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Annu. Rev. Astron. Astrophys. 2005. 43:139–94

doi: 10.1146/annurev.astro.43.112904.104850

Copyright c ? 2005 by Annual Reviews. All rights reserved

First published online as a Review in Advance on June 16, 2005

DIGITAL IMAGE RECONSTRUCTION:

Deblurring and Denoising

R.C. Puetter,1,4T.R. Gosnell,2,4and Amos Yahil3,4

1Center for Astrophysics and Space Sciences, University of California, San Diego,

La Jolla, CA 92093

2Los Alamos National Laboratory, Los Alamos, NM 87545

3Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794

4Pixon LLC, Stony Brook, NY 11790; email: Rick.Puetter@pixon.com,

Tim.Gosnell@pixon.com, Amos.Yahil@pixon.com

Key Words

regularization, wavelets

image processing, image restoration, maximum entropy, Pixon,

■ Abstract

lying images hidden in blurry and noisy data can be revealed. The main challenge is

sensitivitytomeasurementnoiseintheinputdata,whichcanbemagnifiedstrongly,re-

sultinginlargeartifactsinthereconstructedimage.Thecureistorestrictthepermitted

images.Thisreviewsummarizesimagereconstructionmethodsincurrentuse.Progres-

sivelymoresophisticatedimagerestrictionshavebeendeveloped,including(a)filtering

the input data, (b) regularization by global penalty functions, and (c) spatially adap-

tive methods that impose a variable degree of restriction across the image. The most

reliable reconstruction is the most conservative one, which seeks the simplest underly-

ing image consistent with the input data. Simplicity is context-dependent, but for most

imagingapplications,thesimplestreconstructedimageisthesmoothestone.Imposing

the maximum, spatially adaptive smoothing permitted by the data results in the best

image reconstruction.

Digital image reconstruction is a robust means by which the under-

1. INTRODUCTION

Digital image processing of the type discussed in this review has been developed

extensively and now routinely provides high-quality, robust reconstructions of

blurryandnoisydatacollectedbyawidevarietyofsensors.Thefieldexistsbecause

itisimpossibletobuildimaginginstrumentsthatproducearbitrarilysharppictures

uncorrupted by measurement noise. It is, however, possible mathematically to

reconstruct the underlying image from the nonideal data obtained from real-world

instruments,sothatinformationpresentbuthiddeninthedataisrevealedwithless

blur and noise. The improvement from raw input data to reconstructed image can

be quite dramatic.

0066-4146/05/0922-0139$20.00

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PUETTER ? GOSNELL ? YAHIL

Our choice of nomenclature is deliberate. Throughout this review, “data” refers

to any measured quantity, from which an unknown “image” is estimated through

the process of image reconstruction.1The term image denotes either the estimated

solution or the true underlying image that gives rise to the observed data. The

discussion usually makes clear which context applies; in cases of possible ambi-

guity we use “image model” to denote the estimated solution. Note that the data

and the image need not be similar and may even have different dimensionality,

e.g., tomographic reconstructions seek to determine a 3D image from projected

2D data.

Image reconstruction is difficult because substantial fluctuations in the image

may be strongly blurred, yielding only minor variations in the measured data.

This causes two major, related problems for image reconstruction. First, noise

fluctuations may be mistaken for real signal. Overinterpretation of data is always

problematic, but image reconstruction magnifies the effect to yield large image

artifacts. The high wave numbers (spatial frequencies) of the image model are par-

ticularly susceptible to these artifacts, because they are suppressed more strongly

by the blur and are therefore less noticeable in the data. In addition, it may be

impossible to discriminate between competing image models if the differences in

the data models obtained from them by blurring are well within the measurement

noise. For example, two closely spaced point sources might be statistically indis-

tinguishable from a single, unresolved point source. A definitive resolution of the

image ambiguity can then only come with additional input data.

Image reconstruction tackles both these difficulties by making additional as-

sumptions about the image. These assumptions may appeal to other knowledge

about the imaged object, or there may be accepted procedures to favor a “reason-

able” or a “conservative” image over a “less reasonable” or an “implausible” one.

The key to stable image reconstruction is to restrict the permissible image mod-

els, either by disallowing unwanted solutions altogether, or by making it much

less likely that they are selected by the reconstruction. Almost all modern im-

age reconstructions restrict image models in one way or another. They differ

only in what they restrict and how they enforce the restriction. The trick is not

to throw out the baby with the bath water. The more restrictive the image re-

construction, the greater its stability, but also the more likely it is to eliminate

correct solutions. The goal is therefore to describe the allowed solutions in a

sufficiently general way that accounts for all possible images that may be encoun-

tered, and at the same time to be as strict as possible in selecting the preferred

images.

1Historically, the problem of deblurring and denoising of imaging data was termed image

restoration, a subtopic within a larger computational problem known as image reconstruc-

tion. Most contemporary workers now use the latter, more general term, and we adopt this

terminology in this review. Also, some authors use “image” for what we call “data” and

“object” for what we call “ image.” Readers familiar with that terminology need to make a

mental translation when reading this review.

