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In highly computationally intensive fields such as signal processing, image processing, computer graphics and visualization, much of the CPU time is spent in computing various transforms on the typically large data sets which may also contain noise. Though extensive work has been reported on the de-noising and subsequent compression of the data, little of it has been reported on the de-noising and subsequent compression of the operators. Moreover, in the work reported so far, thresholding, which is essential for achieving denoising, has not been based on a specific criterion. In the present work, we propose modifications to this approach which result in significant savings in the computational cost of the associated transformations. We present wavelet based approaches to compress different operators. We present two methods to accomplish this. In the first method, we first apply a non-standard wavelet transform on the operator represented in its matrix form. This step is followed by an adaptive thresholding scheme of Donohoe and Johnstone, which results in the de-noised form of the operator. In the second method, the original matrix representation of the operator is split into two sparse diagonal dominant matrices, one in the spatial domain and the other in the wavelet domain. Although, there is a need to use the original signal in the spatial domain, the resulting decomposition actually requires only a portion of the operators. More importantly, the decomposition results in representations with very little total error. We find that the computational complexity of the transformation, using these methods, reduces to O(N2) (where N is the size of the data vector) observed with using the original, denser representation of the operator. In particular, Method 2 allows even more expedient processing of signals with greater accuracy. Hence, many transformations with operators can be represented in a diagonally dominant matrix form resulting in significant savings.
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