Wavelet based multiresolution expectation maximization ...raheja/publications/wmrem_cmig.pdf ·...

Wavelet based multiresolution expectation maximization imagereconstruction algorithm for positron emission tomography

A. Rahejaa, A.P. Dhawanb,*aDepartment of Computer Science, Philadelphia University, Philadelphia, PA 19144, USA

bDepartment of Electrical & Computer Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA

Received 16 December 1999; accepted 23 May 2000

Abstract

Maximum Likelihood (ML) estimation based Expectation Maximization (EM) [IEEE Trans Med Imag, MI-1 (2) (1982) 113] reconstruc-tion algorithm has shown to provide good quality reconstruction for positron emission tomography (PET). Our previous work [IEEE TransMed Imag, 7(4) (1988) 273; Proc IEEE EMBS Conf, 20(2/6) (1998) 759] introduced the multigrid (MG) and multiresolution (MR) conceptfor PET image reconstruction using EM. This work transforms the MGEM and MREM algorithm to a Wavelet based Multiresolution EM(WMREM) algorithm by extending the concept of switching resolutions in both image and data spaces. The MR data space is generated byperforming a 2D-wavelet transform on the acquired tube data that is used to reconstruct images at different spatial resolutions. Wavelettransform is used for MR reconstruction as well as adapted in the criterion for switching resolution levels. The advantage of the wavelettransform is that it provides very good frequency and spatial (time) localization and allows the use of these coarse resolution data spaces inthe EM estimation process. The MR algorithm recovers low-frequency components of the reconstructed image at coarser resolutions in feweriterations, reducing the number of iterations required at finer resolution to recover high-frequency components. This paper also presents thedesign of customized biorthogonal wavelet filters using the lifting method that are used for data decomposition and image reconstruction andcompares them to other commonly known wavelets.q 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Multiresolution reconstruction; Wavelets; Expectation maximization; Positron emission tomography; Lifting scheme

1. Introduction

Expectation Maximization (EM) algorithm has been usedfor Positron Emission Tomography (PET) image recon-struction with several variants [4–6]. Recently, waveletprocessing has been used [7–12] to deduce a multiresolution(MR) representation to solve inverse problems similar tothis problem of image reconstruction. Use of wavelets forMR tomographic reconstruction using the Filtered Backpro-jection operator has been investigated [9–12]. In imageanalysis and reconstruction approaches, MR methods haveproduced better results than conventional approaches [13].A similar approach for representation of the EM operator inthe wavelet domain is not so trivial because of the complexnature of the operator and the sparse nature of the probabil-ity matrix (weight matrix which entirely depends on thegeometry of the imaging system).

We have used standard EM estimation method [1] for MRreconstruction using wavelet analysis to improve efficiency

and reconstruction quality. The wavelets are used toconstruct a MR data space, which are then used in the esti-mation process. The beauty of the wavelet transform toprovide localized frequency-space representation of thedata allows us to perform the estimation using these decom-posed components. The advantage of this method lies withthe fact that the noise in the acquired data becomes localizedin the high–high or diagonal frequency bands and thus notusing these bands for estimation at coarser resolution helpsspeed up the recovery of various frequency componentswith reduced noise estimation. Hence, it is very importantto have a wavelet basis that allows for good spatialfrequency localization into the various subbands at differentlevels. To ensure the above, biorthogonal wavelet filterswere custom-designed using the lifting scheme [14–16]that provides a robust method of creating new biorthogonalwavelets from existing biorthogonal wavelets. The newwavelet filters have a frequency response that ideally loca-lizes the various frequency components of the tube data forreconstruction purposes.

A new and robust wavelet based stopping criterion and awavelet spline based interpolation [3] method used for grid

Computerized Medical Imaging and Graphics 24 (2000) 359–376PERGAMON

ComputerizedMedical Imaging

and Graphics

0895-6111/00/$ - see front matterq 2000 Elsevier Science Ltd. All rights reserved.PII: S0895-6111(00)00035-5

www.elsevier.com/locate/compmedimag

* Corresponding author. Tel.:11-419-530-8267; fax:11-419-530-7392.E-mail address:[email protected] (A.P. Dhawan).

mapping is presented for this iterative algorithm. Thisapproach can be generalized to other image reconstructionmethods that use the concept of EM. The main goal of thispaper is to introduce the MR concept in the EM frameworkand to demonstrate the improved reconstruction by generat-ing a wavelet basis appropriate for frequency localization oftomographic PET data.

The rest of the paper is organized as follows. Section 2reviews the wavelet theory in general and the design ofbiorthogonal wavelets using the method of lifting. Thewavelet based MR approach to EM is described in Section3. Section 4 gives a brief overview of the design of thecustom wavelet filters for the problem of tomographicimage reconstruction and compares the new wavelet filterswith commonly known wavelet filters. Section 5 discussesthe results of the reconstruction using phantom PET dataand compares the effect of various wavelet bases in recon-struction. Finally, Section 6 presents conclusions anddiscusses the future direction of this work.

2. Wavelets and MR analysis

2.1. Orthogonal and biorthogonal wavelets

Assuming most readers to be familiar with wavelets, onlya brief introduction to the concept of discrete wavelet trans-form is presented here. Details on wavelets can be looked upin Refs. [17–19].

A scaling functionf (t) in time t, also known as the“father wavelet,” can be defined as

fj;k�t� � 2j=2f�2j t 2 k� �1�wherej is a scaling parameter andk a translation parameterand j; k [ Z; set of all integers. The scaling and translationof which generates a family of functions that span by usingthe following “dilation” equations:

f�t� � ��2p X

n

hnf�2t 2 n� �2�

wherehn is a set of filter coefficients and��2p

maintains thenorm of the wavelet function with the scale of two. Toinduce a MR analysis ofL2(R), whereR is the space ofall real numbers, it is required to have a nested chain ofclosed subspaces defined as

… , V21 , V0 , V1 , V2 , … , L2 �3�Specifically, there exists a functionc (t) in time t, the“mother wavelet,” such that

c j;k�t� � 2j=2c�2 j t 2 k� �4�form an orthonormal basis ofL2(R). The wavelet basisinduces an orthogonal decomposition ofL2(R), i.e. onecan write

… , W21 , W0 , W1 , W2 , … , L2 �5�

whereWj is a subspace spanned byc (2jt 2 k) for all integersj, k [ Z. This basic requirement of a MR analysis is satisfiedby nesting of the spanned subspaces similar to scaling func-tions i.e.c (t) can be expressed as a weighted sum of theshiftedc (2t) as

c�t� � ��2p X

n

gnf�2t 2 n� �6�

wheregn is a set of filter coefficients and��2p

maintains thenorm of the wavelet function with the scale of two.

