# Admin stuff. 4 Image Pyramids. Spatial Domain. Projects. Fourier domain 2/26/2008. Fourier as a change of basis

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1 Admin stuff 4 Image Pyramids Change of office hours on Wed 4 th April Mon 3 st March 9.3.3pm (right after class) Change of time/date t of last class Currently Mon 5 th May What about Thursday 8 th May? Time to pick! Projects Basis functions: Spatial Domain Every group must come and see my in the next couple of weeks during office hours!.. Tells you where things are. but no concept of what it is Fourier domain Basis functions: Tells you what is in the image. but not where it is Fourier as a change of basis Discrete Fourier Transform: just a big matrix But a smart matrix!

2 Low pass filteringhttp:// High pass filteringhttp:// Image Analysis Why Pyramid? Want representation that combines what and where. Image Pyramids.equivalent to. Keep filters same size Change image size Scale factor of 2 Practical uses Compression Capture important structures with fewer bytes Denoising Model statistics i of pyramid sub bands b Image blending Total number of pixels in pyramid? + ¼ + /6 + /32.. = 4/3 Over complete representation 2

3 Gaussian Laplacian Wavelet/QMF Steerable pyramid Image pyramids The computational advantage of pyramids Sampling without smoothing. Top row shows the images, sampled at every second pixel to get the next; bottom row shows the magnitude spectrum of these images. Slide credit: W.T. Freeman 3

4 Sampling with smoothing. Top row shows the images. We get the next image by smoothing the image with a Gaussian with sigma pixel, then sampling at every second pixel to get the next; bottom row shows the magnitude spectrum of these images. Sampling with more smoothing. Top row shows the images. We get the next image by smoothing the image with a Gaussian with sigma.4 pixels, then sampling at every second pixel to get the next; bottom row shows the magnitude spectrum of these images. Slide credit: W.T. Freeman Slide credit: W.T. Freeman D Convolution as a matrix operation x f = C f x where f = (f_ f_n) and C = ( f_n f_(n-) f_(n-2) f_.. f_n f_(n-) f_2 f_. f_n f_(n-). f_2 f_) Size of C is x - f + by x 2D Convolution as a matrix operation X g = C g X(:) where g = (g_ g_n g_2 g_2n g_m. g_mn) Size of X is I x J Size C g is IJ MN + by IJ (for valid convolution) Convolution and subsampling as a matrix multiply (-d case) Next pyramid level For 6 pixel -D image U2 = U = 6 pixels 8 pixels 8 pixels Im_ Im_2 Im_3.. Im_6 4 pixels

5 >> U2 * U ans = b * a, the combined effect of the two pyramid levels Im_ Im_2 Im_3.. Im_6 Gaussian Laplacian Wavelet/QMF Steerable pyramid Image pyramids Gaussian Laplacian Wavelet/QMF Steerable pyramid Image pyramids The Laplacian Pyramid Synthesis preserve difference between upsampled Gaussian pyramid level and Gaussian pyramid level band pass filter - each level represents spatial frequencies (largely) unrepresented at other levels Analysis reconstruct Gaussian pyramid, take top layer Laplacian pyramid algorithm

6 Why use these representations? Handle real-world size variations with a constant-size vision algorithm. Remove noise Analyze texturet Recognize objects Label image features Efficient search Image Blending 6

7 Feathering Affect of Window Size + = Encoding transparency I(x,y) = (αr, αg, αb, α) I blend = I left + I right left right Affect of Window Size Good Window Size Optimal Window: smooth but not ghosted What is the Optimal Window? To avoid seams window >= size of largest prominent feature To avoid ghosting window <= 2*size of smallest prominent feature Natural to cast this in the Fourier domain largest frequency <= 2*size of smallest frequency image frequency content should occupy one octave (power of two) FFT What if the Frequency Spread is Wide FFT Idea (Burt and Adelson) Compute F left = FFT(I left ), F right = FFT(I right ) Decompose Fourier image into octaves (bands) F left = F left + F left2 + Feather corresponding octaves F i left with F i right Can compute inverse FFT and feather in spatial domain Sum feathered octave images in frequency domain Better implemented in spatial domain 7

8 Pyramid Blending Left pyramid blend Right pyramid Pyramid Blending laplacian level 4 laplacian level 2 laplacian level left pyramid right pyramid blended pyramid Laplacian Pyramid: Region Blending General Approach:. Build Laplacian pyramids LA and LB from images A and B 2. Build a Gaussian pyramid GR from selected region R 3. Form a combined pyramid LS from LA and LB using nodes of GR as weights: LS(i,j) = GR(I,j,)*LA(I,j) + (-GR(I,j))*LB(I,j) 4. Collapse the LS pyramid to get the final blended image Blending Regions 8

9 Horror Photo Simplification: Two-band Blending Brown & Lowe, 23 Only use two bands: high freq. and low freq. Blends low freq. smoothly Blend high freq. with no smoothing: use binary mask david dmartin (Boston College) 2-band Blending Linear Blending Low frequency (λ > 2 pixels) High frequency (λ < 2 pixels) 2-band Blending Spatial Gaussian pyramid Fourier Laplacian pyramid Fourier Spatial 9

