ECE468_18 - ECE 468: Digital Image Processing Lecture 18...

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Unformatted text preview: ECE 468: Digital Image Processing Lecture 18 Prof. Sinisa Todorovic sinisa@eecs.oregonstate.edu Outline • • Multiresolution image processing (Textbook 7.1) Image Pyramids (Textbook 7.1.1) Multiresolution Image Processing • • • Informal motivation: Images may show both very large and very small objects. It may be useful to process the images at different resolutions. Multiresolution Image Processing • • • A more formal motivation: An image is a 2D random process with locally varying statistics of pixel intensities Analysis of statistical properties of pixel neighborhoods of varying sizes may be useful Histogram of Small Pixel Neighborhoods Image Pyramids • A representation of the image that allows its multiresolution analysis Example: Image Pyramids Steps to Construct the Image Pyramid 1. Given an image at level j 2. Filter the input and and downsample the filtered result by a factor of 2; This gives the image at level j-1 3. Goto 1 4. Upsample and filter the image at level j-1; this gives an approximation of the image at level j 5. Subtract this result from the image at level j; this give the prediction residual at level j 6. Goto 1 Typical Filters • For the multiresolution pyramid, we use spatial filters: • • Neighborhood averaging Lowpass Gaussian filter • For the residual pyramid, we use interpolation filters: • • bilinear bicubic Upsampling/Downsampling • Upsampling = Inserting zeros f2↑ (x, y ) = ￿ f (x/2, y/2) , x, y are even 0 , o.w. • Downsampling = Discarding pixels f2↓ (x, y ) = f (2x, 2y ) Subband Image Coding Example: Analysis Filter Bank Subband Image Coding for perfect reconstruction: ˆ f (n) = f (n) g0 (n) = (−1) h1 (n) n g1 (n) = (−1)n+1 h0 (n) h0, h1, g0, g1 -- cross-modulated filters Subband Image Coding for perfect reconstruction: g1 (n) = (−1)n+1 h0 (n) g0 (n) = (−1)n h1 (n) Subband Image Coding for perfect reconstruction: g1 (n) = (−1)n+1 h0 (n) ˆ flp (n) = ￿ ￿ f (2n) ￿ h0 (2n) , 2n 0 , 2n + 1 g0 (n) = (−1)n h1 (n) ˆ fhp (n) = f (2n + 1) ￿ h1 (2n + 1) , 2n + 1 0 , 2n Subband Image Coding for perfect reconstruction: g1 (n) = (−1)n+1 h0 (n) ˆ flp (n) = ￿ ￿ f (2n) ￿ h0 (2n) , 2n 0 , 2n + 1 g0 (n) = (−1)n h1 (n) ˆ fhp (n) = f (2n + 1) ￿ h1 (2n + 1) , 2n + 1 0 , 2n ˆ f = f (2n) ￿ h0 (2n) ￿ g0 (2n) + f (2n + 1) ￿ h1 (2n + 1) ￿ g1 (2n + 1) Subband Image Coding for perfect reconstruction: g1 (n) = (−1)n+1 h0 (n) ˆ flp (n) = ￿ ￿ f (2n) ￿ h0 (2n) , 2n 0 , 2n + 1 g0 (n) = (−1)n h1 (n) ˆ fhp (n) = f (2n + 1) ￿ h1 (2n + 1) , 2n + 1 0 , 2n ˆ f = f (2n) ￿ h0 (2n) ￿ g0 (2n) + f (2n + 1) ￿ h1 (2n + 1) ￿ g1 (2n + 1) ˆ f = f (2n) ￿ h0 (2n) ￿ h1 (2n) + f (2n + 1) ￿ h1 (2n + 1) ￿ h0 (2n + 1) Subband Image Coding for perfect reconstruction: g1 (n) = (−1)n+1 h0 (n) ˆ flp (n) = ￿ ￿ f (2n) ￿ h0 (2n) , 2n 0 , 2n + 1 g0 (n) = (−1)n h1 (n) ˆ fhp (n) = f (2n + 1) ￿ h1 (2n + 1) , 2n + 1 0 , 2n ˆ f = f (2n) ￿ h0 (2n) ￿ g0 (2n) + f (2n + 1) ￿ h1 (2n + 1) ￿ g1 (2n + 1) ˆ f = f (2n) ￿ h0 (2n) ￿ h1 (2n) + f (2n + 1) ￿ h1 (2n + 1) ￿ h0 (2n + 1) ˆ f = f (n) ￿ [h0 (n) ￿ h1 (n)] Subband Image Coding for perfect reconstruction: g1 (n) = (−1)n+1 h0 (n) ˆ flp (n) = ￿ ￿ f (2n) ￿ h0 (2n) , 2n 0 , 2n + 1 g0 (n) = (−1)n h1 (n) ˆ fhp (n) = f (2n + 1) ￿ h1 (2n + 1) , 2n + 1 0 , 2n ˆ f = f (2n) ￿ h0 (2n) ￿ g0 (2n) + f (2n + 1) ￿ h1 (2n + 1) ￿ g1 (2n + 1) ˆ f = f (2n) ￿ h0 (2n) ￿ h1 (2n) + f (2n + 1) ￿ h1 (2n + 1) ￿ h0 (2n + 1) ˆ f = f (n) ￿ [h0 (n) ￿ h1 (n)] ˆ f = f (n) Vector Inner Product Given sequences f1 (n), f2 (n) ￿f1 , f2 ￿ = ￿ n ∗ f1 (n)f2 (n) Subband Image Coding g0 (n) = (−1) h1 (n) n g1 (n) = (−1)n+1 h0 (n) h0, h1, g0, g1 are biorthogonal ￿hi (2n − k ), gj (k )￿ = δ (i − j )δ (n), i, j = {0, 1} Subband Image Coding g0 (n) = (−1) h1 (n) n g1 (n) = (−1)n+1 h0 (n) h0, h1, g0, g1 are biorthogonal ￿hi (2n − k ), gj (k )￿ = δ (i − j )δ (n), i, j = {0, 1} Example: ￿h0 (2n − k), g1 (k)￿ = 0 Subband Image Coding g0 (n) = (−1) h1 (n) n g1 (n) = (−1)n+1 h0 (n) h0, h1, g0, g1 are orthonormal ￿hi (2n − k ), gj (k )￿ = δ (i − j )δ (n), i, j = {0, 1} + ￿gi (n), gj (n + 2m)￿ = δ (i − j )δ (m), i, j = {0, 1} Orthonormal Filter Bank Keven is the number of filter coefficients that must be even g1 (n) = (−1)n g0 (Keven − 1 − n) hi (n) = gi (Keven − 1 − n), i = {0, 1} Orthonormal filter bank can be obtained from a single filter -- prototype Example: Orthonormal Filter Bank g1 (n) = (−1)n g0 (Keven − 1 − n) hi (n) = gi (Keven − 1 − n), i = {0, 1} Example: Orthonormal Filter Bank g1 (n) = (−1)n g0 (Keven − 1 − n) hi (n) = gi (Keven − 1 − n), i = {0, 1} = g0 (n) Example: Orthonormal Filter Bank g1 (n) = (−1)n g0 (Keven − 1 − n) hi (n) = gi (Keven − 1 − n), i = {0, 1} = g0 (n) = g1 (n) Example: Orthonormal Filter Bank g1 (n) = (−1)n g0 (Keven − 1 − n) hi (n) = gi (Keven − 1 − n), i = {0, 1} = g0 (n) = h0 (n) = g1 (n) Example: Orthonormal Filter Bank g1 (n) = (−1)n g0 (Keven − 1 − n) hi (n) = gi (Keven − 1 − n), i = {0, 1} = g0 (n) = h0 (n) = h1 (n) = g1 (n) Next Class • • Sub-band Decomposition (Textbook 7.1.2) Haar Transform (Textbook 7.1.3) ...
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