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There are great arguments on how image restriction should be accomplished.

After decades of development, the literature on the subject is still unusually edito-

rialandevencontentiousintone.Attimesitsoundsasthoughimagereconstruction

isanart,amatteroftasteandsubjectivepreference,insteadofanobjectivescience.

We take a different view. For us, the goal of image reconstruction is to come as

close as possible to the true underlying image, pure and simple. And there are

objective criteria by which the success of image reconstruction can be measured.

First, it must be internally self-consistent. An image model predicts a data model,

and the residuals—the differences between the data and the data model—should

be statistically consistent with our understanding of the measurement noise. If we

see structure in the residuals, or if their statistical distribution is inconsistent with

noisestatistics,thereissomethingwrongwiththeimagemodel.Thisleavesimage

ambiguity that cannot be statistically resolved by the available data. The image

reconstruction can then only be validated externally by additional measurements,

preferably by independent investigators. Simulations are also useful, because the

true image used to create the simulated data is known and can be compared with

the reconstructed image.

Thereareseveralexcellentreviewsofimagereconstructionandnumericalmeth-

ods by other authors. These include: Calvetti, Reichel & Zhang (1999) on itera-

tive methods; Hansen (1994) on regularization methods; Molina et al. (2001) and

Starck, Pantin & Murtagh (2002) on image reconstruction in astronomy; Narayan

& Nityananda (1986) on the maximum-entropy method; O’Sullivan, Blahut &

Snyder (1998) on an information-theoretic view; Press et al. (2002) on the inverse

problem and statistical and numerical methods in general; and van Kempen et al.

(1997) on confocal microscopy. There are also a number of important regular con-

ferences on image processing, notably those sponsored by the International Soci-

ety for Optical Engineering (http://www.spie.org), the Optical Society of America

(http://www.osa.org), and the Computer Society of the Institute of Electrical and

Electronics Engineers (http://www.computer.org).

Our review begins with a discussion of the mathematical preliminaries in

Section 2. Our account of image reconstruction methods then proceeds from the

simple to the more elaborate. This path roughly follows the historical develop-

ment, because the simple methods were used first, and more sophisticated ones

were developed only when the simpler methods proved inadequate.

Simplest are the noniterative methods discussed in Section 3. They provide

explicit, closed-form inverse operations by which data are converted to image

models in one step. These methods include Fourier and small-kernel deconvolu-

tions, possibly coupled with Wiener filtering, wavelet denoising, or quick Pixon

smoothing. We show in two examples that they suffer from noise amplifica-

tion to one degree or another, although good filtering can greatly reduce its

severity.

The limitations of noniterative methods motivated the development of iterative

methods that fit an image model to the data by using statistical tests to determine

how well the model fits the data. Section 4 launches the statistical discussion by

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introducing the concepts of merit function, maximum likelihood, goodness of fit,

and error estimates.

Fitting methods fall into two broad categories, parametric and nonparametric.

Section 5 is devoted to parametric methods, which are suitable for problems in

which the image can be modeled by explicit, known source functions with a few

adjustableparameters.“Clean”isanexampleofaparametricmethodusedinradio

astronomy. We also include a brief discussion of parametric error estimates.

Section 6 introduces simple nonparametric iterative schemes including the van

Cittert,Landweber,Richardson-Lucy,andconjugate-gradientmethods.Thesenon-

parametric methods replace the small number of source functions with a large

number of unknown image values defined on a grid, thereby allowing a much

larger pool of image models. But image freedom also results in image instability,

which requires the introduction of image restriction. The two simplest forms of

imagerestrictiondiscussedinSection6aretheearlyterminationofthemaximum-

likelihoodfit,beforeitreachesconvergence,andtheimpositionoftherequirement

that the image be nonnegative. The combination of the two restrictions is surpris-

ingly powerful in attenuating noise amplification and even increasing resolution,

but some reconstruction artifacts remain.

To go beyond the general methods of Section 6 requires additional restrictions

whose task is to smooth the image model and suppress the artifacts. Section 7 dis-

cusses the methods of linear (Tikhonov) regularization, total variation, and max-

imum entropy, which impose global image preference functions. These methods

were originally motivated by two differing philosophies but ended up equivalent

to each other. Both state image preference using a global function of the image

and then optimize the preference function subject to data constraints.

Global image restriction can significantly improve image reconstruction, but,

because the preference function is global, the result is often to underfit the data

in some parts of the image and to overfit in other parts. Section 8 presents spa-

tially adaptive methods of image restriction, including spatially variable entropy,

wavelets,MarkovrandomfieldsandGibbspriors,atomicpriorsandmassiveinfer-

ence,andthefullPixonmethod.Section8endswithseveralexamplesofbothsim-

ulatedandrealdata,whichservetoillustratethetheoreticalpointsmadethroughout

the review.

