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Introduction to the Curvelet Transform By Zvi Devir and Yanai Shpinner

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Introduction Curvelet Transform is a new multi-scale representation most suitable for objects with curves. Developed by Candès and Donoho (1999). Still not fully matured. Seems promising, however.

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Approximation Rates Having an object in the domain [0,1] [0,1], how ‘fast’ can we approximate it using certain system of functions? Using the Fourier Transform: Using the Wavelet Transform: Using the Curvelet Transform:

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Point and Curve Discontinuities A discontinuity point affects all the Fourier coefficients in the domain. Hence the FT doesn’t handle points discontinuities well. Using wavelets, it affects only a limited number of coefficients. Hence the WT handles point discontinuities well. Discontinuities across a simple curve affect all the wavelets coefficients on the curve. Hence the WT doesn’t handle curves discontinuities well. Curvelets are designed to handle curves using only a small number of coefficients. Hence the CvT handles curve discontinuities well.

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Curvelet Transform The Curvelet Transform includes four stages: Sub-band decomposition Smooth partitioning Renormalization Ridgelet analysis

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Sub-band Decomposition Dividing the image into resolution layers. Each layer contains details of different frequencies: P 0 – Low-pass filter. 1, 2, … – Band-pass (high-pass) filters. The original image can be reconstructed from the sub-bands: Energy preservation

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Sub-band Decomposition

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Low-pass filter 0 deals with low frequencies near | | 1. Band-pass filters 2s deals with frequencies near domain | | [2 2s, 2 2s+2 ]. Recursive construction – 2s (x) = 2 4s (2 2s x). The sub-band decomposition is simply applying a convolution operator:

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Sub-band Decomposition The sub-band decomposition can be approximated using the well known wavelet transform: Using wavelet transform, f is decomposed into S 0, D 1, D 2, D 3, etc. P 0 f is partially constructed from S 0 and D 1, and may include also D 2 and D 3. s f is constructed from D 2s and D 2s+1.

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Sub-band Decomposition P 0 f is “smooth” (low-pass), and can be efficiently represented using wavelet base. The discontinuity curves effect the high-pass layers s f. Can they be represented efficiently? Looking at a small fragment of the curve, it appears as a relatively straight ridge. We will dissect the layer into small partitions.

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Smooth Partitioning

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A grid of dyadic squares is defined: Q s – all the dyadic squares of the grid. Let w be a smooth windowing function with ‘main’ support of size 2 -s 2 -s. For each square, w Q is a displacement of w localized near Q. Multiplying s f with w Q ( Q Q s ) produces a smooth dissection of the function into ‘squares’.

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Smooth Partitioning The windowing function w is a nonnegative smooth function. Partition of the energy: The energy of certain pixel (x 1,x 2 ) is divided between all sampling windows of the grid. Example: An indicator of the dyadic square (but not smooth!!). Smooth window function with an extended compact support: Expands the number of coefficients.

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Smooth Partitioning Partition of the energy: Reconstruction: Parserval relation:

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Renormalization Renormalization is centering each dyadic square to the unit square [0,1] [0,1]. For each Q, the operator T Q is defined as: Each square is renormalized:

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Before the Ridgelet Transform The s f layer contains objects with frequencies near domain | | [2 2s, 2 2s+2 ]. We expect to find ridges with width 2 -2s. Windowing creates ridges of width 2 -2s and length 2 -s. The renormalized ridges has an aspect ratio of width length 2. We would like to encode those ridges efficiently Using the Ridgelet Transform.

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The Ridgelet Transform Ridgelet are an orthonormal set { } for L 2 ( 2 ). Developed by Candès and Donoho (1998). Ridge in Square 2-s2-s 2 -2s Ridge in Square 1 2-s2-s It’s Fourier Transform 2s2s Ridgelet Tiling radius 2 s 2 s divisions It’s Fourier TransformRidgelet Tiling Divides the frequency domain to dyadic coronae | | [2 s, 2 s+1 ]. In the angular direction, samples the s -th corona at least 2 s times. In the radial direction, samples using local wavelets. Fourier Transform within Tiling

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The Ridgelet Transform The ridgelet element has a formula in the frequency domain: where, i,l are periodic wavelets for [- , ). i is the angular scale and l [0, 2 i-1 –1] is the angular location. j,k are Meyer wavelets for . j is the ridgelet scale and k is the ridgelet location.

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Ridgelet Analysis Each normalized square is analyzed in the ridgelet system: The ridge fragment has an aspect ratio of 2 -2s 2 -s. After the renormalization, it has localized frequency in band | | [2 s, 2 s+1 ]. A ridge fragment needs only a very few ridgelet coefficients to represent it.

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Digital Ridgelet Transform ( DRT ) Unfortunately, the (current) DRT is not truly orthonormal. An array of n n elements cannot be fully reconstructed from n n coefficients. The DRT uses n 2n coefficients for almost perfect reconstruction Still a lot of research need to be done…

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Curvelet Transform The four stages of the Curvelet Transform were: Sub-band decomposition Smooth partitioning Renormalization Ridgelet analysis

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Image Reconstruction The Inverse of the Curvelet Transform: Ridgelet Synthesis Renormalization Smooth Integration Sub-band Recomposition

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Example: Roy Lichtenstein: “In The Car” 1963 Original Image (256 256) Approximation with only 64 wavelets and 256 curvelets (about 0.5% of the coefficients)

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Original Adding Gaussian Noise WT + Thresholding WT + k- Thresholding Curvelet Transform Example: Noise Reduction using Curvelet transform.

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Original Adding Gaussian Noise WT + Thresholding WT + k- Thresholding Curvelet Transform Example: Noise Reduction using Curvelet transform.

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References [1]D.L. Donoho and M.R. Duncan. Digital Curvelet Transform: Strategy, Implementation and Experiments; Technical Report, Stanford University 1999 [2]E.J. Candès and D.L. Donoho. Curvelets – A Surprisingly Effective Non-adaptive Representation for Objects with Edges; Curve and Surface Fitting: Saint Malo 1999 [3]Lenna examples from

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