Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

Directional Filter Banks 25-27suppressed version of the noisy original. Since the forward and inverse transforms amount to an identitysystem, it is evident that the noise suppression can be directly attributed to the nonlinear operationsperformed on the subbands.Typically, the nonlinearity is a shrinking or coring operation which takes a subband coefficient andmodifies its magnitude. Small coefficients tend to be suppressed and large coefficients maintain theirvalues. A commonly used operator for images is soft thresholding, whereby a subband coefficientx(n 0 , n 1 ) is modified to ^x(n 0 , n 1 ) ¼ sgn(x(n 0 , n 1 ))(jx(n 0 , n 1 )j T) when the coefficient magnitude isgreater than T, and the coefficient is set to zero otherwise. The value of the threshold is set explicitlywithin each subband or adapted individually to each coefficient based on some criterion, such as energyor statistical characteristics [46,47].This approach was first explored with maximally decimated filter banks and later with undecimatedtransforms. It turns out that better results are generally achieved with undecimated (shift-invariant)decompositions.Since the Bamberger pyramid, whose decomposition is shown in Figure 25.22, provides both radialand angular subband resolution, one might imagine that it would perform well in a denoising application[47,48]. In fact such is the case as shown by the comparisons in Figure 25.25. Bamberger pyramids canprovide better directional selectivity across resolutions along with shift invariance when the UDFB isused. Shown in Figure 25.25c is the denoising result for an undecimated Bamberger pyramid withfrequency plane partitioning similar to the Steerable pyramid [31]. The midband pyramid levels in thisparticular system are decomposed with an eight-band UDFB.For threshold selection, the spatially adaptive wavelet thresholding (SAWT) algorithm [46] was used,where a threshold is computed for each subband coefficient using local statistics under a Bayesianframework.(a)(b)(c)FIGURE 25.25 Denoising results using Lenna (a) Image with additive white Gaussian noise with s ¼ 22:5. (b)Denoised image using the UDWT. (c) Denoised image using Bamberger pyramids.