We consider the problem of reconstructing a discrete-time signal (sequence) with continuous-valued components corrupted by a known memoryless channel. When performance is measured using a per-symbol loss function satisfying mild regularity conditions, we develop a sequence of denoisers that, although independent of the distribution of the underlying “clean” sequence, is universally optimal in the limit of large sequence length. This sequence of denoisers is universal in the sense of performing as well as any sliding-window denoising scheme which may be optimized for the underlying clean signal. Our results are initially developed in a “semi-stochastic” setting, where the noiseless signal is an unknown individual sequence, and the only source of randomness is due to the channel noise. It is subsequently shown that in the fully stochastic setting, where the noiseless sequence is a stationary stochastic process, our schemes universally attain optimum performance. The proposed schemes draw from nonparametric density estimation techniques and are practically implementable. We demonstrate efficacy of the proposed schemes in denoising Gray-scale images in the conventional additive white Gaussian noise (AWGN) setting, with additional promising results for less conventional noise distributions.

We revisit the problem of denoising a discrete-time continuous-amplitude signal corrupted by a known memoryless channel. By modifying our earlier approach to the problem, we obtain schemes that are much more tractable than the original ones, while retaining their universal optimality properties. The schemes involve a simple preprocessing step of quantizing the noisy symbols to generate quantized contexts which (according to the quantized context value of each symbol) are then used to partition the unquantized symbols to subsequences. A universal context-free denoiser (of zero context length for the unquantized sequences) is then separately employed on each of the subsequences. We identify a rate in which the context length and quantization resolution should be increased so that the resulting scheme is universal in both the semi-stochastic and fully stochastic settings. The proposed family of schemes is computationally attractive, having linear complexity with a proportionality constant that is independent of the context length and the quantization resolution. Experimental results show that these schemes are not only superior from a computational viewpoint, but also achieve better denoising in practice.

We consider the problem of image denoising wherein the statistical characterization of the noise corruption mechanism is known. We make no assumptions on the nature or statistics of the underlying noise-free signal. A denoiser is proposed which, although ignorant of the statistical properties of the noise-free image, does essentially as well as a scheme with full knowledge of the statistics. The solution is presented as a sequence of schemes that progressively consider larger neighborhoods (contexts) around a pixel being denoised, to achieve optimum performance under a user-defined distortion measure.

We consider the problem of reconstructing a discrete-time continuous-amplitude signal corrupted by a known memoryless channel with a general output alphabet. We develop a sequence of denoisers that asymptotically achieve optimum performance in a semi-stochastic setting of an unknown individual noiseless signal, where the quality of reconstruction is measured with respect to a general given loss function satisfying mild conditions. We also extend this to the fully stochastic setting and show that our denoiser is asymptotically optimal for any stationary noiseless source. We conclude with some experimental validations of the proposed theory

This paper describes a uniform transform domain video codec where the motion estimation/compensation (ME) is performed in the transform domain. The estimation technique discussed here is a subpixel transform domain ME based on the dual tree complex wavelet transform (DT CWT) and a maximum phase correlation technique. The DT CWT is a multiresolution fine-to-coarse bandpass filtered decomposition of each still frame and has desirable properties of shiftability, directional selectivity and perfect reconstruction (PR). Estimation is first performed at the finest resolution and successively proceeds to the coarse resolutions using a fine-to-coarse strategy. This gives multiresolution motion estimates enabling estimation of current frame transform coefficients from the corresponding ones in the previous frame. The key difference of this approach is the transform domain error frames-a uniform transform domain video codec. This further simplifies the encoder and decoder resulting in computational savings with comparable performance to the standards.