The wavelet-spanned subspace is such that it satisfies therelation

Vm11 � Vm % Wm whereL2 �… % W22 % W21 % W0

% W1 % W2 % … (7)

Hence, the wavelets are functions whose dilations and trans-lations form an orthonormal basis ofL2(R). Since, the wave-lets span the “difference” or orthogonal complement spaces�W21 ' W22 ' W23 ' V21�; the orthogonality requiresthe scaling and wavelet filter coefficients to be relatedthrough the following:

gn � �21�nh12n �8�In the case of orthogonal wavelets as described above,hn

andgn are called a pair of quadrature-mirror low-pass andhigh-pass filters. The biorthogonal wavelet system gener-alizes the classical orthogonal case. For biorthogonal wave-lets, a dual scaling function is similar to the MR approach ofthe scaling function is defined as

~f �t� � ��2p X

n

~hn~f �2t 2 n� �9�

Similarly, one can define the dual wavelet as

~c �t� � ��2p X

n

~gn~f �2t 2 n� �10�

The following relations can now relate the dual scalingand wavelet filters:

gn � �21�n ~h12n �11�

~gn � �21�nh12n �12�The subspaces spanned by the scaling functions, dual

scaling functions and wavelet and dual wavelet functionsfollow the relationship

Vj ' ~Wj �13�

~Vj ' Wj �14�

2.2. Wavelet expansion and reconstruction of 1D and 2Dsignals

Mallat [20] has showed using a pyramid algorithmhow orthogonal wavelets could be used to create an MR

A. Raheja, A.P. Dhawan / Computerized Medical Imaging and Graphics 24 (2000) 359–376360

representation of a signal and Unser et al. [21] extended thisto the case of non-orthogonal wavelets using splines. Usingthe definitions in the section above, any signalaj11�t� [L2�R� can be written as

aj11�t� �Xj;k

aj11�t�;c j;k�t�D E

c j;k�t� �15�

where the brackets denote the inner product. In the MRanalysis, the function is successively approximated in thesubspaceVj. From the representation in Fig. 1(a), one canwrite

aj11�t� �X

k

aj0�k�fj0;k�t�1X

k

XJ 2 1

j�j0

bj�k�c j;k�t� �16�

wherej0 � 2 in Fig. 2(b).Similar to the analysis, the reverse process of synthesis to

get a perfect reconstructed signal is seen in Fig. 1(b). Thesynthesis of any wavelet-decomposed signal should give aperfectly reconstructed signal if orthogonal or biorthogonalwavelet filters are used.

In case of a 2D signal, the same concept discussed abovecan be extended toL2(R2). The wavelet MR analysis can beconstructed using a separable scaling function that can bewritten as

f�x; y� � f�x�f�y� �17�This implies that the wavelet transform can be performed asdescribed above for the 1D case to the 2D signal. This isachieved by first applying the transform to the rows of theimage and then to the columns of the image.

2.3. Multiresolution analysis

Mallat [20] showed how separable functions can be used

A. Raheja, A.P. Dhawan / Computerized Medical Imaging and Graphics 24 (2000) 359–376 361

h

g

aj+1

bj

aj

bj-1

aj-1

2

2

h

g

2

2

bj-1

aj-1

aj+1

bj

aj h~

g~

h~

g~

2

2

2

2

(a)

(b)

Fig. 1. (a) Two-stage two-band analysis tree.h andg are the low-pass and high-pass decomposition wavelet filters as mentioned in the previous section. (b)Two-stage two-band synthesis tree.h̃ andg̃ are the low-pass and high-pass reconstruction wavelet filters as mentioned in the previous section.

(a)

(b)

g

h

h

g

h

gAj+1

Aj

2 1

2 1

1 2

1 2

1 2

1 2

Dj1

Dj2

Dj3

J-12j0

Fig. 2. (a) Single level decomposition of an imageAj11. g andh are the high-and low-pass filters, 2# 1: keep one column out of two, and 1# 2: keep onerow out of two. (b) Schematic representation of the 2D-wavelet transformcoefficients of an image.Aj ; refers to the approximation at resolutionj whileD1

j ; D2j ; andD3

j ; refer to the details at resolutionj.

to create a MR representation of a 2D signal. In brief, a one-stage 2D wavelet transform can be performed as seen in Fig.2(a). Performing the wavelet transform on an image result infour sub-images, each one of them is half the size of theoriginal because of the downsampling of two for both rowsand columns. These subimagesAj, D1

j , D2j and D3

j arereferred to as the low–low (LL), low–high (LH), high–low (HL) and high–high (HH) subbands, respectively.Unlike a complete analysis, where each subband is furtherdecomposed to the desired level, a wavelet tree analysisdecomposes only the LL band further to the desired level.Synthesis for an analyzed image is done similar to that of asignal in Fig. 1(b).

2.4. Design of biorthogonal wavelet filters using lifting

The idea of lifting as a method of biorthogonal waveletconstruction was introduced by Wim Sweldens [15]. Liftingis also a simple method that is used to increase the numberof vanishing moments or dual vanishing moments of awavelet or dual wavelet. This section defines and discussesbriefly the design considerations while using the method oflifting. The reader is pointed towards Sweldens’s work [14–16,22] for an in depth understanding of the lifting schemefrom which this entire section has been derived.

