eﬃcientopmizaon basedonlocalshiinvariance( for(adap+ve ...€¦ · larar.$v.$pato$ bert...

Lara R. V. Pato Bert Vandeghinste, Stefaan Vandenberghe, Roel Van Holen

MEDISIP, Ghent University -‐ iMinds -‐ IBiTech, Ghent, Belgium

Workshop on Small Animal SPECT November 2012, Tucson, Arizona

FACULTY OF ENGINEERING AND ARCHITECTURE

Efficient Op+miza+on Based on Local Shi7-‐Invariance for Adap+ve SPECT Systems

Overview •  IntroducFon

•  Method

•  Image Quality

•  OpFmizaFon

•  ApplicaFon •  Case 1: Single-‐Pinhole RotaFng SPECT

•  Case 2: MulF-‐Pinhole StaFonary SPECT (U-‐SPECT-‐II)

•  Conclusion


Evaluate Image Quality for a given

purpose

Evaluate Image Quality for a given

purpose

Adap+ve SPECT

OpFmize system seRngs to

increase Image Quality

OpFmize system seRngs to

increase Image Quality

Need to be fast for adaptaFon in

real +me

Take informaFon from object under

study

Alter system seRngs

accordingly

3 IntroducFon Method ApplicaFon Conclusion


IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 58, NO. 5, OCTOBER 2011 2205

Adaptive Angular Sampling for SPECT ImagingNan Li and Ling-Jian Meng

Abstract—This paper presents an analytical approach forperforming adaptive angular sampling in single photon emis-sion computed tomography (SPECT) imaging. It allows for arapid determination of the optimum sampling strategy thatminimizes image variance in regions-of-interest (ROIs). Theproposed method consists of three key components: (a) a set ofclose-form equations for evaluating image variance and resolutionattainable with a given sampling strategy, (b) a gradient-basedalgorithm for searching through the parameter space to find theoptimum sampling strategy and (c) an efficient computation ap-proach for speeding up the search process. In this paper, we havedemonstrated the use of the proposed analytical approach with asingle-head SPECT system for finding the optimum distributionof imaging time across all possible sampling angles. Compared tothe conventional uniform angular sampling approach, adaptiveangular sampling allows the camera to spend larger fractions ofimaging time at angles that are more efficient in acquiring usefulimaging information. This leads to a significantly lowered imagevariance. In general, the analytical approach developed in thisstudy could be used with many nuclear imaging systems (such asSPECT, PET and X-ray CT) equipped with adaptive hardware.This strategy could provide an optimized sampling efficiency andtherefore an improved image quality.

Index Terms—Adaptive angular sampling, non-uniform object-space pixelation (NUOP) approach, single photon emission com-puted tomography (SPECT).

I. INTRODUCTION

S INGLE photon emission computed tomography (SPECT)is a commonly used nuclear imagingmodality for small an-

imal studies [1], [2]. One of the recent emphases in SPECT in-strumentation is to push for higher spatial resolution. Examplesof recent developments include the SemiSPECT reported byKastis et al. [4], the SiliSPECT under development by Petersonet al. [5], the MediSPECT proposed (and evaluated) by Accorsiet al. [6] and the U-SPECT-III proposed by Beekman et al. [7],a low-cost ultra-high resolution imager based on the second-generation image intensifier [8] and the use of a pre-existingSPECT camera, arranged in an extreme focusing geometry forultra-high resolution small animal SPECT imaging applications[9]. We have recently developed a prototype ultra-high reso-lution single photon emission microscope (SPEM) system formouse brain studies [10], [11]. This system delivers an ultra-high imaging resolution of around 100 in phantom studies.It was demonstrated that the current dual-headed SPEM systemis capable of visualizing a very small number of radiolabeled

Manuscript received April 03, 2011; revised July 26, 2011; accepted August05, 2011. Date of current version October 12, 2011.The authors are with the Department of Nuclear, Plasma, and Radiological

