# A New Method for 2D-Adapted Wavelet Construction:
## 1 Introduction
Daubechies-8 is the most successful choice of wavelet. A review of published techniques related to the detection and segmentation of mammography masses of any shape, margin, and size is presented in [9]. Detection is defined as the identification of potential lesions within all the parenchymal background. Usually, these methods generate a marker/prompt at a suspicious region in a mammogram. In contrast, segmentation is defined as a method able to detect the precise outline of the potential lesion. The quantitative performance of seven mass detection methods were compared: (i) based on a detection of concentric layers; (ii) based on a Laplacian edge detector approach; (iii) based on threshold; (iv) based on Iris filter; (v) based on a classifier approach; (vi) based on a Difference of Gaussian's; and (vii) based on a pattern matching approach, which starts by defining a template--in our case, a tumor-like template. The definition of the template is based on the approach of [12], who defined the tumor by three characteristics: brightness contrast, uniform density, and circular shape. They are compared using two different mammography databases: a public digitized database (MIAS) and a local full-field digital database (Trueta). The obtained results show a dependence on the breast density and on the lesion size and shape. The analysis of the FROC curves shows that algorithms based on the Laplacian, the Iris filter, the pattern matching, and the gradient analysis provide the best results, obtaining a sensitivity better than 0.8. However, the number for false positive per image is more than five at that sensitivity. None of the investigated approaches provide the overall best detection performance. The authors in [10] proposed a simple algorithm for mass enhancement and detection in mammographic images using wavelet processing and an adaptive threshold technique. The algorithm includes preprocessing, wavelet-based denoising, enhancement, and finally mass segmentation using adaptive thresholding. Thus, comparisons of the obtained results for mass detection have been carried out with ground truth provided by MIAS and DDSM databases, using the standard Jaccard similarity measure. The true positive fraction of 90.9 and 91% was achieved at the rate of a 2.35 and 2.1 average of false positive per image (FP/I) among 45 and 85 mammograms for MIAS and DDSM databases, respectively, with the proposed method. Our goal in this contribution is to use custom-built wavelet bases to detect mass-like abnormalities (mass-like pattern) in mammograms. The signal representation and analysis using basis functions such as wavelets has been a booming research topic for more than 20 years due to their attractive properties for various purposes [13; 14; 15]. Pattern recognition and detection belong to this field. Success in detecting patterns in a signal depends, to a great extent, on the wavelet basis used for the transform. The more the wavelet resembles the pattern at a certain position of the signal, the greater the value of the transform coefficient, and vice versa [16]. If the pattern to be detected is very different from the wavelet, then the detection will be less accurate. It is possible that a suitable wavelet cannot be found among the known options for certain applications. Although there are various types of wavelets in the literature, there is no universal rule that determines which wavelet is most suitable for each application. Instead, the researcher is expected to try different wavelets, more or less arbitrarily, in order to find the right one [17]. So, it is natural to try to build a new wavelet that is suited to the problem. This requires efficient numerical algorithms where the wavelet is as similar as possible to the pattern given a priori and allows for an accurate detection of said pattern in a signal. This approach is called adapted wavelet, matched wavelet, or data-driven wavelet [18; 19; 20; 21]. Taking a close approach to the Daubechies discrete wavelet transform (DWT), [PERSON] et al.