Detection Segmentation Stage

The terms segmentation and detection may be confusing for the reader not so familiar with the medical imaging vernacular. In some instances these terms may be used interchangeably, but other times not. We might consider segmentation as being a more refined or specialized type of detection. For instance, we may gate a receiver for some time increment and make a decision as to whether or not a signal of interest was present within the total time duration, but not care about exactly where the signal is within the time window; this may be defined as a detection task with a binary output of yes or no. Segmentation takes this a step further. With respect to the image processing, the detection task makes a decision if the abnormality is present, which in this case is a calcification. If, in addition, the detection provides some reasonable estimate as to the spatial location and extent of the abnormality, then we would say that the calcification has been segmented. Thus, the segmentation process in mammography often results in a binary-labeled image with the probable calcification areas marked.

Before getting into the details of the techniques we implemented for the detection/segmentation stage of our CADiagnosis algorithm, a brief discussion of related bibliography is in order for the novice or beginner in the field to allow for a heads-up on study material of the suitable level. The list that follows is in no-way complete, or totally contemporary, but is comprised of useful citations (textbooks generally) that we have used extensively in our research and algorithm development.

Tolstov [30], Bracewell [31], andBrigham [32] are excellent sources for studying Fourier series and Fourier transforms—a prerequisite to understanding the wavelet transforms used in our approach. In particular, Brigham [32] provides a comprehensible treatment of the relations with the continuous Fourier transform, the discrete Fourier transform, and sampling theory. Similarly, Bracewell [33] gives a well-balanced treatment of standard imaging processing techniques.

Noise and filtering will be discussed in the following sections. Generally, the study of noise processes comes under many subject headings such as stochastic analysis, random signal analysis, or probability analysis. Again there are many diverse resources in this area and several provide many useful examples of random variable transformations and Fourier analysis of random signals [3437]. An excellent treatment of transforms and probability analysis applications is given by Giffin [38].

Wavelet analysis may be looked at from a simple filtering approach as well as from an elegant mathematical framework that involves understanding multiresolution functional spaces. Again, there is massive published work in this area. Strang and Nguyen [39], Akansu and Haddad [40], and Vetterli and Ko-vacevic [41] are excellent sources for understanding wavelets from a filtering approach, which also include the multiresolution framework. The seminal work in wavelet theory may be found in the more mathematically sophisticated work of Daubechies [42].

Finally, in the sections below, we will discuss how mammograms are associated with power spectra that obey an inverse power law. This characteristic is associated with self-similarity, fractals, and chaos. We are not aware of any traditional textbooks that address power laws specifically but the work of Peitgen et al. [43], Wornell [44], and Turner et al. [45] may be useful; Peitgen et al. [43] cover many types of phenomena, while Wornell [44] and Turner et al. [45] are specific to wavelet-based signal processing and 2-D image analysis, respectively. Note that the idea of self-similarity implies that things or events are invariant under a scale change. Wavelets have this property. Thus, it would seem natural to study self-similar noise fields (such as mammograms) with a self-similar analyzing transform (wavelets).

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