Engineering, University of Illinois at Urbana Champaign, Urbana-Champaign,IL 61801 USA (e-mail: [email protected]; [email protected]).Digital Object Identifier 10.1109/TNS.2011.2164935

cells in mouse brain [12]. Despite these promising results, theperformance of the SPEM system is limited by the common bot-tleneck for ultra-high resolution SPECT instrumentations—thelimited detection efficiency. Even with the large number of pin-holes in the entire system, the overall detection efficiency forthe SPEM system is typically or lower. The long imagingtime required for ultra-high resolution studies could precludemany interesting applications.This problem could be partially alleviated by using the adap-

tive imaging concept proposed by Barrett et al. [13], Clarksonet al. [14] and Freed et al. [15]. In an adaptive SPECT system,the system hardware could vary in real-time to maximize theefficiency for collecting useful imaging information regardinga given task, and therefore provide an optimum imaging perfor-mance. In [12] (Fig. 12), we have proposed the use of a vari-able aperture system with four sets of apertures that can be in-terchanged during an imaging study. Therefore, apertures withlarger pinholes can be used for localizing the target-region andhighly focusing ultra-high resolution apertures can then be usedfor a closer examination of the target region.In this paper, we propose an analytical approach for adaptive

angular sampling in SPECT imaging. This approach assigns anon-uniform distribution of imaging times across all possiblesampling angles, and the actual sampling strategy will be deter-mined based on the relative importance of each angle for col-lecting useful imaging information regarding a given imagingtask. With the adaptive angular sampling approach, an imagingstudy could start with a uniform time distribution across all pos-sible angles. During the study, the imaging information beingacquired and the input from the user (e.g., the target-region tobe examined) will be used to determine the optimum time dis-tribution based on the expected system performance measuredwith certain analytical performance indices.The adaptive angular sampling approach requires an efficient

computation method for searching through the parameter spaceto find the optimum time distribution in real time. For thispurpose, we have proposed a search algorithm that utilizes thegradient function of certain system performance indices, suchas image variance, with respect to imaging times at individualangles. To allow for a rapid optimization process, we have alsoincorporated a non-uniform object-space pixelation (NUOP)scheme that uses different pixel sizes adaptively according tothe characteristics of the object and the input from the user[3]. By combining the gradient-based search algorithm and theNUOP approach, one can refine the angular sampling strategyadaptively during an imaging study in near real time to achievean improved image quality.Although we have focused on the problem of adaptive

angular sampling in SPECT, the basic approach developed in

0018-9499/$26.00 © 2011 IEEE

•  RotaFng single-‐head SPECT

•  Adapt Fme t(k) spent at each angle k∈{1,…,K}, for a given total imaging Fme

t (1)

[1] N. Li and L.-J. Meng, IEEE Trans. Nucl. Sci. 58 (2011) [2] B. L. Franc et al., J. Nucl. Med. 49 (2008)



•  H

•  For angle k:

is determined by a constant t0, which if too big can leadto too much increase/decrease such that the CNR actuallydecreases with the new time sampling. To garantee that the CNRalways increased with each iteration, whenever the new timedistribution would lead to an decrease in the CNR we wouldreject the new t(k)’s and reduce t0. Another consideration is thata higher t0 speeds up the convergence to the maximum CNRvalue, and so when the variance values would start becomingvery close we would increase t0.

Also include that if delta(CNR)<0 in succession for 10 times,stop.

C. USPECT

y(k)= t

(k) ⇧H(k) ⇧ x (32)

y(k)= t

(k) ⇧H ⇧ x(k) (33)

H ⇧ x(k)= H ⇧T(k) ⇧ x(1)

= H(k) ⇧ x(1) (34)

x(k)= T(k) ⇧ x(1) (35)

H(k)= H ⇧T(k) (36)

F(k)e

j

= H(k)Tdiag

1

�

H(k)x(1)�

i

!

H(k)e

j (37)

y = H ⇧ x (38)

H =

0

B

B

@

t

(1) ⇧H(1)

t

(2) ⇧H(2)

...

t

(K) ⇧H(K)

1

C

C

A

(39)

t

(k) 2 R, k 2 {1, . . . ,K}H(k) the matrix formed by the rows k to k +

N

K

of H. Wecan then separate the projection data 1 in K components

y(k)= t

(k)H(k)x (40)

With this decomposition it can be shown [2], using 40 in thelikelihood expression 2, that the FIM is given by

F =

K

X

k=1

t

(k)F(k) (41)

with

F(k)= H(k)T

diag

1

�

H(k) ⇧ x�

i

!

