Adaptive Discriminant Wavelet Packet Transform (ADWPT)

Since the invention of best-basis selection algorithm by Coifman and Wicker- hauser [119] many new algorithms have been proposed for the selection of a wavelet basis. Some of these will be discussed in the next section. Different opti- mal bases have been computed for various applications. In this thesis, we present a new algorithm for wavelet packet basis selection designed to obtain a wavelet packet basis that is optimal for differentiating between signals and textures being analysed.

Our novel algorithm, the Adaptive Discriminant Wavelet Packet Transform (ADWPT), seeks to obtain an optimal wavelet representation by optimising the discrimination power of the wavelet-bases. The ADWPT as a result obtains the optimal wavelet basis that best differentiates between signals under study. To define the ADWPT algorithmically, we define a discriminant function ¯D which may be represented as

2.4 Wavelet Packets for Texture Analysis

¯

D(Υ,B) (2.20) where Υ represents the training data-set which in this instance comprises of the four meningioma subtypes included in this study and B represents the wavelet packet bases. The objective of our computation would be to find a wavelet packet decomposition B∗ _{that has the maximum discriminant function ¯D representing}

the overall discrimination power of a best basis B∗_.

B∗ _{= argmax}

B

¯

D(Υ,B) (2.21) The above equation represents the computation function over a set of all possi- ble wavelet packet trees to compute the wavelet packet representation that best differentiates between the different meningioma textures. The ADWPT thus ob- tained contains the subbands that are most discriminant. Hence, the aim is to obtain a set of subbands that collectively have the highest discrimination power. Various stages in the computation of the ADWPT are described in the next chap- ter. Next, we present an overview of the various wavelet packets based techniques used for texture analysis. The literature on meningioma classification and the use of wavelets in histological analysis was reviewed in Chapter 1.

2.4

Wavelet Packets for Texture Analysis

Over the last few decades, a large body of research work has been conducted in the area of texture analysis. Zhang and Tan [120] presented a review of various tex- ture analysis methods. Various mechanisms and techniques have been employed ranging from statistical approaches to transforms and model based methods. En- tirely new paradigms have been envisaged and efforts have been made to mimic the process of human perception in machines. Hence, the techniques employed for texture discrimination and classification are varied and bring together knowl- edge from various domains and fields of study. In this thesis, we are mainly concerned with techniques pertaining to the Wavelet Packet Transform (WPT).

2.4 Wavelet Packets for Texture Analysis

In this section, the aim is to present a synopsis of the work done in texture anal- ysis by employing wavelet packet transform and the various best bases selection techniques that have been used.

Many subband selection algorithms for wavelet packets have been presented. Some researchers perform subband selection whereas others have chosen to use all the subbands obtained in a wavelet packet decomposition. Many criteria have been used for subband selection, since the entropy based criterion proposed by Coifman and Wickerhauser [119]. Chang and Kuo [121] used energy as the criterion for selection of subbands by comparing subbands on the same scale and decomposing those subbands that have comparatively high energy. Pun and Lee [122] chose subbands that have high energy and achieve rotation and scale invariance using log polar wavelet signatures. Chen and Kundu [123] used wavelet packets and Hidden Markov Models (HMMs) for rotation and gray-scale invariant texture identification for Brodatz textures. The lowest level subbands in the wavelet packet tree were used to extract features. Laine and Fan [124] also performed Brodatz texture classification using wavelet packets but chose to select subbands arbitrarily by selecting all subbands from levels 1 to 3, only level 3, level 2 to 3 and only level 4, producing comparative results. Senguret al. [125] combined a multi-layer perceptron classifier with wavelet packets for Brodatz texture classification. A 2-level wavelet packet decomposition was obtained for various textures in the Brodatz album and all subbands were used for the analysis. The concept of wavelet frames was introduced by Unser [126] which was ex- tended by Kim and Kang [127] to wavelet packets. Kim and Kang [127] used wavelet packet frames for Brodatz texture classification and segmentation. The frames with the greatest variance were selected producing high accuracies. Ra- jpoot [128] employed the local discriminant wavelet packet bases for subband se- lection which involved determining the discrimination power of a subband using various distance functions, namely Kullback-Leibler divergence, Jensen Shannon, Euclidean and Hellinger distance, for subband selection. The principle is that the higher the discrimination power the better the subband. A comparison is made between the parent and child nodes in the wavelet packet tree. Bhalerao and Rajpoot later extended the work in [129] to show that the discrimination power is an important measure that can indicate the usefulness of a subband in

2.4 Wavelet Packets for Texture Analysis

differentiating between various textures. They add Bhattacharya distance to the set of distance functions and show that it performs better in most instances.

Some recent work on wavelet packet subband selection has been carried out by Huang and Aviyente [130, 131]. They propose a new mechanism for selec- tion of subbands based upon mutual information amongst the various subbands. Two algorithms are presented, namely Mutual Information based Subband Se- lection (MISS) and Subband Grouping and Selection (SGS). Their algorithms are shown to perform better than the traditional subband selection paradigms described above. The techniques aim to compare all subbands at the same time based upon their energy values. The aim is to obtain a sparse representation whose coefficients are as independent as possible. This technique has been shown to produce promising results in comparison with other techniques for subband selection but their method compares low-pass subbands directly with high-pass subbands which may be flawed, since the frequency information they represent is very different. Moreover, the technique may result in selection of the parent subband with the children which would not have information exactly indepen- dent of the parents, a criterion which the technique aims to maximize i.e. mutual independence.

Some comparisons between wavelet packets and other techniques have also been carried out. An elaborate comparison of various filters namely Law masks, ring/wedge filters, Gabor filters, wavelet transforms, wavelet packets and wavelet frames, quadrature mirror filters, discrete cosine transform, linear predictors and finite impulse response filters was performed by Randen and Husoy [132]. The textural data used for the analysis was obtained from the Brodatz album, the MIT Vision Texture database and the MeasTex Image Texture Database. The results showed that no one filter performs better for all data sets and the study was important in the sense that the test and training data is completely separated with no overlap. This is very much the case in our study for meningioma subtype classification.

Arivazhagan et al. [133] uses Gabor wavelets for rotation invariant texture classification of Brodatz textures and compares the results with those of discrete wavelet transforms, indicating that Gabor wavelets perform better. Livens et al.