2-dimensional laplacian smoothing by iterative methods with ...

2-DIMENSIONAL LAPLACIAN SMOOTHING BY ITERATIVE METHODS WITH APPLICATION TO A AND B-MODE IMAGE RESTORATION

by

Finbarr O'Sullivan

TECHNICAL REPORT No. 118

January 1988

Department of Statistics, GN-22 University of Washington Seattle, Washington 98195 USA

2-Dimensional Laplacian Smoothing by Iterative Methods with Application to A and B-mode Image Restoration Finbarr O'Sullivan!

Department of Statistics and Biostatistics University of Washington Seattle. WA 98195.

ABSTRACT

Smoothing has b.ecome· an essential tool in modem graphically oriented data analysis. \Vhile there are a number of fast reliable univariate cross-validated smoothing algorithms comparable· methods are not available in higher dimensions. The paper develops an iterative approach to2-dimensional Laplacian spline smoothing. The method involves multi-grid iteration, block successive over relaxation and conjugate gradient acceleration. Special matrix structure is exploited and some asymptotic approximations are developed which allow for efficient computation of generalized cross-validation scores. The algorithm is naturally suited to 2-dimensional scatter plot smoothing. However the algorithm also has potential value for image restoration, and we illustrate this on a lOOxlOO multi-color image example. Borrowing form the terminology of Ultrasound both A and B-mode restorations are obtained. Along . with these restorationsw.e introduce image Wlcertainty •assessment Which is based on the entropy < of the:m.arginal p0steri()r pixelvalue distribution. This analysis particularly highlights the statistical uncertainty in resolving object boundaries in the image.

an

AMS 1980 subject classifications. Primary, 62-G05, Secondary, 62-J05, 41-A35, 41-A25, 47-A53, 45-LIO, 45-M05. Key words and phrases. A and B-mode imaging, bicubic B-splines, conjugate gradient acceleration, cross-validation, Laplacian smoothing, leverage, multigrid, SSOR, posterior pixel entropy

Running Head: 2-d Laplacian Smoothing and Image Restoration

Research ,sup~d in part by the Department of Energy under Grant No. DE-FGQ6..85ER25006.

January 4,

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2-Dimensional Laplacian Smoothing by Iterative Methods with Application to A and B-mode Image Restoration Finbarr O' Sullivan 1 Department of Statistics and Biostatistics University of Washington Seattle. WA 98195.

1. Introduction Data smoothing is an essential part of modern statistical methodology. Techniques such as ACE[8] and Projection Pursuit[17] use smoothing algorithrris as building blocks for more elaborate models. At the moment however these methodologies are restricted to using 1dimensional smoothers essentially because it is only here that there are fast cross-validated algorithms.••Of course this means that· the ultimate accuracy of ACE and Projection. Pursuit models depends on how well one can hope to approximate a function of several variables by rudimentary combinations of functions of single variables. Fast 2- and 3-dimensional smoothing algorithms will greatly enhance the power of modem statistical m.ethodology. that there is

role

smoothing in image restoration. This compliments recent developments in

the field which have emphasized Markov random fields combined with stochastic relaxation algorithms,

Geman and Geman[18] for example. is an eX1tensiv'e literature on mlllti,·dimensi.onal data smoothing.

four

Kridging[24], Local Interpolation[2, 29], Moving Average report focuses on

Research supponed ill part by the DelllirtlDell1Ot' EI111rg,' Ull.,der ·Gr.tl1t 1'10. DE-FG06-85ER25006.

-2-

as ,xi ,Yi ,wi , f = 1,2, ... n}, with

which

,Yi

I (Xi ,Yi)] = f (Xi ,Yi) Var(z; I (Xi ,Y;)] = wr 1cr2 E(Zi

where f

:.n --? R

(1.1)

and cr is the unknown noise level. The Laplacian smooth h . is defined as the

minimizer of

(1.2)

J m (/) is the m 'thorder Laplacian smoothing; functional, see Wahba and Wendelberger[34]. For m = 2, the case which we consider here,

J 2(/) =

UifJ + 2 f~ + f ~ ]dxdy .

(1.3)

A is the regularization parameter; as A --? co we obtain the smoothest solution (a weighted least squares planar solution), A --? 0 produces the roughest data interpolation fit. Wahba and Wende1berger[34] developed an algorithm to evaluate the Laplacian smooth and simultaneously adjusts the regularization parameter by cross-validation.

This algorithm involves the

manipulation and singular valued~compositionoffullO en) matrices and so becomes impractical

developed by Bates

Wahba[35] the resulting algorithms in GCVPACK(5] are

still practically limited to data sets on the order of 1000. the Laplacian smOOI:.tlirtg were HutchittS9n[21] and TorJ!5:eISl~n mesh with bi-linear

int~~rpis lj.nifQrmly low (light colored) artd only becomes large on the rim of the image. Appreciable uncertainties in the B-mode restoration correspond to areas where the restoration fails to agree with the true image. Object boundaries are immediately recognizable as places with high uncertainty.

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