Introduction to the Curvelet Transform By Zvi Devir and Yanai Shpinner

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.

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:

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.

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

Sub-band Decomposition Dividing the image into resolution layers. Each layer contains details of different frequencies: P0 – Low-pass filter. 1, 2, … – Band-pass (high-pass) filters. The original image can be reconstructed from the sub-bands: Energy preservation

Sub-band Decomposition

Sub-band Decomposition Low-pass filter 0 deals with low frequencies near ||1. Band-pass filters 2s deals with frequencies near domain ||[22s, 22s+2]. Recursive construction – 2s(x) = 24s (22sx). The sub-band decomposition is simply applying a convolution operator:

Sub-band Decomposition The sub-band decomposition can be approximated using the well known wavelet transform: Using wavelet transform, f is decomposed into S0, D1, D2, D3, etc. P0 f is partially constructed from S0 and D1, and may include also D2 and D3. s f is constructed from D2s and D2s+1.

Sub-band Decomposition P0 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.

Smooth Partitioning

Smooth Partitioning A grid of dyadic squares is defined: Qs – 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, wQ is a displacement of w localized near Q. Multiplying s f with wQ (QQs) produces a smooth dissection of the function into ‘squares’.

Smooth Partitioning The windowing function w is a nonnegative smooth function. Partition of the energy: The energy of certain pixel (x1,x2) 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.

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

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

Before the Ridgelet Transform The s f layer contains objects with frequencies near domain ||[22s, 22s+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  length2. We would like to encode those ridges efficiently Using the Ridgelet Transform.

The Ridgelet Transform Ridgelet are an orthonormal set {} for L2(2). Developed by Candès and Donoho (1998). Ridgelet Tiling radius 2s 2s divisions Ridge in Square 2-s 2-2s Ridge in Square 1 2-s It’s Fourier Transform It’s Fourier Transform 2s Ridgelet Tiling Fourier Transform within Tiling Divides the frequency domain to dyadic coronae ||[2s, 2s+1]. In the angular direction, samples the s-th corona at least 2s times. In the radial direction, samples using local wavelets.

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, 2i-1–1] is the angular location. j,k are Meyer wavelets for . j is the ridgelet scale and k is the ridgelet location.

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 ||[2s, 2s+1]. A ridge fragment needs only a very few ridgelet coefficients to represent it.

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…

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

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

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

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

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

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 http://www-stat.stanford.edu/~jstarck/comp.html