Abstract. We introduce two new methods of deriving the classical PCA in the framework of minimizing the mean square error upon performing a lower-dimensional approximation of the data. These methods are based on two forms of the mean square error function. One of the novelties of the presented methods is that the commonly employed process of subtraction of the mean of the data becomes part of the solution of the optimization problem and not a pre-analysis heuristic. We also derive the optimal basis and the minimum error of approximation in this framework and demonstrate the elegance of our solution in comparison with an existing solution in the framework.

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-In this paper, we propose the Fukunaga-Koontz Transform (FKT) as applied to Small-Sample Size (SSS) problems and formulate a feature scatter matrix based equivalent of the FKT. We establish the classical Linear Discriminant Analysis (LDA) analogy of the FKT and apply it to a SSS situation. We demonstrate the significant computational savings and robustness associated with our approach using a multi-class face detection problem.Keywords -Fukunaga-Koontz Transform, Linear Discriminant Analysis, Karhunen-Loève Transform, Feature Scatter Matrix I IntroductionIn high-dimensional signal processing applications such as state-space analysis, image and array processing, etc. it is necessary to analyse the data in a low dimensional subspace. Algorithms based on the Karhunen-Loève Transform (KLT) or eigen analysis are widely used in subspace signal processing, e.g. PCA, MUSIC, ESPRIT etc. These algorithms are computationally efficient in generating a low-dimensional eigen subspace from the available signal samples to which new signal samples could be projected to process them quickly and efficiently. a) The KLT in Two-Class ProblemsIn signal processing and pattern recognition, we often encounter the two-class problem where it is necessary to separate or classify the two classes of either the signal, noise or noise corrupted signal from one another. The KLT as used in two-class problems take the following general form: using the samples from one of the two classes C 1 or C 2 , an eigen subspace is generated which is expected to capture almost all the variability of that particular class of samples, say C 1 . A new signal belonging to C 1 is statistically expected to fit well into this eigen subspace. In many problems such as face recognition, direction-of-arrival estimation, etc., the signal belonging to C 1 would respond very differently from that of C 2 enabling good signal classification performance. The KLT does not tell where in the eigen subspace created using the samples from C 1 , would the projections of samples from C 2 might fall. As a result, in many cases when there is considerable overlap between the projection of samples from C 1 and C 2 in the low-dimensional eigen subspace, the KLT-based signal classification might fail. b) Organisation of the PaperIn this paper we analyse the Fukunaga-Koontz Transform (FKT) which uses the KLT to generate a shared eigenspace for both C 1 and C 2 where their principal eigen subspaces are orthogonal complements of each other. Since the principal eigen subspaces generated using FKT do not overlap, it could be used for effective classification in many difficult two-class problems. Throughout signal and image processing literature, we can see numerous successful applications of FKT. In this paper, our emphasis is on the small sample size (SSS) problems which are situations where the dimensionality of the signal is larger than the number of signal samples. Such situations are very common in image processing where it is often not possible to get, say for e.g., over ten t...

This chapter presents a comprehensive scheme for automated detection of colorectal polyps in computed tomography colonography (CTC) with particular emphasis on robust learning algorithms that differentiate polyps from non-polyp shapes. The authors’ automated CTC scheme introduces two orientation independent features which encode the shape characteristics that aid in classification of polyps and non-polyps with high accuracy, low false positive rate, and low computations making the scheme suitable for colorectal cancer screening initiatives. Experiments using state-of-the-art machine learning algorithms viz., lazy learning, support vector machines, and naïve Bayes classifiers reveal the robustness of the two features in detecting polyps at 100% sensitivity for polyps with diameter greater than 10 mm while attaining total low false positive rates, respectively, of 3.05, 3.47 and 0.71 per CTC dataset at specificities above 99% when tested on 58 CTC datasets. The results were validated using colonoscopy reports provided by expert radiologists.

This chapter presents a comprehensive scheme for automated detection of colorectal polyps in computed tomography colonography (CTC) with particular emphasis on robust learning algorithms that differentiate polyps from non-polyp shapes. The authors’ automated CTC scheme introduces two orientation independent features which encode the shape characteristics that aid in classification of polyps and non-polyps with high accuracy, low false positive rate, and low computations making the scheme suitable for colorectal cancer screening initiatives. Experiments using state-of-the-art machine learning algorithms viz., lazy learning, support vector machines, and naïve Bayes classifiers reveal the robustness of the two features in detecting polyps at 100% sensitivity for polyps with diameter greater than 10 mm while attaining total low false positive rates, respectively, of 3.05, 3.47 and 0.71 per CTC dataset at specificities above 99% when tested on 58 CTC datasets. The results were validated using colonoscopy reports provided by expert radiologists.