Sharpening Enhancement of Digitized Mammograms with Complex Symmetric Daubechies Wavelets

{Some complex symmetric Daubechies wavelets provide a natural way to calculate zero-crossings because of a hidden "Laplacian operator" in the imaginary part of the scaling function. We propose a simple multiscale sharpening enhancement algorithm based on this property. The algorithm is tested on low-contrast digitized mammograms. Many breast cancers cannot be detected on the basis of mammographic images. This is partly due to the fact that mammograms are poor quality images with low-contrast resulting from the small diierences in X-ray attenuation between breast tissues. A very actual challenge is to improve the visual quality of mammograms by numerical processing in order to help in the early detection of cancer. The standard method to improve the local contrast of an image is to subtract the Laplacian of the signal from the original signal. Some very promising wavelet-based algorithms have been proposed that extend this method to allow multiscale processing using wavelet representations 1,2]. The wavelets used in these studies are of various kinds: discrete separable, discrete non-separable (hexagonal) or continuous. The aim of the present paper is to show that complex Symmetric Daubechies Wavelets (SDW) have also a great potential in such processing 3]. II. SYMMETRIC DAUBECHIES WAVELETS We recall that a Daubechies wavelet (x) is constructed from a (in general complex) scaling function '(x) = h(x) + ig(x), through the multiresolution analysis procedure 4]. In the construction, three constraints are imposed on the function '(x): compact support inside the interval ?J; J +1] (for some non-negative integer J), orthogonality of the discrete translates and regularity. The SDW are obtained by imposing an additional symmetry constraint on '(x). This constraint restricts J to Work supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada. even numbers and implies that '(x) and (x) are even and odd functions respectively (with respect to x = 1=2) 3]. SDW have many interesting properties for numerical simulations and image processing 5-7]. One of them is g(x) = d 2 h(x)=dx 2 (1) where is a real parameter (for example, =0.154 for J = 2 and =0.102 for J = 4). Thus, the imaginary part of '(x) inherits a Laplacian operator from the symmetry constraint. It has been shown that the approximation (1) is very accurate for the rst values of J and on most of the frequency range 0; ], where is the normalized Nyquist frequency (sampling steps are normalized to …