We end with a summary in Section 9. Let us state our conclusion outright. The

future, in our view, lies with the added flexibility enabled by spatially adaptive

image restriction coupled with strong rules on how the image restriction is to be

applied. On the one hand, the larger pool of permitted image models prevents

underfitting the data. On the other hand, the stricter image selection avoids over-

fitting the data. Done correctly, we see the spatially adaptive methods providing

the ultimate image reconstructions.

The topics presented in this review are by no means exhaustive of the field,

but limited space prevents us from discussing several other interesting areas of

image reconstruction. A major omission is superresolution, a term used both for

subdiffraction resolution (Hunt 1995, Bertero & Boccacci 2003) and subpixel

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resolution. Astronomers might be familiar with the “drizzle” technique developed

attheSpaceTelescopeScienceInstitute(Fruchter&Hook2002)toobtainsubpixel

resolution using multiple, dithered data frames. There are also other approaches

(Borman & Stevenson 1998; Elad & Feuer 1999; Park, Park & Kang 2003).

Other areas of image reconstruction left out include: (a) tomography (Natterer

1999); (b) vector quantization, a technique widely used in image compression and

classification (Gersho & Gray 1992, Cosman et al. 1993, Hunt 1995, Sheppard

et al. 2000); (c) the method of projection onto convex sets (Biemond, Lagendijk

& Mersereau 1990); (d) the related methods of singular-value decomposition,

principal components analysis, and independent components analysis (Raykov

& Marcoulides 2000; Hyv¨ arinen, Karhunen & Oja 2001; Press et al. 2002); and

(e) artificial neural networks, used mainly in image classification but also useful

in image reconstruction (D´ avila & Hunt 2000; Egmont-Petersen, de Ridder &

Handels 2002).

Wealsoconfineourselvestophysicalblur—instrumentaland/oratmospheric—

thatspreadstheimageovermorethanonepixel.Lossofresolutionduetothefinite

pixelsizecaninprinciplebeovercomebybetteroptics(strongermagnification)ora

finerfocalplanearray.Inpractice,theuserisusuallylimitedbytheopticsandfocal

plane array at hand. The main recourse then is to take multiple, dithered frames

and use some of the superresolution techniques referenced above. Ironically, if the

physicalblurextendsovermorethanonepixel,onecanalsorecoversomesubpixel

resolution by requiring that the image be everywhere nonnegative (Section 2.4).

This technique does not work if the physical blur is less than one pixel wide.

Another area that we omit is the initial data reduction that needs to be per-

formed before image reconstruction can begin. This includes nonuniformity cor-

rections,eliminationofbadpixels,backgroundsubtractionwhereappropriate,and

the determination of the type and level of statistical noise in the data. Major astro-

nomical observatories often have online manuals providing instructions for these

operations, e.g., the direct imaging manual of Kitt Peak National Observatory

(http://www.noao.edu/kpno/manuals/dim).

2. MATHEMATICAL PRELIMINARIES

2.1. Blur

The image formed in the focal plane is blurred by the imaging instrument and

the atmosphere. It can be expressed as an integral over the true image, denoted

symbolically by ⊗:

M(x) = P ⊗ I =

For a 2D image, the integration is over the 2D y-space upon which the image

is defined. In general, the imaging problem may be defined in a space with an

?

P(x,y)I(y)dy.

(1)

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PUETTER ? GOSNELL ? YAHIL

arbitrary number of dimensions, and the dimensionalities of x and y need not even

bethesame(e.g.,intomography).Wavelengthand/ortimemightprovideadditional

dimensions. There may also be multiple focal planes, or multiple exposures in the

same focal plane, perhaps with some dithering.

The kernel of the integral, P(x, y), is called the point-spread function. It is the

probability that a photon originating at position y in the image plane ends up at

position x in the focal plane. Another way of looking at the point-spread function

is as the image formed in the focal plane by a point source of unit flux at position

y, hence, the name point-spread function.

2.2. Convolution

In general, the point-spread function varies independently with respect to both x

and y, because the blur of a point source may depend on its location in the image.

Opticalsystemsthatsufferstronggeometricaberrationsbehaveinthisway.Often,

however, the point-spread function can be accurately written as a function only of

the displacement x − y, independent of location within the field of view. In that

case, Equation 1 becomes a convolution integral:

?

When necessary, we use the symbol*to specify a convolution operation to make

clearthedistinctionwiththemoregeneralintegraloperation⊗.Mostoftheimage

reconstruction methods described in this review, however, are general and are not

restricted to convolving point-spread functions.

Convolutions have the added benefit that they translate into simple algebraic

products in the Fourier space of wave vectors k (e.g., Press et al. 2002):

M(x) = P ∗ I =

P(x − y)I(y)dy.

(2)

˜M(k) =˜P(k)˜I(k).

