On a variational theory of image amodal completion

Simon Masnou^{(1)} Jean-Michel Morel^{(2)}

April 26, 2005

1 Introduction

Digital images can be represented as grey level functions u(x, y) defined on a simple open subset of ℝ^{2} (usually a rectangle), which we call “image domain”. Of course, digital images are given as a discrete set of samples, but there are standard interpolation methods to get back to a smooth image, e.g. a trigonometric polynomial by Shannon interpolation (also called zero-padding ^{\underline{EdenYaroslavski}} [?]). There is no substantial difference between digital images and what we know of retinal images as rough data: in both cases, images are band-limited by an optical device and then sampled on a grid. So most questions in visual perception theory are easily translated into “computer vision” problems. This opens the way to a mathematical formalization and numerical experiments.

In this paper, we shall deal with the counterpart in image processing of the “amodal completion” phenomenon that arises in human vision. This phenomenon has been widely studied by the phenomenologist Gaetano Kanizsa ^{\underline{Kanizsa}} [?], who tried to give a consistent answer to one of the major enigmas of visual perception. Its understanding starts with the straightforward observation that the objects that we see in all day life are partially occulting each other, so that we only see parts of them. Georges Matheron ^{\underline{Matheron}} [?] actually proved that, under a simple and realistic stochastic model of object occultation, the so called “dead leaves model”, we only see half of the objects in sight. To be more explicit, in any all day life image or photograph, whenever we distinguish some object, we only see, on the average, half of it. Mathematically precise versions of this statement can be found in the aforementioned book and in ^{\underline{GousseauPhD, Gousseau}} [?, ?]. So we only see (significant) pieces of all shapes we perceive, but these pieces change constantly according to our position with respect to all objects present in the scene. Our perception, however, does not even notice this problem: we perceive objects as though they were complete. The mechanism of this visual illusion was formalized by Kanizsa who formulated two geometric laws, under the names of “amodal completion” and “good continuation”.

The amodal completion principle applies when a perceived curve stops on another perceived curve, entailing the perception of a “T-junction”. The stopped curve is the leg of the T and the other curve is represented by the horizontal bar of the T. In such a situation, our perception tends to interpret the interrupted curve as the boundary of some object undergoing an occlusion. The leg of the T is then mentally extrapolated and, whenever possible, connected to another leg in front. This fact is illustrated in figure ^{\underline{fig-completion-amodale}} ?? and is called “amodal completion”. The connection of two T-legs in front obeys the “good continuation” principle, according to which the reconstructed amodal curve must be similar to the pieces of curve it interpolates (same direction, curvature, etc.)

(1)Lab. J.-L. Lions, Univ. Pierre et Marie Curie, B.C. 187, 75252 Paris Cedex 05, France, masnou@ann.jussieu.fr (2)CMLA, ENS Cachan, 61 Av. du Prsident Wilson, 94235 Cachan, France, Jean-Michel.Morel@cmla.ens-cachan.fr

Figure 1: T-junctions entail an amodal completion and a completely different interpretation of the

image.  fig-completion-amodale