We study the problem related to the reconstruction of an image from a perceptual segmentation based on the geometric classification of its points. The classification of these points is accomplished by non-linear operators based on the curvatures of Monges patch associated to the image. A theoretical proof is presented that an image can be reconstructed solely from its points with non-zero Gaussian curva-tures. This result provides theoretical background to a method of non-linear two-dimensional signal processing proposed by C. Zetzche, E. Barth e B. Wegmann. Curvature operators are use to detect edges and vertices from images and we show that it is possible to reconstruct them from these elements.
We study the perceptual problem related to image quantization from an opti-mization point of view, using different metrics on the color space. A consequence of the results presented is that quantization using histogram equalization provides optimal perceptual results. This fact is well known and widely used but, to our knowledge, a proof has never appeared on the computer graphics literature.