(3)

Foraconvolvingpoint-spreadfunctionthereisthusadirectk-by-kcorrespondence

betweenthetrueimageandtheblurredimage.Thereasonforthissimplicityisthat

the Fourier spectral functions exp(ιk · x) are eigenfunctions of the convolution

operator, i.e., convolving them with the point-spread function returns the input

function multiplied by the eigenvalue˜P(k).

The separation of the blur into decoupled operations on each of the Fourier

components of the image points to a simple method of image reconstruction.

Equation 3 may be solved for the image in Fourier space:

˜I(k) =˜M(k)/˜P(k),

(4)

and the image is the inverse Fourier transform of˜I(k). In practice, this method is

limited by noise (Section 3.1). We bring it up here as a way to think about image

reconstructionand,inparticular,aboutsamplingandbiasing(Sections2.3and2.4).

Eveninthemoregeneralcaseinwhichimageblurisnotapreciseconvolution,itis

still useful to conceptualize image reconstruction in terms of Fourier components,

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because the coupling between different Fourier components is often limited to a

small range of k.

2.3. Data Sampling

Imagingdetectorsdonotactuallymeasurethecontinuousblurredimage.Inmodern

digital detectors, the photons are collected and counted in a finite number of pixels

with nonzero widths placed at discrete positions in the focal plane. They typically

form an array of adjacent pixels. The finite pixel width results in further blur,

turning the point-spread function into a point-response function. Assuming that

all the pixels have the same response, the point-response function is a convolution

of the pixel-responsivity function S and the point-spread function:

H(x,y) = S ∗ P.

(5)

The point-response function is actually only evaluated at the discrete positions of

the pixels, xi, typically taken to be at the centers of the pixels. The data expected

to be collected in pixel i—in the absence of noise (Section 2.7)—is then

?

WealsorefertoMiasthedatamodelwhentheimageunderdiscussionisanimage

model.

Image reconstruction actually only requires knowledge of the point-response

function, which relates the image to the expected data. There is never any need to

determine the point-spread function, because the continuous blurred image is not

measureddirectly.Butitisnecessarytodeterminethepoint-responsefunctionwith

sufficient accuracy (Section 2.9). Approximating it by the point-spread function is

often inadequate.

Mi=

H(xi,y)I(y)dy = (H ⊗ I)i.

(6)

2.4. How to Overcome Aliasing Due to Discrete Sampling

Another benefit of the Fourier representation is its characterization of sampling

andbiasing.Thesamplingtheoremtellsuspreciselywhatcanandcannotbedeter-

mined from a discretely sampled function (e.g., Press et al. 2002). Specifically, the

sampled discrete values completely determine the continuous function, provided

that the function is bandwidth-limited within the Nyquist cutoffs:

−

1

2∆= −kc≤ k ≤ kc=

1

2?.

(7)

(The Nyquist cutoffs are expressed in vector form, because the grid spacing ∆

need not be the same in different directions.) By the same token, if the continuous

function is not bandwidth-limited and has significant Fourier components beyond

the Nyquist cutoffs, then these components are aliased within the Nyquist cutoffs

and cannot be distinguished from the Fourier components whose wave vectors

really lie within the Nyquist cutoffs. This leaves an inherent ambiguity regarding

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PUETTER ? GOSNELL ? YAHIL

the nature of any continuous function that is sampled discretely, thereby limiting

resolution.

The point-spread function is bandwidth-limited and therefore so is the blurred

image. The bandwidth limit may be a strict cutoff, as in the case of diffraction, or

a gradual one, as for atmospheric seeing (in which case the effective bandwidth

limit depends on the signal-to-noise ratio). In any event, the blurred image is as

bandwidth-limitedasthepoint-spreadfunction.Itisthereforecompletelyspecified

by any discrete sampling whose Nyquist cutoffs encompass the bandwidth limit

of the point-spread function. The true image, however, is what it is and need

not be bandwidth-limited. It may well contain components of interest beyond the

bandwidth limit of the point-spread function.

Thepreviousdiscussionaboutconvolution(Section2.2)suggeststhattherecon-

structed image would be as bandwidth-limited as the data and little could be done

to recover the high-k components of the image. This rash conclusion is incorrect.

We usually have additional information about the image, which we can utilize. Al-

most all images must be nonnegative. (There are some important exceptions, e.g.,

complex images, for which positivity has no meaning.) Sometimes we also know

somethingabouttheshapesofthesources,e.g.,theymayallbestellarpointsources.

Taking advantage of the additional information, we can determine the high-k im-

age structure beyond the bandwidth limit of the data (Biraud 1969). For example,

wecantellthatanonnegativeimageisconcentratedtowardonecornerofthepixel

because the point-spread function preferentially spreads flux to pixels around that

corner. (Autoguiders take advantage of this feature to prevent image drift.)

