Jean-Michel Morel

Formalization and computational aspects of image analysis

In this article we shall present a unified and axiomatized view of several theories and algorithms of image multiscale analysis (and low level vision) which have been developed in the past twenty years. We shall show that under reasonable invariance and assumptions, all image (and shape) analyses can be reduced to a single partial differential equation. In the same way, movie analysis leads to a single parabolic differential equation. We discuss some applications to image segmentation and movie restoration. The experiments show how accurate and invariant the numerical schemes must be and we compare several (old and new) algorithms by discussing how well they match the axiomatic invariance requirements.

Morphological Approach to Multiscale Analysis: From Principles to Equations

A numerical image can be modelized as a real function I 0 (x) defined in ℝN (In practice, N = 2 or 3). The main concept of vision theory and image analysis is multiscale analysis (or “scale space”). Multiscale analysis associates with I(0) = I 0 is a sequence of simplified (smoothed) images I(t, x) which depend upon an abstract parameter t > 0, the scale. The image I(t, x) is called analysis of the image I 0 at scale t. The formalization of scale-space has received a lot of attention in the past ten years; more than a dozen of theories for image, shape or “texture” multiscale analysis have been proposed and recent mathematical work has permitted a formalization of the whole field. We shall see that a few formal principles (or axioms) are enough to characterize and unify these theories and algorithms and show that some of them simply are equivalent. Those principles are causality (a concept in vision theory which can be led back to a maximum principle), the Euclidean (and/or affine) invariance, which means that image analysis does not depend upon the distance and orientation in space of the analysed image, and the morphological invariance which means that image analysis does not depend upon a contrast change.

Runge property and multiplicity of solutions for elliptic equations in ℝN with a monotone nonlinearity

In this paper, we describe a method for extending (in some approximated sense) solutions of a nonlinear P.D.E. on a domain Ω, to solutions in a domain Ω′ containing Ω. Such an extension property, the Runge property, is well known for a large class of linear problems including elliptic equations. We prove here the Runge property for semilinear problems of the kind -Δu+g(u)=f, with f ∈ Lloc1(ℝN). (As a consequence, we get infinitely many solutions for these problems). The proof is based on a “homotopy method”, and requires a refinement of the linear results: We prove that the “Runge extension” v on Ω′ of a solution u in Ω for a linear elliptic equation Lu=f can be choosen in order to depend continuously on u and the coefficients of L.

4.15 – Shape Smoothing and PDEs

One of the aims of Computer Vision in the past thirty years has been to recognize shapes with numerical algorithms. In this chapter, we describe five curve smoothing algorithms, of growing sophistication and invariance. We give a detailed implementation and link these algorithms to the curve evolution PDEs they implement. We let the five algorithms undergo a practical invariance testing. The tested invariance requirements face no less than five classes of perturbations, namely noise, geometric or affine distortions, contrast changes, occlusion and figure/background reversal. Quite contrary to the main stream idea that curve evolution schemes should be implemented by level set methods, we describe very fast and accurate direct curve evolution implementations. We give precise bibliographical links to the mathematical and image analysis literature justifying these curve evolution algorithms and their relationship to mathematical morphology, scale space theory, classical filtering. We finally give some hints on the role of shape smoothing in actual shape recognition systems.