H(k) (42)

We applied the local shift-invariant approximation to theF(k) matrices, F̃. F̃(j), with �

i

=

P

k

t

(k)�

F

(k)

i

(�F

(k)

i

theeigenvalues of F̃(k)

(j)).

y(k)= t

(k) ⇧H ⇧ x(k) (43)

H ⇧ x(k)= H ⇧T(k) ⇧ x(1)

= H(k) ⇧ x(1) (44)

x(k)= T(k) ⇧ x(1) (45)

H(k)= H ⇧T(k) (46)

F(k)e

j

= H(k)Tdiag

1

�

H(k)x(1)�

i

!

H(k)e

j (47)

III. SIMULATIONS

To validate our approach we used similar settings to the onesused in [2], as those has been shown to provide better imagequality using the non-uniform time sampling approach(/as thosehave been shown to benefit from the non-uniform samplingapproach). We had a 64⇥64⇥64 image with 0.2mm⇥0.2mm⇥0.2mm voxels; within this image there was a large spherewith a uniform low activity, a high activity ellipsoid in thecenter, and finally an off-center small sphere with intermediateactivity levels, which constituted the region of interest (ROI)/ thephantom consisted of a large sphere with a uniform low activity(background), a high activity ellipsoid in the center (25:1),and an off-center small sphere with intermediate activity levels(5:1), which was the region of interest (ROI). We simulateda single-head single-pinhole SPECT system, which scans theobject along 32 equally-spaced angles between 0

� and 360

�,and a total imaging time of 64 minutes / for a total of 64minutes. The collimator was made out of Tungsten and thepinhole was 0.3mm in diameter, located at a distance of 15mm

from the center of the image. The detector was 64 ⇥ 64 with0.7mm ⇥ 0.7mm pixels, located at 45mm from the pinhole /so that we get magnification of the object.

We used a 7-ray tracer [10] (chack Vunckx’s reference),taking sensitivity and resolution correction into account, toget the system matrix, which is needed to computed the F(k)

matrices used in the optimization algorithm, and also for thereconstructions which validated the approach. The operator Pwas chosen as a 3-D Gaussian filter with a 0.8mm FWHM.

To obtain the optimum imaging time / t(k)’s, we used theCNR at the center of the ROI as figure of merit / at the VOI(center of the ROI). We started from a constant/uniform imagingtime, with the time step t0 = 10; during the optimization, if theupdated CNR decreased its value, t0 ! 0.99t0; when the CNRvalues between iterations would not change more than 10

�i,with i = 5, 6, 7..., then t0 ! 2t0 - purely pragmatic approach,just to speed up the process and refine the optimization. ; t0

remained constant in the other cases. As the CNR values were ⇠10

�4, we considered convergence when the absolute differencebetween consecutive CNR values was lower than 10

�10.To validate both the optimization and the approximations

made to achive the CNR value, we also performed actualreconstructions, with both he uniform (UT) and the optimumnon-uniform time (NUT) sampling. To compute the CNR, thevariance was calculated based on 300 (600?) noise realizationsof the projection data and the CRC based on a reconstructionwith and without an extra impulse on voxel j of 200% of itsvalue. The number of reconstruction iterations was chosen tobe 500 (800?), as it was seen that the variance value was moreor less stable from that point on.

[1] N. Li and L.-‐J. Meng, IEEE Trans. Nucl. Sci. 58 (2011)

y(k)= t

(k)H(k)x (40)

With this decomposition it can be shown [2], using 40 in the likelihood expression 2, that the FIM is given by

F =

K

X

k=1

t

(k)F(k), F(k)

= H(k)Tdiag

1

�

H(k)x�

i

!

H(k) (41)

with

F(k)= H(k)T

diag

1

�

H(k)x�

i

!

