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For example [5-7], suppose that the measurement of luminance in eachpixel is quantized to (in some units) an integer value. LetU=MXuµ(18.7.7)µ=1be the total number of luminance quanta in the whole image. Then we can base our“prior” on the notion that each luminance quantum has an equal a priori chance ofbeing in any pixel.
(See [8] for a more abstract justification of this idea.) The numberof ways of getting a particular configuration u is"!#XX1U!∝ exp −ln U −(18.7.8)uµ ln(uµ /U ) +ln uµu1 !u2 ! · · · uM !2µµSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).Prob(u|cI) = Prob(c|uI)82118.7 Maximum Entropy Image RestorationHere the left side can be understood as the number of distinct orderings of allthe luminance quanta, divided by the numbers of equivalent reorderings withineach pixel, while the right side follows by Stirling’s approximation to the factorialfunction.
Taking the negative of the logarithm, and neglecting terms of order log Uin the presence of terms of order U , we get the negentropyMXuµ ln(uµ /U )(18.7.9)µ=1From equations (18.7.5), (18.7.6), and (18.7.9) we now seek to maximize1 2Prob(u|c) ∝ exp − χ exp[−H(u)]2(18.7.10)or, equivalently,minimize:MX1 21 2uµ ln(uµ /U )− ln [ Prob(u|c) ] = χ [u] + H(u) = χ [u] +22µ=1(18.7.11)This ought to remind you of equation (18.4.11), or equation (18.5.6), or in fact any ofour previous minimization principles along the lines of A + λB, where λB = H(u)is a regularizing operator. Where is λ? We need to put it in for exactly the reasondiscussed following equation (18.4.11): Degenerate inversions are likely to be ableto achieve unrealistically small values of χ2 .
We need an adjustable parameter tobring χ2 into its expected narrow statistical range of N ± (2N )1/2 . The discussion atthe beginning of §18.4 showed that it makes no difference which term we attach theλ to. For consistency in notation, we absorb a factor 2 into λ and put it on the entropyterm. (Another way to see the necessity of an undetermined λ factor is to note that itis necessary if our minimization principle is to be invariant under changing the unitsin which u is quantized, e.g., if an 8-bit analog-to-digital converter is replaced by a12-bit one.) We can now also put “hats” back to indicate that this is the procedurefor obtaining our chosen statistical estimator:u] + λH(bu) = χ2 [bu] + λminimize: A + λB = χ2 [bMXubµ ln(buµ )(18.7.12)µ=1(Formally, we might also add a second Lagrange multiplier λ0 U , to constrain thetotal intensity U to be constant.)It is not hard to see that the negentropy, H(bu), is in fact a regularizing operator,similar to bu·bu (equation 18.4.11) or bu·H·bu (equation 18.5.6).
The following ofits properties are noteworthy:1. When U is held constant, H(bu) is minimized for ubµ = U/M = constant, so itsmooths in the sense of trying to achieve a constant solution, similar to equation(18.5.4). The fact that the constant solution is a minimum follows from the factthat the second derivative of u ln u is positive.Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).H(u) =822Chapter 18.Integral Equations and Inverse Theory2. Unlike equation (18.5.4), however, H(bu) is local, in the sense that it does notdifference neighboring pixels. It simply sums some function f, heref(u) = u ln u(18.7.13)jµ(cf. equation 18.4.7). Setting the formal derivative with respect to ubµ to zero givesX∂H= f 0 (buµ ) =λj Rjµ∂buµ(18.7.15)jor defining a function G as the inverse function of f 0 ,Xλj Rjµubµ = G (18.7.16)jThis solution is only formal, since the λj ’s must be found by requiring that equation(18.7.16) satisfy all the constraints built into equation (18.7.14). However, equation(18.7.16) does show the crucial fact that if G is linear, then the solution bu contains onlya linear combination of basis functions Rjµ corresponding to actual measurementsj.
This is equivalent to setting unmeasured cj ’s to zero. Notice that the principalsolution obtained from equation (18.4.11) in fact has a linear G.Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).over all pixels; it is invariant, in fact, under a complete scrambling of the pixelsin an image.
This form implies that H(bu) is not seriously increased by theoccurrence of a small number of very bright pixels (point sources) embeddedin a low-intensity smooth background.3. H(bu) goes to infinite slope as any one pixel goes to zero.
This causes it toenforce positivity of the image, without the necessity of additional deterministicconstraints.4. The biggest difference between H(bu) and the other regularizing operators thatwe have met is that H(bu) is not a quadratic functional of bu, so the equationsobtained by varying equation (18.7.12) are nonlinear. This fact is itself worthyof some additional discussion.Nonlinear equations are harder to solve than linear equations.
For imageprocessing, however, the large number of equations usually dictates an iterativesolution procedure, even for linear equations, so the practical effect of the nonlinearityis somewhat mitigated. Below, we will summarize some of the methods that aresuccessfully used for MEM inverse problems.For some problems, notably the problem in radio-astronomy of image recoveryfrom an incomplete set of Fourier coefficients, the superior performance of MEMinversion can be, in part, traced to the nonlinearity of H(bu).
One way to see this [5]is to consider the limit of perfect measurements σi → 0. In this case the χ2 term inthe minimization principle (18.7.12) gets replaced by a set of constraints, each withits own Lagrange multiplier, requiring agreement between model and data; that is,"#XXu)(18.7.14)λj cj −Rjµubµ + H(bminimize:18.7 Maximum Entropy Image Restoration823Is MEM Really Magical?How unique is the negentropy functional (18.7.9)? Recall that that equation isbased on the assumption that luminance elements are a priori distributed over thepixels uniformly. If we instead had some other preferred a priori image in mind, onewith pixel intensities mµ , then it is easy to show that the negentropy becomesH(u) =MXuµ ln(uµ /mµ ) + constant(18.7.17)µ=1(the constant can then be ignored).
All the rest of the discussion then goes through.More fundamentally, and despite statements by zealots to the contrary [7], thereis actually nothing universal about the functional form f(u) = u ln u. In someother physical situations (for example, the entropy of an electromagnetic field in thelimit of many photons per mode, as in radio-astronomy) the physical negentropyfunctional is actually f(u) = − ln u (see [5] for other examples). In general, thequestion, “Entropy of what?” is not uniquely answerable in any particular situation.(See reference [9] for an attempt at articulating a more general principle that reducesto one or another entropy functional under appropriate circumstances.)The four numbered properties summarized above, plus the desirable sign fornonlinearity, f 000 (u) < 0, are all as true for f(u) = − ln u as for f(u) = u ln u.√Infact these properties are shared by a nonlinear function as simple as f(u) = − u,which has no information theoretic justification at all (no logarithms!).
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