = ± k,l x(k,l)h(1-k, -l) = 0 0 0 0 0 -2 5 0 0 0 0 0 = 3 1 5 5 1 3 10 5 2 2 3 -2 -3 m n y(m,n)= verify! 8 Spatial frequency and Fourier transforms A discrete image can be thought of as a regular sampling of a 2D continuous function – The basis function used in sampling is, conceptually, an impulse function, shifted to various image locations – Can be implemented as a convolution
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9 Spatial frequency and Fourier transforms • We could use a different basis function (or basis set) to sample the image • Let’s instead use 2D sinusoid functions at various frequencies (scales) and orientations
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L8(LinFilt) - Linear Filtering CS / ECE 181B Convolution,...

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