We can see how this works for a nonnegative image by considering the Fourier

reconstruction of an image blurred by a convolving point-spread function (Section

2.2). The reconstructed Fourier image˜I(k) is determined by Equation 4 within

the bandwidth limit of the data. But the image obtained from it by the inverse

Fourier transformation may be partly negative. To remove the negative image

values, we must extrapolate˜I(k) beyond the bandwidth limit. We are free to do

so, because the data do not constrain the high-k image components. Of course, we

cannot extrapolate too far, or else aliasing will again cause ambiguity. But we can

establish some k limit on the image by assuming that the image has no Fourier

components beyond that limit. The point is that the image bandwidth limit needs

to be higher than the data bandwidth limit, and the reconstructed image therefore

has higher resolution than the data. The increased resolution is a direct result of

therequirementofnonnegativity,withoutwhichwecannotextrapolatebeyondthe

bandwidth limit of the data. In the above example of subpixel structure, we can

deduce the concentration of the image toward the corner of the pixel because the

image is restricted to be nonnegative. If it could also be negative, the same data

couldresultfromanynumberofimages,asthesamplingtheoremtellsus.Subpixel

resolution is enabled by nonnegativity. On the other hand, if we also know that the

image consists of point sources, we can constrain the high-k components of the

image even further and obtain yet higher resolution.

Note that the pixel-responsivity function is not bandwidth-limited in and of

itself because it has sharp edges. If the point-spread function is narrower than a

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pixel, the data are not Nyquist sampled. The reconstructed image is then subject to

additional aliasing, and it may not be possible to increase resolution beyond that

setbythepixelationofthedata.Ifthepoint-spreadfunctionismuchnarrowerthan

the pixel, a point source could be anywhere inside the pixel. We cannot determine

its position with greater accuracy, because the blur does not spill enough of its flux

into neighboring pixels.

2.5. Background Subtraction

Theuseofnonnegativityasamajorconstraintinimagereconstructionpointstothe

importance of background subtraction. Requiring a background-subtracted image

to be nonnegative is much more restrictive, because when an image sits on top of a

significant background, negative fluctuations in the image can be absorbed by the

background. The user is therefore well advised to subtract the background before

commencingimagereconstruction,ifatallpossible.Astronomicalimagesusually

lend themselves to background subtraction because a significant area of the image

is filled exclusively by the background sky. It may be more difficult to subtract the

background in a terrestrial image.

The best way to subtract background is by chopping and nodding, alternating

betweenthetargetandanearbyblankfieldandrecordingonlydifferencemeasure-

ments.Buttheimaginginstrumentmustbedesignedtodoso.(Seethedescriptions

of such devices at major astronomical observatories, e.g., on the thermal-region

cameraspectrographoftheGeminiSouthtelescope,http://www.gemini.edu/sciops

/instruments/miri/T-ReCSChopNod.html.)Absentsuchcapability,thebackground

can only be subtracted after the data are taken. Several methods have been pro-

posed to subtract background (Bijaoui 1980; Infante 1987; Beard, McGillivray &

Thanisch1990;Almoznino,Loinger&Brosch1993;Bertin&Arnouts1996).The

background may be a constant or a slowly varying function of position. It should

inanyeventnotvarysignificantlyonscalesoverwhichtheimageisknowntovary,

or else the background subtraction may modify image structures. There are also

several ways to clip sources when estimating the background (see the discussion

by Bertin & Arnouts 1996).

2.6. Image Discretization

In general, an image can either be specified parametrically by known source func-

tions (Section 5) or it can be represented nonparametrically on a discrete grid

(Section6),inwhichcasetheintegralinEquation1isconvertedtoasum,yielding

a set of linear equations:

?

in matrix notation M=HI. Here M is a vector containing n expected data values, I

is a vector of m image values representing a discrete version of the image, and H is

an n × m matrix representation of the point-response function. Note that each of

what we here term vectors is in reality often a multidimensional array. In a typical

Mi=

j

HijIj,

(8)

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PUETTER ? GOSNELL ? YAHIL

2D case, n = nx × ny, m = mx × my, and the point-response function is an

(nx × ny) × (mx × my) array.

Note also for future reference that one often needs the transpose of the point-

response function HT, which is the m × n matrix obtained from H by transposing

its rows and columns. HTis the point-response function for an optical system

in which the roles of the image plane and the focal plane are reversed (known

in tomography as back projection). It is not to be confused with the inverse of

the point-response function H−1, an operator that exists only for square matrices,

n = m. When applied to the expected data, H−1provides the image that gave rise

to the expected data through the original optical system at hand.

Thediscussionofsamplingandaliasing(Section2.4)showsthattheimagemust,

under some circumstances, be determined with better resolution than the data to

ensure that it is nonnegative. This requires the image grid to be finer than the data

grid,sotheimageisNyquistsampledwithintherequisiteimagebandwidth.Inthat

case, the number of image values is not equal to the number of data points, and the

point-response function is not a square matrix. Conversely, if the image is known

to be more bandwidth-limited than the data, or if the signal-to-noise ratio is low,

so the high-k components are uncertain, we may choose a coarser discretization

of the image than the data.

