Noise Analysis of a New Singularity Index

We analyze the noise sensitivity of a new singularity index that was designed to detect impulse singularities in signals of arbitrary dimensionality while rejecting step-like singularities see Muralidhar, IEEE Signal Process. Lett., vol. 20, no. 1, pp. 7-10, 2013 and Muralidhar , Proc. IEEE Int. Conf. Image Process., 2012. For example, the index responds strongly to curvilinear masses (ridges) in images, while weakly to jump discontinuities (edges). We analyze the detection power of the index in the presence of noise. Our analysis is geared towards answering the following questions: a) in the presence of noise only, what is the probability of falsely detecting an impulse given a threshold; b) given an impulse submerged in noise, what is the probability of detecting it given a threshold; and c) since the index is designed to be edge suppressing, what is the probability of incorrectly detecting an edge submerged in noise given a threshold. We compare the detection power of the index with that of a nominal impulse detector, the second derivative operator. Simulations and example applications in 1-D and 2-D reveal the efficacy of the new singularity index for correctly detecting impulses submerged in noise, while suppressing edges. A software version of the 2-D singularity index can be downloaded from: http://live.ece.utexas.edu/research/SingularityIndex/SingularityIndexCode.zip.