Design Of The Orthogonal Filter In Laplacian Pyramid Structure

In recent years, Laplacian Pyramid structure has been widely applied in the field of signal processing as a multi-scale, multi-resolution tool. The Contourlet transform based on this structure is very practical in the field of image processing. In Contourlet transform, Laplacian Pyramid structure acted as the part of extracting the low-frequency information is very important. And the repeated extraction of low-frequency information make the Contourlet transform as a multi-scale, multi-resolution tool. However, in the conventional methods, the filter within the Laplacian Pyramid structure is always obtained from the filter bank, which leads some good properties of the filter to be omitted, such as orthogonality and linear-phase.The orthogonal property and linear-phase of filters are very important. In any linear transformation, the best basis function is the orthogonal one. When the filter is orthogonal, the signal is approximated best. And the advantage of the linear-phase is shown in the proccess of video and image. The linear-phase filter makes the image decomposed into the sub-band signal without phase distortion.Aiming at these problems, this paper proposes a method of designing the Laplacian Pyramid filter individually, with which the filter can get rid of the constraints caused by constructing filter banks and be designed with linear-phase and orthogonal property. This paper studies the orthogonality of the filter and gives the condition of the orthogonal filter in the direction of matrix transformation, and proposes the individual design conditions of the orthogonal filter in Laplacian Pyramid structure. Under these conditions, the filter can also be of linear-phase and its bandwidth and length can be adjusted flexibly. That is to say, the filter whose bandwidth is π/3, π/4 or narrower can be designed with both orthogonality and linear-phase. Experiments show that the designed filter make the single channel be a perfect reconstruction subsysterm with the characteristics of orthogonality and linear-phase. For the application of image processing, any eligible band information can be extracted directly. As an example, in this paper, we take the nonlinear approximation experiments with the filter which can extract the low-frequency of π/4. Compared with the other methods where the existing filters are employed, the proposed method with the designed filter can be used to improve the PSNR of the reconstructed image obviously, leading to a better ability of image representation.