In the case of a convolution, the discretization is greatly simplified by using

Equation 2 to give:

?

Note that Equation 9 assumes that the data and image grids have the same spacing,

and the point-response function takes a different form than in Equation 8. Like the

expected data and the image, H is also an n-point vector, not an n × n matrix.

(Recall that n refers to the total number of grid points, e.g., for a 2D array n =

nx× ny.) Discrete convolutions of the form of Equation 9 have a discrete Fourier

analog of Equation 3 and can be computed efficiently by fast Fourier transform

techniques (e.g., Press et al. 2002). When the image grid needs to be more finely

spaced than the data, the convolution is performed on the image grid and then

sampled at the positions of the pixels on the coarser grid.

Mi=

j

Hi−jIj.

(9)

2.7. Noise

A major additional factor limiting image reconstruction is noise due to measure-

ment errors. The measured data actually consist of the expected data Miplus

measurement errors:

Di= Mi+ Ni= (H ⊗ I)i+ Ni=

?

H(xi,y)I(y)dy + Ni.

(10)

The discrete form of Equation 10 is obtained in analogy with Equation 8.

Measurement errors fall into two categories. Systematic errors are recurring

errors caused by erroneous measurement processes or failure to take into account

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physical effects that modify the measurements. In addition, there are random,

irreproducible errors that vary from one measurement to the next. Because we do

notknowandcannotpredictwhatarandomerrorwillbeinanygivenmeasurement,

we can at best deal with random errors statistically, assuming that they are random

realizationsofsomeparentstatisticaldistribution.Inimaging,themostcommonly

encounteredparentstatisticaldistributionsaretheGaussian,ornormal,distribution

and the Poisson distribution (Section 4.2).

To be explicit, consider a trial solution to Equation 10,ˆI(y), and compute the

residuals

Ri= Di− Mi= Di−

?

H(xi,y)ˆI(y)dy.

(11)

Theimagemodelisanacceptablesolutionoftheinverseproblemiftheresidualsare

consistentwiththeparentstatisticaldistributionofthenoise.Thedatamodelisthen

our estimate of the reproducible signal in the measurements, and the residuals are

our estimate of the irreproducible noise. There is something wrong with the image

model if the residuals show systematic structure or if their statistical distribution

differs significantly from the parent statistical distribution. Examples would be if

its mean is not zero, or if the distribution is skewed, too broad, or too narrow. After

the fit is completed, it is therefore imperative to apply diagnostic tests to rule out

problems with the fit. Some of the most useful diagnostic tools are goodness of fit,

analysis of the statistical distribution of the residuals and their spatial correlations,

and parameter error estimation (Sections 4 and 5).

2.8. Instability of Image Reconstruction

Image reconstruction is unfortunately an ill-posed problem. Mathematicians con-

sider a problem to be well posed if its solution (a) exists, (b) is unique, and (c)

is continuous under infinitesimal changes of the input. The problem is ill posed

if it violates any of the three conditions. The concept goes back to Hadamard

(1902, 1923). Scientists and engineers are usually less concerned with existence

and uniqueness and worry more about the stability of the solution.

In image reconstruction, the main challenge is to prevent measurement errors

in the input data from being amplified to unacceptable artifacts in the recon-

structed image. Stated as a discrete set of linear equations, the ill-posed nature

of image reconstruction can be quantified by the condition number of the point-

response-function matrix. The condition number of a square matrix is defined as

the ratio between its largest and smallest (in magnitude) eigenvalues (e.g., Press

et al. 2002).2A singular matrix has an infinite condition number and no unique

solution. An ill-posed problem has a large condition number, and the solution is

sensitive to small changes in the input data.

2If the number of data and image points is not equal, then the point-response function is not

square, and its condition number is not strictly defined. We can then use the square root of

the condition number of HTH, where HTis the transpose of H.

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PUETTER ? GOSNELL ? YAHIL

Howlargeistheconditionnumber?Arealisticpoint-responsefunctioncanblur

the image over a few pixels. In this case, the highest k components of the image

are strongly suppressed by the point-response function. In other words, the high-

k components correspond to eigenfunctions of the point-response function with

very small eigenvalues. Hence, there is no escape from a large condition number.

Equations 8 or 9 can therefore not be solved in their present, unrestricted forms.

Either the equations must be modified, or the solutions must be projected away

fromthesubspacespannedbytheeigenfunctionswithsmalleigenvalues.Thebulk

of this review is devoted to methods of image restriction (Sections 3–8).

