Impact of scale on signed, unsigned, authentic and phantom zero crossings in a noisy environment

Abstract

Scale-space description of an image refers to the descriptions of the same image at different resolutions. One popular scale-space representation, the topic of this research, is obtained by passing an image through a bank of smoothing filters (each tuned to a different scale) and detecting the zero crossings (z.c.s) of the second derivative (Laplacian) of the outputs. Any z.c. based algorithm will be affected by the corruption of the z.c.s due to the input noise. Statistical analysis of the z.c.s is used to determine the effect of scale change on the different types of z.c.s. We also study the computational and performance trade-offs involved in choosing scale. Statistical analysis of the z.c.s is nontrivial. The 2 main reasons are: (i) detection of z.c.s is a nonlinear operation, and (ii) the noise at different scales is correlated. In this paper we identify different types of z.c.s and compute the densities of their occurrence in the presence of white and colored noise. The formulae for computing the z.c. densities are applicable to any smoothing filter. The special case of Gaussian smoothing filters is investigated in depth. It is demonstrated how thea priori knowledge about the input image is reflected in the performance of an algorithm which uses the z.c.s as the primitives. Other applications of the statistical analysis of z.c.s include design of optimum smoothing filter, performance analysis and active multiscaling. It is quantitatively demonstrated how the performance of a scale-space system improves as more and morea priori knowledge about the scene is incorporated.