The lifting scheme was inspired by two theorems, onefrom the work of Cohen et al. [23] and the other fromChui [24]. Sweldens’s corollary [16] uses these two theo-rems to define the following.

Take an initial set of finite biorthogonal filters{ h; ~h0

;g0; ~g} : Then a new set of finite biorthogonal filters

{ h; ~h; g; ~g} can be found as

~h�v� � ~h0�v�1 ~g�v�s�2v� �18�

g�v� � g0�v�2 h�v�s�2v� �19�wheres(2v ) is a trigonometric polynomial. As seen fromthe above definition, the scaling filter and the dual wave-let filter remain unchanged. The lifting scheme theorem isdefined in terms of biorthogonal functions as follows.Take an initial set of finite biorthogonal scaling functionsand wavelets {f; ~f 0

;c 0; ~c 0} : Then a new set

{f; ~f ;c; ~c } ; which is formally biorthogonal can befound as:

c�x� � c 0�x�2X

k

skf�x 2 k� �20�

~f �x� � 2X

k

~h0kf�2x 2 k�1

Xk

s2k~c �x 2 k� �21�

~c �x� � 2X

k

~gk~f �2x 2 k� �22�

where the coefficientssk can be freely chosen.

The sequences provides the ability to manipulate thewavelets and dual functions that can be constructed froma simple scaling function. This sequence can also be used toprovide a wavelet with increased number of vanishingmoments or have a desired shape.

Sweldens’s work [14] provides examples using the finiteimpulse response (FIR) filter coefficients based upon thework of Deslauriers and Dubuc’s [25] as the initial set ofbiorthogonal filters that are derived from interpolating scal-ing functions. Sweldens’s definition [16] of an interpolatingscaling function implied that every dual interpolating scal-ing function has as its dual, a Dirac function as its scalingfunction. This concept is combined with the lifting methodto generate new wavelets with higher vanishing moments.In Ref. [14], the linear spline hat functions are used with twolifting parameters to create a new dual wavelet and a scalingfunction. The equation for the dual wavelet can be written as

~c �x� � ~f �2x 2 1�2 a ~f �x�2 b ~f �x 2 1� �23�The coefficients or lifting parametersa and b are chosensuch that the resulting dual wavelet is symmetric and hasa vanishing moment. sym_ab implies the new wavelet to besymmetric i.e.

sym_ab� a� b �24�Thepth vanishing moment, denoted by momp is defined as

momp�Z

xp ~c �x� � 0 �25�

mom0 for Eq. (23) lead to the symmetry condition. Usingcondition mom1, the result is given asZ1

2 1=2

~c �x� dx�Z1

2 1=2~w�2x 2 1� dx 2 a

Z1

2 1=2~w�x� dx

2 bZ1

2 1=2~w�x 2 1� dx

� 0 (26)

Solving this yields

a� 1=2 2 b �27�Using Eqs. (24) and (27), the lifting coefficients area� b�1=4: These coefficients lead to the following representationof the combination of old wavelet with two scaling func-tions (reduced by a factor of 1/4 each) at the same level toform the new dual wavelet as seen in Fig. 3 [16].

This technique of finding the lifting parameters and thencombining old wavelets with scaling functions from thesame level results in design of new wavelets.

3. Method: wavelet based MREM

The MREM reconstruction method is an extension of themultigrid EM (MGEM) reconstruction concept introduced


by Dhawan et al. [2] but merged with the MR approach ofwavelets. A schematic approach to the wavelet basedMREM method is presented and compared with the MGapproach in Fig. 4. In the MGEM algorithm [2], a set ofgrids Gk, k � 1;…;K represented the same reconstructionspace at different resolutions.G0 was the coarsest grid whileGK was the finest resolution grid. The reconstruction pyra-mid consisted of square grids of sizesSk, k � 1;…;K suchthat the ratio,Sk11 : Sk � 1 : 2: The maximum frequency,fk, of the image on gridGk is given by 1/2Sk, which is twicethe maximum frequency that can be represented byGk21.Using the hierarchical structure of the reconstruction pyra-mid, the low-frequency components of the finest resolutionwere recovered quickly at the coarser grids.

In the MREM algorithm [3], the authors introduced theconcept of varying resolution in the detector space. This wasdone by combining groups of adjacent detectors i.e. rebin-ning the tube data to create new data spaces at coarser

resolutions. This process of rebinning increases the signalto noise ratio of the data at coarser resolutions, but usingthese new data spaces in the MR framework provided fasterand better quality image reconstruction. In this work,wavelets are used to create MR data spaces in order to usethe lower resolution data spaces without using much of thehigh-frequency components of the original tube data. Thenew Wavelet based MREM (WMREM) reconstructionalgorithm aims at using the relatively noise free tube dataat coarser resolutions to recover the low-frequency compo-nents of the image and recover most of the high-frequencycomponents at the original resolution of the data. Thus, thisalgorithm merges the EM algorithm with the wavelet basedMR approach, and also utilizes spatial-frequency localiza-tion analysis obtained from the wavelet processing in theimage reconstruction. To optimize spatial-frequency locali-zation analysis for better reconstruction, the wavelet iscustom designed using the method of lifting [14–16].


(a) (b)

(c)

Combination

(d) (e)

Fig. 3. (a) Piecewise linear spline hat dual scaling functions at levelj. (b) Old dual wavelet function at levelj. (c) Dual scaling functions at neighboring levelj.(d) Combination of old dual wavelets and one-fourth times dual scaling function at the same level. (e) The new dual wavelet created as a result of lifting.


(a) (b)

Final ReconstructedImage = Final

Estimations of theMGEM algorithm

Wavelet Transform:Transition Criterion

andInterpolation

MultigridEstimation

Process (MGEM)

Tube datap(b,d)weight matrix

Tube dataWavelet

Transformand

Analysis

ReconstructedImage = Inverse

Wavelet Transformof the final estimationsat various resolutions

MultiresolutionEstimation

Process (MREM)

p(b,d)weight matrix

Tube data atcoarser

resolutions

Fig. 4. The difference of the concept between the (a) multigrid EM (MGEM) and the (b) wavelet based multiresolution EM (WMREM).