2.9. Accuracy of the Point-Response Function

Image reconstruction is further compromised if the point-response function is not

determined accurately enough. The signal-to-noise ratio determines how accu-

rately it needs to be determined. The goal is that the wings of bright sources will

notbeconfusedwithweaksourcesnearby.Thehigherthesignal-to-noiseratio,the

greater the care needed in determining the point-response function. The residual

errors caused by the imprecision of the point-response function should be well

below the noise.

Thefirstrequirementisthattheprofileofthepoint-responsefunctioncorrespond

to the real physical blur. If the point-response function assumed in the reconstruc-

tion is narrower than the true point-response function, the reconstruction cannot

remove all the blur. The image model then has less than optimal resolution, but

artifactsshouldnotbegenerated.Ontheotherhand,iftheassumedpoint-response

function is broader than the true one, then the image model looks sharper than it

really is. In fact, if the scene has narrow sources or sharp edges, it may not be pos-

sible to reconstruct the image correctly. Artifacts in the form of “ringing” around

sharp objects are then seen in the image model.

Second, the point-response function must be defined on the image grid, which

mayneedtobefinerthanthedatagridtoensureanonnegativeimage(Section2.4).

A point-response function measured from a single exposure of one point source is

then inadequate because it is only appropriate for sources that are similarly placed

within their pixels as the source used to measure the point-response function.

Either multiple frames need to be taken displaced by noninteger pixel widths,

or the point-response function has to be determined from multiple point sources

spanning different intrapixel positions. In either case, the point-response function

is determined on an image grid that is finer than the data grid.

2.10. Numerical Considerations

Weconcludethemathematicalpreliminariesbynotingthatthefullmatrixcontain-

ing the point-response function is usually prohibitively large. A modern 1024 ×

1024detectorarrayyieldsadatasetof106elements,andHcontains1012elements.

Clearly, one must avoid schemes that require the use of the entire point-response

functionmatrix.Fortunately,theydonotallneedtobestoredincomputermemory,

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nor do all need to be used in the matrix multiplication of Equation 8. The number

of nonnegligible elements is often a small fraction of the total, and sparse matrix

storage can be used (e.g., Press et al. 2002). The point-response function may

also exhibit symmetries, such as in the case of convolution (Equation 9), which

enables more efficient storage and computation. Alternatively, because H always

appears as a matrix multiplication operator, one can write functions that compute

themultiplicationontheflywithouteverstoringthematrixvaluesinmemory.Such

computations can take advantage of specialized techniques, such as fast Fourier

transforms (Section 3.1) or small-kernel deconvolutions (Section 3.2).

3. NONITERATIVE IMAGE RECONSTRUCTION

Anoniterativemethodforsolvingtheinverseproblemisonethatderivesasolution

through an explicit numerical manipulation applied directly to the measured data

in one step. The advantages of the noniterative methods are primarily ease of

implementation and fast computation. Unfortunately, noise amplification is hard

to control.

3.1. Fourier Deconvolution

Fourierdeconvolutionisoneoftheoldestandnumericallyfastestmethodsofimage

deconvolution. If the noise can be neglected, then the image can be determined

using a discrete variant of the Fourier deconvolution (Equation 4), which can be

computed efficiently using fast Fourier transforms (e.g., Press et al. 2002). The

technique is used in speckle image reconstruction (Jones & Wykes 1989; Ghez,

Neugebauer & Matthews 1993), Fourier-transform spectroscopy (Abrams et al.

1994, Prasad & Bernath 1994, Serabyn & Weisstein 1995), and the determination

of galaxy redshifts, velocity dispersions, and line profiles (Simkin 1974, Sargent

et al. 1977, Bender 1990).

Unfortunately,theFourierdeconvolutiontechniquebreaksdownwhenthenoise

may not be neglected. Noise often has significant contribution from high k, e.g.,

white noise has equal contributions from all k. But ˜ H(k), which appears in the

denominatorofEquation4,fallsoffrapidlywithk.Theresultisthathigh-knoisein

thedataissignificantlyamplifiedbythedeconvolutionandcreatesimageartifacts.

The wider the point-response function, the faster˜H(k) falls off at high k and the

greater the noise amplification. Even for a point-response function extending over

only a few pixels, the artifacts can be so severe that the image is completely lost

in them.

3.2. Small-Kernel Deconvolution

Fast Fourier transforms perform convolutions very efficiently when used on stan-

dard desktop computers but they require the full data frame to be collected before

the computation can begin. This is a great disadvantage when processing raster

video in pipeline fashion as it comes in, because the time to collect an entire

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data frame often exceeds the computation time. Pipeline convolution of raster data

streamsismoreefficientlyperformedbymassivelyparallelsummationtechniques,

even when the kernel covers as much as a few percent of the area of the frame.

In hardware terms, a field-programmable gate array (FPGA) or an application-

specific integrated circuit (ASIC) can be much more efficient than a digital signal

processor(DSP)oramicroprocessorunit(MPU).FPGAsorASICsavailablecom-

mercially can be built to perform small-kernel convolutions faster than the rate at

which raster video can straightforwardly feed them, which is currently up to ∼150

megapixels per second.

