# A New Method for 2D-Adapted Wavelet Construction:
## 1 Introduction  
Breast cancer is a highly prevalent disease in the female population worldwide. In Cuba, it ranks second in incidence and mortality (National Directorate of Medical Records and Health Statistics, 2020). The analysis of X-ray images of the breast, known as mammography, is the most popular and widely used breast cancer screening modality because the tumor can be detected before it can be physically felt [1]. Analyzing and interpreting mammograms requires practice, experience, and training because these images are inherently noisy and have poor contrast and lower sensitivity dense breast tissues [2] due to the low radiation doses used and shorter exposure time necessary for the safety of human health. To ensure consistent and accurate mammograms diagnosis, some radiologists use intelligent automated systems, but others still do it with the naked eye. There are a large number of different types of mammography abnormality. Some types of breastcancer are often associated with asymmetry between the right and left breasts [3; 4; 5], serving as early indications of the disease. In [3], a computer system solely based on computer vision and morphological features is developed for asymmetry detection in mammography images. This approach aims to support advances in Artificial Intelligence (AI) to complement existing methods for early breast cancer detection. The authors in this paper focused on morphological analysis and breast skin segmentation, which can serve as essential building blocks that could enhance current AI models. They have utilized dynamic time warping (DTW) for shape analysis and the growing seed region (GRS) method for breast skin segmentation. In [6], skin segmentation is also addressed using multi-class partition, which allows us to achieve a more refined delineation of skin regions. The segmentation could also be dealt with the new active contour model based on the level set method and Kullback-Leibler divergence presented in [7]. However, the most common breast abnormalities that may indicate breast cancer are masses and calcifications. Breast abnormalities are defined with a wide range of features and may be easily missed or misinterpreted by radiologists while reading large amounts of mammographic images provided in screening programs [8]. It is generally accepted that mass detection is a more challenging problem than the detection of micro-calcifications, not only due to the large variation in size and shape in which masses can appear in a mammogram but also because masses often exhibit poor image contrast (see [9; 10] and references therein). Mammogram features extraction is widely studied from the perspective of pattern recognition analysis to detect the best features that could represent a mammogram. The features of a digital image could be extracted directly from the spatial data or using a different space via special data transform, such as the Fourier transform or wavelets transform, which could be helpful in separating special data that contain specific scale-dependent characteristics, as is done in [11]. In this paper, an effective supervised classifier for mammograms is proposed using the discrete wavelet transform decomposition. The Daubechies-4, Daubechies-8, and Daubechies-16 wavelets are used in a four-level decomposition process. In each level of decomposition, a percentage of the low-frequency coefficients is used to represent the corresponding mammogram (i.e., feature vector). The classification is done by measuring the Euclidean distance between class prototype and the desired mammogram coefficient vector. The achieved numerical results indicate the effectiveness of the classifier in solving three basic problems in mammogram diagnosis: (i) classification of cancerous versus cancerous-free patterns; (ii) classification of abnormality indicator (i.e., microcalcifications, circumscribed masses, spiculated lesions, ill-defined lesions, and normal tissues); (iii) classification of the risk level of cancerous cells (i.e., benign versus malignant). The MIAS database is used.