# A New Method for 2D-Adapted Wavelet Construction:
## 4 Conclusions  
A novel strategy was proposed to design a 2D-adapted wavelet that allows for the locating of a pattern within a digital image. The proposed strategy consisted of using the equation of the diagonal coefficients (which depend only on the high pass filter) corresponding to the tensor product to build 2D wavelet bases. A 1D filter was then estimated such that conditions of finite energy, vanished moments, orthogonality, and four new detection conditions are met. These ensure that when performing 2D DWT, the filter detects the pattern by taking the diagonal coefficients with values of the measure \(\mathbb{S}(\alpha=0.1)>0.7\). The positions of these coefficients are used to estimate the position of the pattern in the original image. To find the aforementioned filter, the already-validated 1D strategy is used. The 2D strategy requires that in order to guarantee the convergence of the numerical iteration, the filters do not have a support size greater than 26 samples. For pattern sizes larger than 26 samples, convergence is not achieved with any of the numerical algorithms used using the convergence criterion considered. This is in agreement with the experimental (not theoretical) results of [PERSON] [53]. This limit on the pattern size to achieve convergence of the numerical methods would surely change if better hardware or a parallel computing scheme were used, which will be considered in future research. This is happening because, as the size of the system increases, the convergence of the numerical method may become more difficult or require more iterations to reach an accurate solution. High-dimensional systems of non-linear equations can have multiple solutions, which makes the convergence of the numerical method more difficult. The computational complexity increases with the number of variables in the system, which can lead to numerical stability problems and rounding errors. On the other hand, the detection conditions of the proposed 2D extension are stiff in the sense that these conditions seek a perfect match between the shapelet filter and the pattern to be detected. This fact is difficult to occur in the case of mammographic images, given the high self-similarity of the regions in the breast tissue, the slow variations of pixel intensity in these images and the presence of fuzzy edges. This opens the door for further research, where fuzzy constraints can be used to model such correspondence or a probabilistic and Bayesian approach. The presented results show that the proposal allows for the identification of regions of the image between which the pattern is found, which would help specialists, after effective validation, to focus their gaze on those regions instead of looking at the whole image, with the consequent visual fatigue and time delay that this causes. The results would need to be refined and evaluated on a larger number of digital mammography images annotated by medical specialists. This would allow for the proposal to be used in the medium to long term as a feature vector for training machine learning models in lesion classification and malignancy, given the importance of early diagnosis of the disease. Practical benefits can be found in other less-complex domains with respect to the nature of the image, such as the detection of characters in documents or small dimensional patterns such as microcalcifications in digital mammograms. This work has shown the possibility of using DST-II to localize shapes and masses in digital mammograms. There have been experiments that support our starting hypothesis. However, we have not performed systematic experimentation to determine the best threshold and the best alpha value associated with the normalized similarity measure. The use of statistical learning techniques to determine such parameters could help to improve the specificity values. High sensitivity and positive predictive value in detection were achieved but not high specificity or negative predictive value, contrary to what occurred in the 1D strategy.