(a)

(b)

n0*(d)Tube data128x128

h

g

h

g

h

g

h

g

h

g

h

g

n1*(d) 32x32

n6*(d) 64x64

n5*(d) 64x64

n4*(d) 32x32

n3*(d) 32x32

n2*(d) 32x32

n7*(d) 64x64

Level 2Level 1Level 0

n1*(d) 32x32n1*(d) 32x32

2

2

2

2

2

22

2

2

22

2

n*0

Original

Tube data

(128x128)

n*6

64x64

n*5

64x64

n*1

32x32

n*3

32x32n*

432x32

n*2

32x32

MultiresolutionRepresentation

n*7

64x64

Fig. 5. (a) Two-channel two-stage analysis of the tube datan0p(d) at the finest resolution using the wavelet low- and high-pass decomposition filtersh andg,

respectively. (b) MR representation of the data spaces.


ReconstructedImage

Wavelet reconstruction

n2*(d) 32x32

n1*(d) 32x32

n3*(d) 32x32

n5*(d) 64x64

n6*(d) 64x64

EM

EM

EM

EM

EM

2

2

2

2

2

h~

h~

h~

g~

g~

Initial estimation

Fig. 6. MR reconstruction scheme for WMREM. EM is the Expectation Maximization process.h̃ andg̃ are the low-pass and high-pass reconstruction filters.

InitializeGrid level, k = 0 (coarsest)Data space, j = 2 (coarsest)

Iterations, i = 0

i = i +1

NO

YES

NO

YES

i = 0

Is gridoptimization

measuresatisfied ?

Is intra-gridlevel

performance measuresatisfied ?

k = k + 1j = j -1

Initial Solutionλ for0

0 *6

*5

*3

*2

*1 n,n,n,n,n

Create data spacesDWT2( ) results in)(n*

0 d

*7

*6

*5

*4

*3

*2

*1 n,n,n,n,n,n,n

λ = DWT-1(λ ,λ ,λ )Wavelet Synthesis

j4 ,k+10 i

j2

,kij1 ,k

ij3 ,k

λ = EM( λ ,n1)λ = EM( λ ,n2)λ = EM( λ ,n3)

j1 , ki+1

j2 , ki+1

j3 , ki+1

ij1 , k

ij3 , k

ij2 , k

WaveletDecomposition

Final ReconstructedImage

NO

YES

Final ReconstructedImage

Is currentgrid resolution

>detector

resolution?

Is J=0K =2

λ = DWT-1(λ ,λ ,λ )Wavelet Synthesis

j8 ,k+10 i

j5 ,kij4 ,k

ij6 ,k

YES

k = k+1

Fig. 7. Flowchart for the WMREM algorithm.DWTn refers to a forward discrete wavelet transform andn refers to the level. Whenn . 0; it is a forwarddiscrete wavelet transform and inverse wavelet transform whenn , 0:

3.1. MR data spaces

The MR data spaces can be constructed using the originaltube data and performing a two-stage or two-level two-bandanalysis as seen in the Fig. 5(a). The reason for choosingtwo levels of decomposition is because it provides optimaldata spaces in terms of information contained in them. Theselection of two levels is discussed in detail under the resultsin a latter section.

Using the notation introduced earlier in Section 2 for theseparable wavelet transform of an image, we can write thefollowing:

np1�d� � A2np

0�d� �28�

np2�d� � D1

2np0�d�; np

3�d� � D22np

0�d�; np4�d� � D3

2np0�d� �29�

np5�d� � D1

1np0�d�; np

6�d� � D21np

0�d�; np7�d� � D3

1np0�d� �30�

Eq. (28) is the low-pass component of the tube data after twolevels of decomposition also called the LLLL component.The three detail components at level 2 in Eq. (29) areusually referred to as the LLLH, LLHL and LLHH compo-nents, respectively. The three detail components at level 1 inEq. (30) are referred as the LH, HL and HH components,respectively. The image representation in Fig. 5(b) displaysthe creation of 7 new data spaces from the original datausing a two-level wavelet-decomposition.

The data spacesnp4�d� andnp

7�d� shall not be used in thereconstruction algorithm since they contain the possiblehigh-frequency noise of the data collection process. Thecoarser resolution data created in this manner is relativelyfree of the high frequency of components of the original data.

3.2. WMREM algorithm

In the WMREM method, the MREM is performed within

the framework of EM using the newly constructed dataspaces and switching grid and data space resolution simul-taneously to perform a faster and better quality reconstruc-tion. The interesting property of spatial localization of thedata after performing wavelet decomposition allows for theuse of these data spaces. The LL data spacenp

1�d� is simply alow-pass filtered image and can be used without any modi-fication. The HL and LH components i.e.np

2�d�; np3�d�; np

5�d�andnp

6�d� are subjected to thresholding or DC shifting to beused as data spaces for the estimation process. Both thesemethods, i.e. thresholding and DC shifting are evaluated inthe framework of this algorithm. The WMREM reconstruc-tion process using wavelet basis is shown in Fig. 6.

