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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENTAL SEMINAR}
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\centerline{4:15 p.m., Tuesday, January 7, 2003}
\centerline{Sequoia Hall Room 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Ery Arias-Castro}
\centerline{\sl Stanford University}
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\centerline{\bf Are those currently fashionable edge-preserving smoothers
quantitatively superior?}
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We consider the image de-noising problem from the point of view 
mathematical statistics. We study anisotropic smoothing methods which
aim to recover images with sharp edges from noisy data. We derive the 
optimal anisotropic behavior, and show that the scale of the smoothing must be
spatially variable - getting smaller near the edge -- but also the 
anisotropy must be spatially variable -getting more anisotropic near edges, 
according to the scaling law that says that the effective width of the 
implicit smoothing kernel is the square of the effective length. It turns 
out that, despite reputation, none of the geometric diffusion techniques 
that we have studied exhibits the required behavior. At the same time, 
there exist several methods based on harmonic analysis with the required 
properties -- including wedgelet thresholding and curvelet thresholding. To 
illustrate our point, we report the results of a careful analysis of median 
filtering, show that it is quantitatively outperformed even simple wavelet 
thresholding. We infer similar conclusions for other standard geometric 
diffusions. The implication is that there is a serious need for future 
developments of new geometric-diffusion de-noising techniques which exhibit 
proper adaptive anisotropy scaling and consequently improved de-noising.

Joint work with David Donoho. 

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