So, we can write
From these equations some conclusions can be extracted. First, the projection of a signal f in a space Vj gives a new signal Pjf, an approximation of the initial signal. Secondly, we have a hierarchy of spaces, then Pj_i f will be a better approximation (more reliable) than Pjf. Since Vj_i can be divided in two subspaces Vj and Wj, if Vj is an approximation space then Wj, which is the complementary orthonormal space, it is the detail space. The less the j, the finer the details.
Vj = Vj+i e Wj+i = Vj+2 e Wj+2 = ■■■ = vl e wl e WL_i e ■■■ e Wj+i
This can be viewed as a decomposition tree (see Fig. 2.10). At the top-left side of the image the approximation can be seen, and surrounding it the successive details. The further the detail is located the finer the information provided. So, the details at the bottom and at the right side of the image have information about the finer details and the smallest structures of the image decomposed. Therefore, we have a feature vector composed by the different detail approaches and the approximation for each of the pixels.
Gabor filters represent another multiresolution technique that relies on scale and direction of the contours [25,37]. The Gabor filter consists of a two-dimensional sinusoidal plane wave of a certain orientation and frequency that is modulated in amplitude by a two-dimensional Gaussian envelope. The spatial representation of the Gabor filter is as follows:
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