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MAY 14, 2007 MULTIMEDIA FRAMEWORK LAB YOON, DAE-IL

THE STEERABLE PYRAMID : A FLEXIBLE ARCHITECTURE FOR MULTI-SCALE DERIVATIVE COMPUTATION. MAY 14, 2007 MULTIMEDIA FRAMEWORK LAB YOON, DAE-IL. Introduction. Architecture for efficient and accurate linear decomposition of an image into scale and orientation subband

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MAY 14, 2007 MULTIMEDIA FRAMEWORK LAB YOON, DAE-IL

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

  2. Introduction • 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

  3. Motivation • 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

  4. The steerable pyramid(1/2) • 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 steerable pyramid(2/2) • 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. ANGULAR DECOMPOSITION • 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. RADIAL DECOMPOSITION(1/3) 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)

  8. RADIAL DECOMPOSITION(2/3)

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

  10. 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 • 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

  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

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