Wavelet and Curvelet Transforms for Biomedical Image Processing

Wavelet and Curvelet Transforms for Biomedical Image Processing

Manas Saha (Siliguri Institute of Technology, India), Mrinal Kanti Naskar (Jadavpur University, India) and B. N. Chatterji (B. P. Poddar Institute of Management and Technology, India)
DOI: 10.4018/978-1-5225-5152-2.ch006
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This chapter introduces two mathematical transforms—wavelet and curvelet—in the field of biomedical imaging. Presenting the theoretical background with relevant properties, the applications of the two transforms are presented. The biomedical applications include heart sound analysis, electrocardiography (ECG) characterization, positron emission tomography (PET) image analysis, medical image compression, mammogram enhancement, magnetic resonance imaging (MRI) and computer tomography (CT) image denoising, diabetic retinopathy detection. The applications emphasize the development of algorithms to diagnose human diseases, thereby rendering fast and reliable support to the medical personnel. The transforms—one classical (wavelet) and another contemporary (curvelet)—are selected to focus the difference in architecture, limitation, evolution, and application of individual transform. Two joint applications are addressed to compare their performance. This survey is also supplemented by a case study: mammogram denoising using wavelet and curvelet transforms with the underlying algorithms.
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2. Wavelet Transform: Background

Wavelet is a wave of finite duration having a mean value of zero. The continuous wavelet transform (CWT) is used for signal analysis. Its partially discrete version, that is, wavelet series and its completely discrete version, that is, discrete wavelet transform are applied for signal coding, image compression and computer vision related tasks [37]. Unlike Fourier transform, its way of localizing information in time-frequency plane is unique. It has the ability to trade one type of resolution for the other. This particular feature makes it befitting for the investigation of non-stationary signals. Fortunately, majority of the biomedical images or signals are dynamic in nature. The application of wavelet in biomedical signal and image processing is very much significant.

Let, 978-1-5225-5152-2.ch006.m01 be a time varying signal. Wavelet transforms engage in evaluating coefficients which are inner products of the given signal and a family of “wavelets”. At a given scale 978-1-5225-5152-2.ch006.m02 with time location 978-1-5225-5152-2.ch006.m03, a CWT can be written from [37] as

(1) where 978-1-5225-5152-2.ch006.m05 denotes “mother” wavelet.978-1-5225-5152-2.ch006.m06 is considered as a band pass function and 978-1-5225-5152-2.ch006.m07 denotes energy preservation. There are different approaches to discretize time-scale parameters 978-1-5225-5152-2.ch006.m08. Individual approach generates a dissimilar kind of wavelet.

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