A Steerable, Multiscale Singularity Index

Abstract

Abstract-We propose a new steerable, multi-scale ratio index for detecting impulse singularities in signals of arbitrary dimensionality. For example, it responds strongly to curvilinear masses (ridges) in images, but minimally to step discontinuities. The ratio index employs directional derivatives of gaussians, making it naturally steerable and scalable. Experiments on real images demonstrate the efficacy of the index for detecting multi-scale curvilinear structures. A software version of the index can be downloaded from: http://live.ece.utexas.edu/research/ SingularityIndex/SingularityIndex.zip.

Noise Analysis of a New Singularity Index

“…We recently designed a new index to detect impulse singularities in signals of arbitrary dimensionality [1], [2]. It is defined as the dimensionless ratio (1) where , is a 1-D signal, and are the first and second derivatives of , respectively, and the constant .…”

“…It is defined as the dimensionless ratio (1) where , is a 1-D signal, and are the first and second derivatives of , respectively, and the constant . As explained in [1], we assume the nominal value .…”

“…It is defined as the dimensionless ratio (1) where , is a 1-D signal, and are the first and second derivatives of , respectively, and the constant . As explained in [1], we assume the nominal value . Hence (2) The singularity index (2) is closely related to an energy operator developed by Teager and Kaiser [11], which has been employed for demodulating AM-FM signals [12].…”

“…As detailed in [1], a principled way to accomplish this uses a gaussian lowpass debiasing filter , where . This choice of the filter space constant ensures that the response to a gaussian impulse will be reduced by no more than a fraction .…”

“…This is not practical in real applications since derivative computations are not stable in the presence of noise. Hence, define the smoothed singularity index (3) 1053-587X © 2013 IEEE where is the shorthand notation for used in [1] and is a unit-area gaussian:…”