# A New Method for 2D-Adapted Wavelet Construction:
## 1 Introduction
[22; 23; 24] present three versions of the so-called siscrete _Shapelet_ transform (DST), a transform created not only to determine the support in time with the frequencies of a signal but also to detect a pattern in it. The DST is built to detect this pattern regardless of its amplitude and presents the same time complexity as any DWT when decomposing a signal for the same support size. The DST is capable of adjusting to a predefined pattern but not to a set of patterns like a machine learning algorithm would. The DST-I [23] uses [PERSON] original approach to build wavelets, but from the time domain, imposing constraints on the high-pass filter coefficients such as orthogonality, unit energy, and a certain number of vanished moments. To adjust the wavelet to fit the desired pattern, a constraint based on the fractal dimension of the pattern is added. Then, the system of nonlinear equations (NLES) formed by these restrictions is solved. Despite the success achieved by the adapted wavelet following this algorithm in the pattern detection of the complex visual system of a fly, its computational cost was relatively high [23]. The DST-II [22] arises as a modification to the DST-I, which decreases its computational cost, favors the solution of the NLES, and facilitates a simplified interpretation of the transformed signal. This replaces the fractality constraint in detecting the pattern with two detection constraints based on the correlation between the pattern and the wavelet filter, which were defined in such a way that the scalar products involved in the calculation of the DST-II, according to the algorithm of [PERSON], capture the target pattern. The inclusion of these two restrictions allows the NLES to be squared. On the other hand, the design of wavelets to detect specific patterns defined a priori is also interesting for the 2D case. [PERSON] and [PERSON] [25] developed a technique to derive a band-limited orthonormal wavelet from the spectrum of the desired signal so that the root mean square error between its spectra would be minimal. The work of [PERSON] and [PERSON] [26]used correlation for edge detection, building in the frequency domain a bank of multiple wavelets at different scales that are optimal for ramp and pulse edges. [PERSON] et al. [27] propose the extraction of regions with text from an image using adapted wavelet filters and texture features in the image. Unlike other methods, this one does not require a priori information about the font, geometric transformation, distortion or, texture of the background. The results for various images verify that the proposal is robust, versatile, and effective for extracting text from images. [PERSON] [28] uses the _lifting_ scheme to fit 2D wavelets in the three possible directions of a separable approach. Experiments were carried out with an artificial pattern. The author mentions that the diagonal _lifting_ is the most complicated one to formalize mathematically, and its estimation is more cumbersome than the one carried out in the horizontal and vertical directions. [PERSON] and [PERSON] [29] take advantage of the transient capture capability and directional filtering of wavelets to improve the quality of an image. To do this, they estimate wavelets to optimize the Michelson contrast measure in the spatial domain. Thus, they are able to detect objects in infrared video streams with fewer false alarms than the standard detection approaches. [PERSON] and [PERSON] [30] design image-adapted wavelet bases for denoising. They carried out the experiments in natural, satellite, and medical images, obtaining an improvement in the images with respect to other classical wavelets. [PERSON] and [PERSON] [31] estimate fitted wavelets to images that are not separable from two channels in the statistical sense. The filter bank is estimated by minimizing the root mean square error. The experiments showed that non-separable matched wavelets provided better coding gain compared to classical wavelets. [PERSON] et al.