# A New Method for 2D-Adapted Wavelet Construction:
## 1 Introduction
[32] build wavelets adapted to images of the human iris (called _irislet_). For this, they use the circular Hough transformation, the conversion to polar coordinates, the determination of the profile of the images of 1D lines, the discrete Daubechies transform, and the method of least squares. The results showed that the _irislet_ had a high recognition rate. [PERSON] et al. [33] proposed a design that uses a convolutional neural network to design a two-channel non-separable wavelet that fits a given image. The network loss function is the total squared error between the input image and the reconstructed image, leading to a perfect reconstruction at the end of the training. [PERSON] and [PERSON] [34] used the _lifting_ scheme to obtain separable wavelets adapted to compressed images. This wavelet provides better reconstruction results compared to other classical wavelets. The detection and reconstruction time were significantly reduced without causing degradation of reconstruction quality. [PERSON] [35] proposes a new filter learning method that considers the 2D TWD as a modified convolutional neural network, which is trained from the data via gradient descent. The learned wavelets are capable of capturing the structure of the training data (which were examples of classical wavelets) without relying on the DFT. According to this author, his model requires relatively few parameters compared to traditional networks. In the particular case of digital mammography images, breast lesions have specific shape characteristics, associated with the probability that they are malignant [1]. Therefore, wavelet prototypes have been built to recognize these lesions, in particular, microcalcifications [36]. As a result of the conceptual theoretical analysis of the research topic, an extension to 2D of the DST and its variants was not found, which would allow for the detection of patterns in images. For this reason, this research aims to numerically and computationally validate a strategy for obtaining filters for the construction of wavelets adapted to two-dimensional patterns in digital images. In this research, the 1D strategy proposed in [37] is extended to estimate 2D separable adapted wavelets, as well as a pattern detection algorithm based on the DST-II coefficients. A novel strategy based on the tensor product of one-dimensional vectors was proposed to perform a 2D DWT but with four new detection conditions to locate the 2D pattern. This was used to detect artificial patterns and mass-like abnormalities in digital mammographyimages. The similarity measure \(\mathbb{S}\) proposed in [22] was used as the detection algorithm, which is applied to the wavelet coefficients of the DST-II. The work is organized as follows: In Section 2, we present some elements of the discrete wavelet theory and a novel algorithm to obtain two-dimensional pattern-adapted wavelets based on the two-dimensional extension of the DST-II. The principles of the pattern detection algorithm are also explained. In Section 3, the performance of the proposed algorithm is shown through numerical experimentation, taking as examples an artificial image and a mammography image. Some remarks are presented in Section 4.