Polar factorization and monotone rearrangement of vector‐valued functions

Abstract

Given a probability space ( X , p ) and a bounded domain R in R d equipped with the Lebesgue measure 1 . I (normalized so that 10 I = I ), it is shown (under additional technical assumptions on X and Q) that for every vector-valued function u E L p ( X , p; R d ) there is a unique "polar factorization" u = V$.s, where $ is a convex function defined on R and s is a measure-preserving mapping from ( X , p ) into ( Q , I . I), provided that u is nondegenerate, in the sense that p ( u -' ( E ) ) = 0 for each Lebesgue negligible subset E of Rd.Through this result, the concepts of polar factorization of real matrices, Helmholtz decomposition of vector fields, and nondecreasing rearrangements of real-valued functions are unified.The Monge-Amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate "Monge-Kantorovich" problem.

Optimal Mass Transport for Registration and Warping

“…In this section, we will employ a natural solution to the optimal transport problem based on the equivalent problem of polar factorization; see [6], [14], [24] and the references therein. We will work with the general case of subdomains in i.e., we can decompose % § into the sum of a curl-free and divergence-free vector field [32].…”

The Monge–Ampère equation and its link to optimal transportation

“…The quadratic cost on R n . In [14,15], Brenier considered the case X = Y = R n and c(x, y) = |x−y| 2 /2, and proved the following theorem (which was also obtained independently by Cuesta-Albertos and Matrán [32] and by Rachev and Rüschendorf [99]). For an idea of the proof, see the sketch of the proof of Theorem 3.6 below, which includes this result as a special case.…”

Mass Preserving Mappings and Image Registration

“…The Monge-Kantorovich functional (2) is seen to place a penalty on the distance the mapũ moves each bit of material, weighted by the material's mass. A fundamental theoretical result [4,10], is that there is a unique optimalũ ∈ M P transporting µ 0 to µ 1 , and that thisũ is characterized as the gradient of a convex function w, i.e.,ũ = ∇w. This theory translates into a practical advantage, since it means that there are no non-global minima to stall our solution process.…”

“…Inspired by [4,9], we consider the family of MP mappings of the form u = u • s −1 as s varies over MP mappings from Ω 0 to itself, and try find an s which yields aũ without any curl, that is, such thatũ = ∇w. Once such an s is found, we will have the Monge-Kantorovich mappingũ.…”

“…Once such an s is found, we will have the Monge-Kantorovich mappingũ. We will also have u =ũ • s = (∇w) • s, known as the polar factorization of u with respect to µ 0 [4].…”