Compensated convexity and its applications

Abstract

We introduce the notions of lower and upper quadratic compensated convex transforms C l 2,λ (f) and C u 2,λ (f) respectively and the mixed transforms by composition of these transforms for a given function f : R n → R and for possibly large λ > 0. We study general properties of such transforms, including the so-called 'tight' approximation of C l 2,λ (f) to f as λ → +∞ and compare our transforms with the well-known Moreau-Yosida regularization (Moreau envelope) and the Lasry-Lions regularization. We also study analytic and geometric properties for both the quadratic lower transform C l 2,λ (dist 2 (x, K)) of the squared-distance function to a compact set K and the quadratic upper transform C u 2,λ (f) for any convex function f of at most quadratic growth. We show that both C l 2,λ (dist 2 (x, K)) and C u 2,λ (f) are C 1,1 approximations of the original functions for large λ > 0 and C u 2,λ (f) remains convex. Explicitly calculated examples of quadratic transforms are given, including the lower transform of squared distance function to a finite set and upper transform for some non-smooth convex functions in mathematical programming.

Compensated convexity and Hausdorff stable geometric singularity extractions

“…Let us first recall the notions of quadratic compensated convex transforms defined in Ref. 48. We will consider functions f : R n → R meeting one of the following conditions for x ∈ R n (L) : f (x) ≥ −A(1 + |x| 2 ), (U) : f (x) ≤ A(1 + |x| 2 ), (B) : |f (x)| ≤ A(1 + |x| 2 ) , (1.1) for some constant A ≥ 0.…”

“…If f : R n → R satisfies (B) in (1.1), then the two quadratic mixed compensated convex transforms 48 (mixed transforms for short) for given λ > A and τ > A are defined respectively by C From definition (1.2), it also follows that a C l λ (f )(x) is the envelope of all the quadratic functions with fixed quadratic term λ|x| 2 that are less than or equal to f , that is, C l λ (f )(x) = sup −λ|x| 2 + (x) : −λ|y| 2 + (y) ≤ f (y) for all y ∈ R n and affine , (1.4) whereas from (1.3) it follows that C u λ (f )(x) is the envelope of all the quadratic functions with fixed quadratic term λ|x| 2 that are greater than or equal to f , that is, C u λ (f )(x) = inf λ|x| 2 + (x) : f (y) ≤ λ|y| 2 + (y) for all y ∈ R n and affine .…”

“…Later we simply refer to (L), (U) and (B) for these restrictions. Due to the locality property of compensated convex transforms 48 and our practical concerns for applications to image and data processing, we in fact only need the function f to be bounded, as data arrays have finite number of entries and we can always extend a function defined in a rectangular domain to the whole space by a constant value zero outside the domain.…”

“…48 and apply this theory to geometric singularity extraction problems for singularities such as ridges, valleys, edges for general functions, images and geometric objects given by their characteristic functions in the Euclidean space R n . These geometric singularity extraction problems arise, for example, from signal, image and data processing, and computational geometry and computer-aided geometric design.…”

“…Suppose f : R n → R satisfies (U). Then the quadratic upper compensated convex transform 48 (upper transform for short) for a given λ > A is defined by…”