It is interesting to note that the interpolation step (formapping from a coarser grid level to a finer one) performedin the MREM [3] using a wavelet spline based method isnow naturally performed by the wavelet reconstructionscheme until the finest data resolution is reached. TheWMREM algorithm, however, does retain the concept ofperforming EM at one resolution finer than the desired gridlevel i.e. it performs EM at the desired resolution and onegrid finer using the finest data space which is the originaldata itself. The mapping from the desired grid level to thenext finer level is done using the wavelet spline method [3].Also, the stopping criterion is similar to the one introducedfor MREM. A flowchart of this algorithm is shown in Fig. 7.The algorithm can be described briefly as follows:

Step 1. Create new data spaces using a separable 2Dwavelet transform on the data.Step2. Threshold/DC shift the data spaces that involvethe LH or HL filtering process.Step3. Initialize the starting solution to a constant valuefor the LL, LH and HL spaces at the coarsest level, i.e.say level� J and the HL and LH components at otherlevels i.e.J 2 1;…;1: In this algorithm,J � 2: Coarsestgrid level starts fromK � 0 which corresponds to coar-sest data level atJ � 2:Step4. Perform EM using these 3 data spaces at levelJ tillthe stopping criterion is satisfied. Grid used has the samedimension as the data spaces.Step5. Perform inverse wavelet transform using the threeimages as a result of step 4. The HH image is taken aszeros for this wavelet synthesis step.Step6. The reconstructed image is used as the startingsolution for the LL component at the next finer resolution,i.e. J 2 1: The LH and HL components use a constantvalue image as the starting solution similar to Step 3. Gridresolution is the same as that of the data space, i.e.K � 1at data levelJ � 1:Step7. Steps 4, 5 and 6 are performed on these dataspaces until the coarsest resolution of the data is reachedi.e. J � 0; which is the original data.Step8. The reconstructed image is used as starting solu-tion for the original data. EM is performed till the transi-tion criterion is satisfied. The resulting solution is grid


Fig. 8. Shepp and Logan Phantom.

mapped to the next finer resolution (i.e.K� 3) using awavelet spline based interpolation method.Step9. The mapped image is used as the starting solutionfor the finest grid and EM is performed until the transitioncriterion is satisfied. The final reconstructed image istaken as the LL component of the wavelet transformedimage reconstructed at levelK � 3 andJ � 0:

3.3. The transition and stopping criterion

Decomposing the image created using the LL component

at level j after each iteration, the frequency content of theimage reconstructed at that resolution can be monitored.The information contained in each decomposed frequencyband is represented by the energy. The energy is defined as:

Energy�XI

i�0

�band_pass_coefficient�i��2 �31�

whereI is the total number of pixels in each frequency band.The energy in each frequency band is monitored with


(a) (b)

(c) (d)

(e) (f)

Fig. 9. Images reconstructed using different wavelet basis in the WMREM algorithm that uses threshold on the high-pass data spaces: (a) biorthogonalspline~N � 1; N � 5; (b) Daubechies orthogonalN � 3; (c) biorthogonal spline~N � 6; N � 8; (d) least asymmetric~N � 4; (e) biorthogonal spline~N � 3; N � 1;and (f) Daubechies orthogonalN � 20:

respect to the normalized RMS error defined as

Error�

��XI

i�1

�l�i�2 l̂ �i��2

XI

i�1

l�i�2

vuuuuuuuut �32�

wherel (i) is the known emission density in boxi, l̂ �i� is thecomputed density, andI is the total number of pixels in theimage.

The energy in the high–high frequency band representsthe high-frequency content of the image at that grid level.The grid level is switched when the energy in this bandceases to increase and starts to decrease. This is done inorder to retrieve the high-frequency components of thedesired grid resolution, which are the low-frequencycomponents of a grid level finer than the desired grid reso-lution. Mathematically, the transition criterion [3] is writtenas

E�u j11�2 E�u j�E�u j�2 E�u j21� , 0:0 and E�u j�2 E�u j21� . 0:0

�33�whereE is the energy operator defined in Eq. (31), andu j is

the image reconstructed at thejth iteration at a certain gridlevel. It is observed that the RMS error decreases andreaches a minimum and then starts increasing. The transi-tion point corresponds to an iteration before the RMS errorreaches its minimum and the change in the RMS errorbecomes very small.

4. Custom design of biorthogonal wavelet filters

As mentioned in Section 2, one can use a set of biortho-gonal wavelet filters {h; ~h0

;g0; ~g} that can be used to create

a new set {h; ~h; g; ~g} with only the dual wavelet and thescaling filter remaining the same. Cohen et al. [23] createda set of compactly supported biorthogonal wavelets thatresemble spline hat functions. This Daubechies wavelet[26] which has ~N � 2 andN � 2 is a special case of thesecond-generation wavelet created by using the liftingcoefficients a� 1=4 and b� 1=4 and Deslauriers andDubuc’s [25] FIR filter coefficients for the first order� ~N �1� interpolating scaling function.

In this paper, we use the second order interpolating splinefunction� ~N � 3� as the dual scaling function. The reason forchoosing this function is discussed later in Section 5. Thefamily of biorthogonal wavelets constructed by Daubechies


(a) (b)

(c) (d)

Fig. 10. Images reconstructed using different wavelet basis in the WMREM algorithm that used DC shifting on the high-pass data spaces: (a) biorthogonalspline ~N � 1; N � 5, (b) biorthogonal spline~N � 6; N � 8; (c) least asymmetricN � 4; and (d) biorthogonal spline~N � 3; N � 1:

[26] are special cases of second generation wavelets that canbe constructed using different orders of interpolating splinefunctions as dual scaling functions and lifting coefficientsderived using symmetry and moment conditions. The set ofFIR filter coefficients for the second order spline function tobe used for the lifting method are given as

~h� {1 =4;3=4;3=4;1=4�= ��2p

and

g� { 21=4;3=4;23=4; 1=4} =��2p �34�

h� old samples and ~g� even samples �35�Using the lifting method, the equations for the dual wavelet

function and the scaling function can be written as

~c �x� � ~f �2x�2 ~f �2x 2 1�2 a ~f �x 1 3�2b ~f �x 1 2�2 c ~f �x 1 1�2 d ~f �x�2e ~f �x 2 1�2 f ~f �x 2 2�2 g ~f �x 2 3� (36)

f�x� � f�2x�1 f�2x 2 1�1 ac�x 1 3�1bc�x 1 2�1 cc�x 1 1�1 dc�x�1ec�x 2 1�1 fc�x 2 2�1 gc�x 2 3� (37)

Table 1 lists the lifting coefficients found using differentsymmetry and momentum conditions. Each set gives riseto a different set of wavelet filters. Three and five liftingparameters were also used but since the wavelet for sevenlifting coefficients gave good results, only a few sets oflifting coefficients have been listed. The wavelet namesas mentioned in the table above have been assignedarbitrarily. The bior3lift7.1 performed best for theWMREM algorithm. The frequency response of thelow-pass and high-pass analysis filters for the newwavelet and for some other commonly used waveletsare shown in Fig. 8.

