Edge Detection in Medical Images Using the Wavelet Transform

Edge detection improves image readability and it is an important part of images preprocessing aimed to their segmentation and automatic recognition of their contents. This paper describes selected methods of edge detection in magnetic resonance images, with the emphasis on the wavelet transform use. The first part briefly describes the mathematical background of the wavelet transform, including its properties and application in image processing. Modulus Maxima Method by Stephane Mallat provides the method for edge detection using wavelet transform. This method is based on finding local maxima of horizontal and vertical wavelet coefficients in the first level of wavelet decomposition. It is supposed that this level represents edges. This method was tested with various wavelet functions both on simulated and real medical images. A complex wavelet function use could help to improve results of edge detection in real images. Presented paper contains a comparison of basic edge detection methods including simple gradient operators and Canny edge detector, and their combination with wavelet transform use. Mathematical principals were studied, as well as application of these methods. All algorithms were developed in the MATLAB environment using Wavelet and Image Processing Toolboxes. 1 2D Discrete Wavelet Transform 2D Discrete Wavelet Transform (2D DWT) [1, 6] is used in image processing as a powerful tool solving to image analysis, denoising, image segmentation and other. 2D DWT can be applied as a convolution of a selected wavelet function with an original image or it can be seen as a set of two matrices of filters, row and column one. Using a separability property of DWT, the first part of decomposition consists of an application of row filters to the original image. The column filter are used for further processing of image resulting from the first step. This image decomposition [1] can by mathematically described by Eq. (1) C = X · I ·Y (1) where C is the final matrix of wavelet coefficients, I represents an original image, X is a matrix of row filters and Y is a matrix of column filters. In the first level of decomposition of 2D DWT, the image is separated into four parts. Each of them has a quarter size of the original image [6]. They are called approximation coefficients (LowLow or LL), horizontal (LowHigh or LH), vertical (HighLow or HL) and detail coefficients (HighHigh or HH) [2, 6], see Fig.1. Approximation coefficients obtained in the first level can be used for the next decomposition level. Inverse 2D Discrete Wavelet Transform used in image reconstruction is defined by Eq. (2) Irec = X−1 ·C ·Y−1 (2) For the orthogonal matrices this formula can be simplified into Eq. (3) Irec = X ·C ·YT (3)