NOISE REMOVAL VIA BAYESIAN WAVELET CORINGEero

The classical solution to the noise removal problem is the Wiener lter, which utilizes the second-order statistics of the Fourier decomposition. Subband decomposi-tions of natural images have signiicantly non-Gaussian higher-order point statistics; these statistics capture image properties that elude Fourier-based techniques. We develop a Bayesian estimator that is a natural extension of the Wiener solution, and that exploits these higher-order statistics. The resulting nonlinear esti-mator performs a \coring" operation. We provide a simple model for the subband statistics, and use it to develop a semi-blind noise-removal algorithm based on a steerable wavelet pyramid. A common technique for noise reduction is known as \coring". An image signal is split into two or more bands; the highpass bands are subjected to a threshold non-linearity that suppresses low-amplitude values while retaining high-amplitude values. Use of such techniques is widespread: for example, most consumer VCR's use a simple one-dimensional coring technique. Many variants of coring have been developed, including two-dimensional coring 1], multi-scale oriented coring 2, 3], pyramid coring 4], and multi-band coring with orthogonal bases 5]. The nonlinear operator is often smoothed to give a \soft" threshold, but the exact choice of function in these techniques has been ad hoc. Similar techniques, based on the statistical concept of \shrinkage", have been recently used with wavelet expansions 6]. The intuition underlying coring is that images typically have spatial structure, consisting of smooth areas interspersed with occasional edges. This notion is evident statistically: the pixels in highpass and bandpass sub-bands of images have signiicantly non-Gaussian probability density functions (pdf's) that are sharply peaked at zero with broad tails. Speciically, the coeecient of kurtosis (fourth moment divided by squared variance) is typically well above the value of 3 that one expects for a Gaussian pdf. Field 7] has shown that kurtosis for subbands of natural scenes varies with lter bandwidth, and is maxi