5. Results

This new algorithm was evaluated quantitatively usingthe normalized RMS error and qualitatively by comparisonof reconstructed images and profiles through the recon-structed and original phantom. The algorithm is contrastedwith the traditional EM method and the Filtered Backpro-jection. Commonly known wavelets were also used in theWMREM algorithm and the results are compared with thebest custom designed wavelet suitable for tomographicimage reconstruction.

The simulated emission phantom data was created usingthe method described by Shepp and Logan [27]. The resultspresented here were generated using the phantom of128× 128 pixels resolution as seen in Fig. 8. It was assumedto be 128 discrete detectors equally spaced around the circleof radius

��2p

circumscribing the image boxes. The totalnumber of emissions was set to 10 million. To demonstratethe transition criterion, the energy in the HH band of thewavelet decomposed image after each iteration is plotted foreach level of reconstruction grid.

Daubechies “least asymmetric” [26] compactlysupported wavelet with maximum number of vanishingmoments forN � 4 was used for analysis of the imageafter each iteration to develop the transition criterion[3]. This wavelet has a spectral response that allowsfor better quantization of the high-frequency componentsof an image compared with some other waveletsinvestigated.


Fig. 11. Frequency response for: (a) low-pass; and (b) high-pass decom-position filter for the bior3lift7.1 compared with other commonly usedwavelets Daubechies biorthogonal bior4.6, bior5.5, bior1.3 and Daubechiesorthogonal d20 and d6.

The choice of the most suitable wavelet for use in thisalgorithm was not trivial. Different wavelets were used andevaluated for the analysis and synthesis in the WMREMalgorithm. We initially tested the algorithm using the popu-lar Daubechies orthogonal wavelets [26] of different orders.As seen in Fig. 9b and f, the system of orthogonal waveletsdoes not perform well for this algorithm. The Daubechies“least asymmetric” wavelets with different number of

vanishing moments were also evaluated in this frameworkand these provided better results as seen in Fig. 9d and d.The Cohen et al. [23] biorthogonal spline wavelet familywas used in this framework and these provided the bestresults (reconstructed images) as seen in Fig. 9c and eamong all the commonly known wavelets evaluated.Hence, this work also brings forth the effect of using differ-ent wavelet bases in the specific problem at hand.


Fig. 12. Decomposition: (a) scaling function; (b) wavelet function and reconstruction; (c) scaling function; and (d) wavelet function for the custom designedbior3lift7.1 wavelet.

Table 1Lifting coefficients using seven lifting parameters (lifting coefficient derived using various conditions of symmetry and vanishing moments)

Wavelet name Conditions used forlifting coefficients

a b c d e f g

bior3lift7.1 sym_ag, sym_bfsym_ce, mom0,mom1, mom3, mom5

152048

231512

4592048

024592048

31512

2152048


22531744

230347507904

350115872

225253952

2350115872

29897507904

222531744


126391715200

21043091715200

963942880

21531715200

276653343040

1033911715200

26243857600


213956381952

21559533102111232

2398947525527808

2139551055616

398947525527808

1562123102111232

13956381952


2141038865248

23525331141843968

57015316616

21410370921984

257015316616

23553537141843968

141038865248

5.1. Evaluation of various wavelet bases for use in WMREMalgorithm

The following two cases discuss the effect of subjectinghigh-pass filtered data spaces to thresholding and DC shift-ing to provide a positive distribution to the EM estimationprocess. Reconstruction results using both these methodsare discussed.

Case IThresholding the coefficients of data spacesn2p(d),

n3p(d) n5

p(d), andn6p(d).

; nj�d� , 0:0 make nj�d� � 0:0; j � 2; 3;5 and 6

Performing this operation on the data spaces, some of thecoefficients are set to zero, i.e. some information in thesedata spaces is lost but it is required to perform this in orderto provide a distribution that can be used for the EM estima-tion process. Some of the reconstructed images in Fig. 9 aredifficult to distinguish qualitatively, hence Table 2 provides

a comparison in terms of the normalized RMS error. As seenfrom the errors, the orthogonal wavelets like the Daubechiesdo not perform well in this scheme of tomographic recon-struction. Also, the least asymmetric wavelet for orderN �4 performs well but with some image degradation along theedges. The biorthogonal spline wavelet family [23,26] for~N � 1;2;3; 4;5 and 6 were also tested for use in theWMREM algorithm and the family of~N � 3 provided the


Fig. 13. Transition curves for different grid levels for the WMREM algorithm: (a) Grid level: 32, transition iteration: 1; (b) Grid level: 64, transition iteration:9; (c) Grid level: 128, transition iteration: 18; and (d) Grid level: 256, transition iteration: 6.

Table 2RMS errors using different wavelet bases for the algorithm. The high-passfiltered data spaces use the method of thresholding

Wavelet name RMS error Wavelet name RMS error

Bior ~N � 1; N � 5 0.151491 DaubechiesN � 3 0.783103Bior ~N � 3; N � 1 0.117206 Least asymmetricN � 4 0.116828Bior ~N � 6; N � 8 0.118325 Bior3lift7.1~N � 3; 7

lifting parameters0.114550

best results. This initiated the use of second order interpo-lating spline function for the custom design using themethod of lifting.

Case IIDC shifting the coefficients of data spacesn2p(d),

n3p(d) n5

p(d), andn6p(d).

nj�d� � abs�min�1 nj�d�; �39�wheremin�minimum �nj�d��: This method finds the mini-mum value for each data space, and DC shifts the completedata space by adding the absolute of the minimum value.The idea behind this method is to preserve the statistics ofthe coefficients even though the values are being changed.The reconstruction was performed on phantom data usingthis method and the results are displayed in Fig. 10. Thequantitative measure using the RMS errors is compared inTable 3.

