Geometry and Color in Natural Images

Abstract

Most image analysis algorithms are defined for the grey level channel, particularly when geometric information is looked for in the digital image. We propose an experimental procedure in order to decide whether this attitude is sound or not. We test the hypothesis that the essential geometric contents of an image is contained in its level lines. The set of all level lines, or topographic map, is a complete contrast invariant image description: it yields a line structure by far more complete than any edge description, since we can fully reconstruct the image from it, up to a local contrast change. We then design an algorithm constraining the color channels of a given image to have the same geometry (i.e. the same level lines) as the grey level. If the assumption that the essential geometrical information is contained in the grey level is sound, then this algorithm should not alter the colors of the image or its visual aspect. We display several experiments confirming this hypothesis. Conversely, we also show the effect of imposing the color of an image to the topographic map of another one: it results, in a striking way, in the dominance of grey level and the fading of a color deprived of its geometry. We finally give a mathematical proof that the algorithmic procedure is intrinsic, i.e. does not depend asymptotically upon the quantization mesh used for the topographic map. We also prove its contrast invariance.