Pipeline techniques can be used in image reconstruction by writing deconvolu-

tions as convolutions by the inverse H−1of the point-response function

I = H−1∗ D,

(12)

which is equivalent to the Fourier deconvolution (Equation 4). But H−1extends

over the entire array, even if H is a small kernel (spans only a few pixels). Not to

be thwarted, one then seeks an approximate inverse kernel G ≈ H−1, which can

be designed to span only ∼3 full widths at half maximum of H.

Moreover, G can also be designed to suppress the high-k components of H−1

and to limit ringing caused by sharp discontinuities in the data, thereby reducing

the artifacts created by straight Fourier methods.

3.3. Wiener Filter

InSection3.1,wesawthatdeconvolutionofthedataresultsinstrongamplification

of high-k noise. The problem is that the signal decreases rapidly at high k, while

the noise is usually flat (white) and does not decay with k. In other words, the

high-k components of the data have poor signal-to-noise ratios.

The standard way to improve k-dependent signal-to-noise ratio is linear filter-

ing, which has a long history in the field of signal processing and has been applied

in many areas of science and engineering. The Fourier transform of the data˜D(k)

is multiplied by a k-dependent filter ? (k), and the product is transformed back to

provide filtered data. Linear filtering is a particularly useful tool in deconvolution,

because the filtering can be combined with the Fourier deconvolution (Equation

4) to yield the filtered deconvolution,

˜I(k) = ?(k)

˜D(k)

˜H(k).

(13)

It can be shown (e.g., Press et al. 2002) that the optimal filter, which minimizes

the difference (in the least squares sense) between the filtered noisy data and the

true signal, is the Wiener filter, expressed in Fourier space as

?(k) =

?|˜D0(k)|2?

?|˜D0(k)|2? + ?|˜N(k)|2?=?|˜D0(k)|2?

?|˜D(k)|2?.

(14)

Here ?|˜N(k)|2? and ?|˜D0(k)|2? are the expected power spectra (also known as

spectral densities) of the noise and the true signal, respectively. Their sum, which

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appears in the denominator of Equation 14, is the power spectrum of the noisy

data ?|˜D(k)|2?, because the signal and the noise are—by definition—statistically

independent, so their power spectra add up in quadrature.

The greatest difficulty in determining ?(k) (Equation 14) comes in estimating

?|˜D0(k)|2?. In many cases, however, the noise is white and easily estimated at high

k, where the signal is negligible. In practice, it is necessary to average over many

k values, because the statistical fluctuation of any individual Fourier component

is large. The power spectrum of the signal is then determined from the difference

?|˜D0(k)|2? = ?|˜D(k)|2? − ?|˜N(k)|2?. Again, averaging is needed to reduce the

statistical fluctuations.

A disadvantage of the Wiener filter is that it is completely deterministic and

doesnotleavetheuserwithatuningparameter.Itisthereforeusefultointroducean

ad hoc parameter β into Equation 14 to allow the user to adjust the aggressiveness

of the filter.

?|˜D0(k)|2?

?|˜D0(k)|2? + β?|˜N(k)|2?.

Standard Wiener filtering is obtained with β = 1. Higher values result in more

aggressive filtering, whereas lower values yield a smaller degree of filtering.

?(k) =

(15)

3.4. Wavelets

WesawinSection3.1and3.3thattheFouriertransformisaveryconvenientwayto

performdeconvolution,becauseconvolutionsaresimpleproductsinFourierspace.

ThedisadvantageoftheFourierspectralfunctionsisthattheyspanthewholeimage

and cannot be localized. One might wish to suppress high k more in one part of

the image than in another, but that is not possible in the Fourier representation.

The alternative is to use other, more localized spectral functions. These functions

arenolongereigenfunctionsofaconvolvingpoint-responsefunction,sotheimage

reconstruction is not as simple as in the Fourier case, but they might still retain

the Fourier characteristics, at least approximately. How are we to choose those

functions?Ontheonehandwewishmorelocalization.Ontheotherhand,wewant

to characterize the spectral functions by the rate of spatial oscillation, because we

know that we need to suppress the high-k components. Of course, the nature of

the Fourier transform is such that there are no functions that are perfectly narrow

in both image space and Fourier space (the uncertainty principle). The goal is to

find a useful compromise.

The functions to emerge from the quest for oscillatory spectral functions with

local support have been wavelets. The most frequently used wavelets are those

belonging to a class discovered by Daubechies (1988). In addition to striking

a balance between x and k support, they satisfy the following three conditions:

(a) they form an orthonormal set, which allows easy transformation between the

spatial and spectral domains, (b) they are translation invariant, i.e., the same func-

tioncanbeusedindifferentpartsoftheimage,and(c)theyarescaleinvariant,i.e.,

they form a hierarchy in which functions with larger wavelengths are scaled-up

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