As seen from these results, the method of thresholdingperforms better than DC shifting. This is primarily due tothe reason that DC shifting actually changes the coefficient

values in the data spaces, which results in distortion ofthe information contained in the data space. Hence, thereconstructed images are of poorer quality. In case ofthresholding, less information is used from the dataspace as a result of equating negative coefficients to zero.The method of thresholding does not change the values ofthe coefficients used for the estimation process i.e. eventhough less information is used, but without distortion ofthe data. Hence, the thresholding method results in better


Fig. 14. NRMS error curves for different grid levels for the WMREM algorithm: (a) 32× 32; (b) 64× 64; (c) 128× 128; and (d) 256× 256.

Table 3RMS errors using different wavelet bases for the algorithm. The high-passfiltered data spaces use the method of DC shifting

Wavelet name RMS error Wavelet name RMS error

Bior ~N � 1; N � 5 0.322867 DaubechiesN � 3 0.897563Bior ~N � 3; N � 1 0.153096 Least AsymmetricN � 4 0.151780Bior ~N � 6; N � 8 0.136050 Bior3lift7.1~N � 3; 7

lifting parameters0.118339

quality reconstructed images as compared with DC shiftingmethod.

5.2. Custom designed wavelet

The importance of the custom design of the wavelet isbetter understood by evaluating the reconstructed images inFigs. 9 and 10. As seen from these figures, these imagesdiffer because the localization of spatial frequency in thevarious data spaces plays an important role in the MRconcept. Hence, it becomes important to custom designwavelets for data of a specific nature. The wavelets arebeing used to localize a certain range of spatial frequencyin the LL, LH and HL bands that are used for reconstruction.The reconstruction can be improved if appropriate frequen-cies can be localized in these data spaces, i.e. include thedesired high frequencies excluding the noise. Many customwavelets were designed using the lifting method as seen inTable 1. The custom design of wavelets is steered by the

frequency response of the wavelet filters. The frequencyresponse of the decomposition low-pass and high-passwavelet filters of a few commonly used wavelets and thebest custom designed wavelet are compared in Fig. 11. Themost suitable custom designed wavelet used for analysis andsynthesis in the WMREM algorithm is shown in Fig. 12.This is the bior3lift7.1 wavelet of Table 1. Fig. 11demonstrates that the cut-off frequency of the low-passdecomposition filter plays a crucial role in choosing thewavelet to generate MR data spaces. Comparing thefrequency response of the low-pass decomposition waveletfilters in this figure, it is seen that the cut-off frequency forbior3lift7.1 is shifted to the right, hence including higherfrequencies as compared to other wavelet filters. Also, themagnitude response for bior3lift7.1 indicates a lowermagnification of the low frequencies, which is a highlydesirable property for these data spaces because the lowfrequencies are already provided by the LL data space.

The best performing WMREM algorithm uses the custom


(a) (b)

(c) (d)

Fig. 15. (a) Original phantom, Phantom data reconstructed using (b) WMREM (using the best custom designed wavelet bior3lift7.1), (c) ML-EM and (d)Filtered Backprojection algorithm.

designed bior3lift7.1 wavelet filters. The reconstructedimage using WMREM algorithm that used custom designedfilter is seen in Fig. 14 and is discussed in the next twosubsections.

5.3. Transition criterion

The coarsest resolution grid level is 32× 32 which usesdata spacesn1

p(d), n2p(d) andn3

p(d), also at the same resolu-tion. Assigning every pixel, a constant value initializes thecoarsest resolution grid. The grid level 64× 64 used by dataspacen5

p(d) andn6p(d) is also initialized in a similar fashion.

The variation of energy in the HH band and RMS error withiterations is plotted for each grid level in Figs. 13 and 14,respectively. As seen from these plots for the energyand errors, the transition point seen in the energy curveslies just before the RMS error reaches a minimum andstarts to increase. The region before the errors reach aminimum is ideal for the transition to the next finerresolution since the region beyond this iteration startscontributing to the noise to the reconstruction. Fig. 13shows that the number of iterations used to reconstructthe phantom was 1 at grid level 32× 32 using dataspacesn1

p(d), n2p(d) and n3

p(d), 9 at grid level 64× 64using data spacesn4

p(d), n5p(d) and n6

p(d), 18 at gridlevel 128× 128 usingn0

p(d) i.e. original data and 6 atgrid level 256× 256 using original data. This transitioncriterion is more stringent and robust as compared tothe maximum likelihood.

5.4. Qualitative and quantitative comparison

The reconstructed phantom image is compared with theMaximum Likelihood SGEM (ML-EM) algorithm and theFiltered Backprojection algorithm. The images in Fig. 15show that the WMREM using custom designed waveletfilters provides a better reconstruction. The ML-EM algo-rithm was run for the same amount of CPU time as theWMREM for a fair comparison. The RMS errors for thevarious algorithms are tabulated in Table 4. The lowestRMS error for the WMREM is a quantitative measure thatcorroborates the performance of the algorithm. A profilethrough the reconstructed phantom and the original phan-tom for qualitative comparisons is seen in Fig. 16.

5.5. Real PET data reconstruction

The WMREM reconstruction algorithm was tested onactual PET camera data. The data was is a brain slicecollected from an ECAT scanner. The data was also recon-structed using the ML-EM algorithm and the Filtered Back-projection algorithms. Fig. 17 demonstrates that theWMREM performs well and gives better quality recon-structed images.


Fig. 16. Intensity profiles for a vertical line 75 pixels from the left side through the phantom fort the WMREM and the single grid EM (ML-EM).

Table 4RMS errors for various reconstruction methods

Phantom WMREM SGEM(ML-EM)

Filteredbackprojection

MGEMlog-likelihood

Shepp 0.114 0.143 0.583 0.165

6. Conclusions

We have presented a new reconstruction algorithm thatutilizes the wavelet representation and spatio-frequencylocalization in EM based tomographic image reconstruc-tion. This algorithm can be extended to solving inverseproblems that use the EM estimation method.

