ICCV 2007 National Laboratory of Pattern Recognition Institute of Automation Chinese Academy of Sciences Half Quadratic Analysis for Mean Shift: with Extension to A Sequential Data Mode-Seeking Method Xiaotong Yuan, Stan Z. Li October 17, 2007

ICCV 2007 Outline Motivation Theoretical Exploration Algorithm Extension Summary

ICCV 2007 Motivation Put Mean Shift on proper grounds Better understand the essence Facilitate the numerical study Fast multiple data mode-seeking Improve exhaustive initialization based method

ICCV 2007 Mean Shift as Half Quadratic Optimization

ICCV 2007 Background of Mean Shift Mean Shift: A fixed point iteration algorithm to find the local maximum of Prior weights Mahalanobis Distance kernel function

ICCV 2007 Prior Understanding Gradient ascent algorithm with an adaptive step size [Cheng 1995] Quadratic bounding optimization [Fashing et al. 2005] Gaussian Mean shift is an EM algorithm [Carreira-Perpinan 2007]

ICCV 2007 Half Quadratic Optimization Theory of convex conjugated functions Non-quadratic convex objective functions is optimized in a quadratic-like way Convergence property is deeply studied [Mila 2005, Allain 2006]

ICCV 2007 Preliminary Facts All the conditions we impose on kernel are summarized as below:

ICCV 2007 HQ Optimization For KDE The supermum is reached at quadratic term conjugated term dual variable

ICCV 2007 Alternate Maximization A new objective function on extended domain: Equivalent to Mean-Shift

ICCV 2007 Relation to Bound Optimization Quadratic bounding optimization [ Fashing et al ] HQ formulation: for a fixed point, denote analytically defines a quadratic lower bound for at each time stamp

ICCV 2007 Relation to EM Gaussian Mean shift is an EM algorithm [Carreira-Perpinan 2007] HQ optimization: the alternate maximization scheme is equivalent to E-Step and M-Step

ICCV 2007 Convergence Rate When is isotropic, the root-convergence of convex kernel mean shift is at least linear with rate

ICCV 2007 Adaptive Mean Shift for Sequential Data Mode-Seeking

ICCV 2007 Multiple Mode-Seeking Exhaustive Initialization Run the Mean Shift in parallel

ICCV 2007 Adaptive Mean Shift Basic idea Properly estimate the starting points near the significant modes Sequentially re-weight the samples to guide the search

ICCV 2007 Algorithm Description Initialization: Repeat - Starting point estimation: - Local Mode estimation: - Sample prior re-weight: Until ever-found mode reappears Global mode estimation by Annealed MS [Shen 2007] By traditional mean shift

ICCV 2007 Four iterations only! Prior weight curve Illustration

ICCV 2007 Advantages Initialization invariant Highly efficient with linear complexity is the number of significant modes

ICCV 2007 Numerical Test: Image Segmentation

ICCV 2007 Color transformation color surfaces under “canonical” illuminant Application: Color Constancy Linear render model [ Manduchi 2006 ] Test Image containing the color surfaces with illumination change Gaussian distribution Problem: How to estimate possible existing ?

ICCV 2007 Existing Method MAP formulation [ Manduchi 2006 ]: Possible existing are estimated with EM algorithm Limitation: number of render vectors should be known a prior All the render vectors should be properly initialized

ICCV 2007 Convex Kernel based Solution Compensation accuracy Ada-MS for sequential modes seeking Prior weight: pixel i on training surface j

ICCV 2007 Results Ground Truth Mapping Compensated image Prior weight images

ICCV 2007 Summary Put the mean-shift on a proper ground: HQ optimization Connection with previous viewpoints: bounding optimization and EM Fast multiple data modes seeking (linear complexity and initialization invariant) Very simple to implement We hope to see more applications