10 Image pyramids Wavelets/QMF s Gaussian Laplacian Wavelet/Quadrature Mirror Filters (QMF) Steerable pyramid transformed image r r F = Uf Vectorized image Fourier transform, or Wavelet transform, or Steerable pyramid transform Orthogonal wavelets (e.g. QMF s) Forward / Analysis Inverse / Synthesis r r F = Uf f r = V F T r The simplest orthogonal wavelet transform: the Haar transform U = - UV T = I Haar basis is special case of Quadrature Mirror Filter family The inverse transform for the Haar wavelet Apply this over multiple spatial positions >> inv(u) U = ans =

11 The high frequencies The low frequencies U = U = Simoncelli and Adelson, in Subband coding, Kluwer, 99. The inverse transform >> inv(u) ans = Simoncelli and Adelson, in Subband coding, Kluwer, 99. Now, in 2 dimensions Horizontal high pass Frequency domain Horizontal low pass Slide credit: W. Freeman

12 Apply the wavelet transform separable in both dimensions Both diagonals Simoncelli and Adelson, in Subband coding, Kluwer, 99. To create 2-d filters, apply the -d filters separably in the two spatial dimensions Horizontal high pass, vertical high pass Horizontal high pass, vertical low-pass Horizontal low pass, vertical high-pass Horizontal low pass, Slide credit: W. Vertical low-pass Freeman Basis Wavelet/QMF representation Simoncelli and Adelson, in Subband coding, Kluwer, 99. Simoncelli and Adelson, in Subband coding, Kluwer, 99. Some other QMF s 9-tap QMF: Better localized li in frequency Good and bad features of wavelet/qmf filters Bad: Aliased subbands Non-oriented diagonal subband Good: Not overcomplete (so same number of coefficients as image pixels). Good for image compression (JPEG 2) 2

13 Compression: JPEG 2 Compression: JPEG Gaussian Laplacian Wavelet/QMF Steerable pyramid Image pyramids Steerable filters Analyze image with oriented filters Avoid preferred orientation Said differently: We want to be able to compute the response to an arbitrary orientation from the response to a few basis filters By linear combination Notion of steerability Steerable basis filters Filters can measure local orientation direction and strength and phase at any orientation. G2 H2 Steerability examples 3

14 Reprinted from Shiftable MultiScale Transforms, by Simoncelli et al., IEEE Transactions on Information Theory, 992, copyright 992, IEEE Fourier construction Slice Fourier domain Concentric rings for different scales Slices for orientation Feather cutoff to make steerable Tradeoff steerable/orthogonal But we need to get rid of the corner regions before starting the recursive circular filtering Simoncelli and Freeman, ICIP 995 Non-oriented steerable pyramid 3-orientation steerable pyramid 4

15 Steerable pyramids Good: Oriented subbands Non-aliased subbands Steerable filters Bad: Overcomplete Have one high frequency residual subband, required in order to form a circular region of analysis in frequency from a square region of support in frequency. Simoncelli and Freeman, ICIP 995 Application: Denoising Application: Denoising Usually zero, sometimes big Usually close to zero, very rarely big How to characterize the difference between the images? How do we use the differences to clean up the image? Application: Denoising Application: Denoising Coring function: Original Noise-corrupted Wiener filter Steerable pyramid coring 5

16 Summary of pyramid representations Gaussian Laplacian Image pyramids Progressively blurred and subsampled versions of the image. Adds scale invariance to fixed-size algorithms. Shows the information added in Gaussian pyramid at each spatial scale. Useful for noise reduction & coding. Wavelet/QMF Bandpassed representation, complete, but with aliasing and some non-oriented subbands. Steerable pyramid Shows components at each scale and orientation separately. Non-aliased subbands. Good for texture and feature analysis. Fourier transform = * Fourier transform Fourier bases are global: each transform coefficient depends on all pixel locations. pixel domain image Slide credit: W. Freeman Gaussian pyramid Laplacian pyramid = * = * Gaussian pyramid pixel image Laplacian pyramid pixel image Overcomplete representation. Low-pass filters, sampled appropriately for their blur. Slide credit: W. Freeman Overcomplete representation. Transformed pixels represent bandpassed image information. Slide credit: W. Freeman 6

17 Wavelet (QMF) transform Steerable pyramid Wavelet pyramid = * Ortho-normal transform (like Fourier transform), but with localized basis functions. pixel image Slide credit: W. Freeman Steerable pyramid Multiple orientations at one scale = * Multiple orientations at the next scale the next scale pixel image Over-complete representation, but non-aliased subbands. Slide credit: W. Freeman Matlab resources for pyramids (with tutorial) Matlab resources for pyramids (with tutorial) Ted Adelson (MIT) Bill Freeman (MIT) 7

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