MAY 14, 2007 MULTIMEDIA FRAMEWORK LAB YOON, DAE-IL THE STEERABLE PYRAMID : A FLEXIBLE ARCHITECTURE FOR MULTI- SCALE DERIVATIVE COMPUTATION

2  Architecture for efficient and accurate linear decomposition of an image into scale and orientation subband  Directional derivative operators of any desired order  Differential algorithms are used in a wide variety of image processing problems  Another widespread tool in signal and image processing is multi-scale decomposition Introduction

3  Many authors have combined multi-scale decompositions with differential measurements  A multi-scale pyramid is constructed, and then differential operators are applied to the subbands of the pyramid  Since both the pyramid decomposition and the derivative operation are linear and shift-invariant  Combine them into a single operation  Advantage  More accurate  Propose a simple, efficient decomposition architecture for combining these two operations Motivation

4  The latest incarnation of “steerable pyramid”  The scale tuning of the filters is constrained by a recursive system diagram  The orientation tuning is constrained by the property of steerability  Designed to be “self-inverting”  Essentially aliasing-free  Most imprtantly, the pyramid can be designed to produce and number of orientation bans, k  The resulting transform will be overcomplete by a factor 4k/3

5  The spectral decomposition performed by a steerable pyramid with k=4  Frequency axes range from - to  Related by translation, dilations and rotation  The Constraints on the two components A(θ) and B(θ)

6  The angular portion of the decomposition, A(θ), is determined by the desired derivative order  A directional derivative operation in the spatial domain corresponds to multiplication by a linear ramp function in the Fourier domain  Rewrite in polar coordinate  Higher-order directional derivatives correspond to multiplication in the Fourier domain by the ramp raised to a power, and thus the angular portion of the filter is for an N th-orther directional derivative

7  The radial function, B(ω), is constrained by both the desire to build the decomposition recursively, and the need to prevent aliasing from occurring during subsampling operations  The filters and are necessary for preprocessing the image in preparation for the recursion  The subsystem decomposes a signal into two portions (lowpass and highpass) RADIAL DECOMPOSITION(1/3)

8 RADIAL DECOMPOSITION(2/3)

9 is strictly bandlimited, and B(ω) is power-complementary RADIAL DECOMPOSITION(3/3)

10  A design with a single band at each scale (k=1) serves as a (self-inverting) replacement for the Laplacian pyramid  A design with two bands (k=2) will compute multi-scale image gradients  computations of local orientation, stereo disparity or optical flow IMPLEMENTATION(1/2)  A 3-level k = 1 (non-oriented) steerable pyramid  Image coding  Self-inverting, and thus the erros introduce by quantization of the subbands will not appear as low frequency distortions upon reconstruction

11 IMPLEMENTATION(2/2)  A 3-level k = 3 (second derivative) steerable pyramid  Can be used for orientation analysis, edge detection, etc  Image enhancement, orientation decomposition and junction identification texture blending, depth-from-stereo, and optical flow