000130323 001__ 130323
000130323 005__ 20181007231108.0
000130323 0247_ $$2doi$$a10.1109/TIP.2006.877390
000130323 02470 $$2DAR$$a9073
000130323 02470 $$2ISI$$a000239774800014
000130323 037__ $$aARTICLE
000130323 245__ $$aPolyharmonic Smoothing Splines and the Multidimensional Wiener Filtering of Fractal-Like Signals
000130323 269__ $$a2006
000130323 260__ $$bIEEE$$c2006
000130323 336__ $$aJournal Articles
000130323 520__ $$9eng$$a Motivated by the fractal-like behavior of natural images, we develop a smoothing technique that uses a regularization functional which is a fractional iterate of the Laplacian. This type of functional was initially introduced by Duchon for the approximation of nonuniformily sampled, multidimensional data. He proved that the general solution is a smoothing spline that is represented by a linear combination of radial basis functions (RBFs). Unfortunately, this is tedious to implement for images because of the poor conditioning of RBFs and their lack of decay. Here, we present a much more efficient method for the special case of a uniform grid. The key idea is to express Duchon's solution in a fractional polyharmonic B-spline basis that spans the same space as the RBFs. This allows us to derive an algorithm where the smoothing is performed by filtering in the Fourier domain. Next we prove that the above smoothing spline can be optimally tuned to provide the MMSE estimation of a fractional Brownian field corrupted by white noise. This is a strong result that not only yields the best linear filter (Wiener solution), but also the optimal interpolation space, which is not bandlimited. It also suggests a way of using the noisy data to identify the optimal parameters (order of the spline and smoothing strength), which yields a fully automatic smoothing procedure. We evaluate the performance of our algorithm by comparing it against an oracle Wiener filter, which requires the knowledge of the true noiseless power spectrum of the signal. We find that our approach performs almost as well as the oracle solution over a wide range of conditions. 
000130323 6531_ $$aWiener Filters
000130323 700__ $$aTirosh, S.
000130323 700__ $$0240173$$aVan De Ville, D.$$g152027
000130323 700__ $$0240182$$aUnser, M.$$g115227
000130323 773__ $$j15$$k9$$q2616–2630$$tIEEE Transactions on Image Processing
000130323 8564_ $$uhttp://bigwww.epfl.ch/publications/tirosh0601.html$$zURL
000130323 8564_ $$uhttp://bigwww.epfl.ch/publications/tirosh0601.ps$$zURL
000130323 8564_ $$s1301496$$uhttps://infoscience.epfl.ch/record/130323/files/tirosh0601.pdf$$zn/a
000130323 909C0 $$0252054$$pLIB$$xU10347
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000130323 937__ $$aLIB-ARTICLE-2006-013
000130323 970__ $$atirosh0601/LIB
000130323 973__ $$aEPFL$$rREVIEWED$$sPUBLISHED
000130323 980__ $$aARTICLE