Wavelets have been used primarily to form data sub-spacesby performing two-channel orthonormal filter analysis on theoriginal tube data. The HH data sub-spaces are not used sincethey contain most of the possible data collection noise. Wave-let synthesis with the remaining data sub-spaces is used to

reconstruct the image with EM estimation performedusing the data sub-spaces and grid levels at the same resolu-tion. The use of wavelet MR approach in this manner elim-inates the interpolation step that was used earlier in theMREM algorithm [3], and provides better reconstruction.

Wavelet-decomposition of estimated image at eachiteration is also used to develop a transition and stoppingcriterion. This method uses Daubechies least asymmetricwavelet �N � 4�; since its frequency response provides agood localization of the high-frequency contents of an image.

The data spaces should have appropriate frequencycomponents of the data for better reconstruction and thesedata spaces should be more de-correlated than the dataitself. This implies that the wavelet filters of the selectedwavelet basis should have a frequency response ideal tolocalize the data into MR data spaces used for reconstruc-tion. Hence, biorthogonal wavelet filters have been customdesigned for use in the wavelet-analysis and synthesis in thisalgorithm. To have an idea regarding the design of waveletfilters better suited for the nature of tomographic data,commonly known wavelets like the Daubechies orthogonal,biorthogonal spline, least asymmetric etc. were used in theframework of this algorithm. The orthogonal wavelets didnot perform well but the Daubechies biorthogonal splinefamily [26] performed quite well. This led to the use ofsecond order interpolating spline function as the dual scalingfunction in the lifting method to create new biorthogonalwavelet filters. The new custom designed wavelet filtersused for the WMREM algorithm have a frequency responseideal for tomographic data. These wavelet filters include theappropriate higher frequencies in the data spaces required forimproved image reconstruction. Also, the filters have a lowergain for the low frequencies, which is helpful in localizingmore high-frequency data in the high-pass filtered data space.

The new algorithm presented in this paper provides betterquality images for PET image reconstruction compared toearlier MG and single grid EM methods. The algorithm canbe extended to incorporate other derivatives of EM methodfor PET image reconstruction such as the Penalized EM [4].The algorithm, in general, can also be used for other appli-cations involving solutions for inverse problems that useEM as the estimation method. The algorithm can be paral-lelized for more efficient implementation. The estimationprocess using the three data spaces at each grid level canbe run in parallel to save time and make the algorithm faster.The MR part of the algorithm before the original data isutilized, helps provide a good initial estimate to the remain-ing MG part of algorithm.

Acknowledgements

The authors wish to thank Dr Kevin Wheeler of theNASA Ames Research Labs for discussion regardingcustom wavelet design. We also thank Sloan KetteringMedical Center, Dayton for providing PET data.


(a)

(b)

(c)

Fig. 17. Brain slice data reconstructed using the (a) WMREM (usingcustom designed wavelet), (b) SGEM and (c) Filtered Backprojection.

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Amar Raheja obtained BS in Physics in 1992 and MS in Physics in1994, both from Indian Institute of Technology, Kharagpur, India. Hewas with the Department of Radiology, University of Texas Southwes-tern Medical Center, Dallas from 1994–1997 as a PhD candidate inRadiological Sciences. He obtained his PhD in Bioengineering atUniversity of Toledo with a specialization in application of waveletsin medical imaging in 1999. He is currently an Assistant Professor ofComputer Science at Philadelphia University. His current researchinterests are image and signal processing using wavelets, medicalimaging and soft computing (neural networks and genetic algorithms).He is an associate member of IEEE, the IEEE engineering in medicineand biology society and the IEEE computer society.

Atam P. Dhawan obtained his BEng and MEng degrees in ElectricalEngineering from the University of Roorkee, Roorkee, India. He was aCanadian Commonwealth Fellow at the University of Manitoba wherehe completed his PhD in Electrical Engineering with specialization inmedical imaging and image analysis in 1985. In 1984, he won the firstprize and the Martin Epstein Award in the Symposium of ComputerApplication in Medical Care Paper Competition at the Eighth SCAMCAnnual Congress in Washington, DC, for his work on developing a 3Dimaging technique to detect early skin-cancer called melanoma. From1985 to 1988, he was an Assistant Professor in the Department ofElectrical Engineering at the University of Houston. Later, in 1988,he joined the University of Cincinnati as an Assistant Professorwhere he became Professor of Electrical and Computer Engineeringand Computer Science, and Radiology (joint appointment). From1990–1996, he was the Director of Center for Intelligent Vision andInformation System. From 1996–1998, he was Professor of ElectricalEngineering at the University of Texas at Arlington, and AdjunctProfessor of Radiology at the University of Texas Southwestern Medi-cal Center at Dallas. He is currently Professor in the Department ofElectrical & Computer Engineering at the New Jersey Institute of Tech-nology and Director of Medical Imaging and Informatics Laboratory.Dr Dhawan has published more than 50 research articles in refereedjournals, and edited books, and 85 research papers in refereed confer-ence proceedings. Dr Dhawan is a recipient of Martin Epstein Award(1984), National Institutes of Health FIRST Award (1988), Sigma-XiYoung Investigator Award (1992), University of Cincinnati FacultyAchievement Award (1994) and the prestigious IEEE Engineering inMedicine and Biology Early Career Achievement Award (1995). He isan Associate Editor of IEEE Transactions on Biomedical Engineering,Associate Editor of IEEE Transactions on Rehabilitation Engineering,and Editor of International Journal of Computing Information andTechnology. He has served on many IEEE EMBS professional commit-tees and has delivered Workshops on Intelligent Biomedical ImageAnalysis in IEEE EMBS International Conferences (1996 and 1997).He is the Chair of the “New Frontiers in Biomedical Engineering”Symposium at the World Congress 2000 on Medical Physics andBiomedical Engineering.His current research interests are medical imaging, multimodality brainmapping, intelligent image analysis, multigrid image reconstruction,wavelets, genetic algorithms, neural networks, adaptive learning andpattern recognition.

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