H(k) (42)

We applied the local shift-invariant approximation to the F(k) matrices, F̃. F̃(j), with �

i

=

P

k

t

(k)�

F

(k)

i

(�F

(k)

i


(j)).

y(k)= t

(k)Hx(k) (43)

y = Hx (44)

Hx(k)= HT(k)x(1)

= H(k)x (45)

x(k)= T(k)x(1) (46)

H(k):= HT(k)

, x := x(1) (47)

Hx(k) ) H(k)x (48)

T(k) shifts the columns of H such that

CRC

j

⇡ ejTPGFej , V ar

j

⇡ ejTPGFGTPT ej ! CNR

j

=

CRC

j

p

V ar

j

(49)

V ar

j

⇡ ejTPGFGTPT ej (50)

CNR

j

=

CRC

j

p

V ar

j

(51)

F(k) ⇡ circ

n

F(k)ejo

= QTdiagh

�

F

(k)

i

i

Q (52)⇣

F(k)ej⌘

i

! W

i

·⇣

F(k)ej⌘

i

(53)

�

F

(k)

i

! max

⇣

0,<⇣

�

F

(k)

i

⌘⌘

(54)

�

F

i

=

X

k

t

(k)�

F

(k)

i

, �

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

(55)

�

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

(56)

F ⇡ QTdiag⇥

�

F

i

⇤

Q, �

F

i

=

X

k

t

(k)�

F

(k)

i

(57)

G ⇡ QTdiag⇥

�

G

i

⇤

Q, �

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

(58)

@V ar

j

@c

(k)⇡ ejTP

n

GF(k)G� 2GF(k)GFGo

PT ej (59)

@CRC

j

@c

(k)⇡ ejTP

n

GF(k) �GF(k)GFo

ej (60)

rotating cammera). For now we assume that the imaging time is the same for all angles and is incorporated in the systemmatrix (in this case it is just a multiplication factor). The log-likelihood function is

L(y|x) := log p(y|x) =M

X

i=1

(y

i

log y

i

(x)� y

i

(x)� log y

i

!) , (2)

where the second equality is due to the fact that we assume the components of the data vector y to be independentPoisson variables, as usually done in emission tomography. We can then compute the Fisher information matrix (FIM)by inserting expression (2) in its definition, and so we have

Fij

:= �E

@

2

@x

i

@x

j

L(y|x)�

=

M

X

m=1

h

mi

h

mj

y

m

, (3)

with Fij

the components of the FIM and E [·] the expectation operator. From (3), the total matrix can be simply writtenas

F = HT

diag

✓

1

y

i

◆

H, (4)

where diag

⇣

1y

i

⌘

is a diagonal matrix with each (i, i) element equal to 1y

i

, i 2 {1, . . . ,M}.In MAP reconstruction (ran to convergence), the image estimator x̂ is given by

ˆxMAP

(y) = argmax

x�0[L(y|x)� �R(x)] , (5)

where R(x) is a penalty function and � is a regularization parameter. The ML estimator is just a special case of expression(5), with � = 0. In both cases, the estimator (5) is shift-variant and nonlinear in y, so we cannot define a global impulseor frequency response, valid for all the image voxels. As such, Fessler et al. [4] defined the local impulse response at voxelj as

l

j

(

ˆx) = lim

�!0

E

⇥

ˆx(y(x+ �ej))⇤

� E [

ˆx(y(x))]

�

, (6)

with ej 2 RN⇥1 the j-th unit vector. In emission tomography this can be further simplified because the expectation valueof a large number of noisy measurements y reconstructed using ML or MAP is well approximated by a reconstruction ofthe noiseless projection data y, so

E [

ˆx(y)] ⇡ ˆx(y), (7)

and substituting (7) in (6) we get the linearized local impulse response at voxel j. For MAP, this approximation to thelocal impulse response takes the form [4]

l

j

(

ˆx) ⇡ [F+ �R]

�1Fej , (8)

where R is the Hessian matrix of the penalty function R(x). If instead of using MAP reconstruction one considerspost-filtered ML (PF-ML), (8) is replaced by [7, 8]

l

j

(

ˆx) ⇡ PGFej , (9)

with G the pseudoinverse (due to the fact that in pinhole SPECT the FIM is in general non-invertible) and P the post-filtering operator (seen as an N ⇥N matrix). The local contrast recovery coefficient (CRC) is defined as the jth elementof the local impulse response

CRC

j

(

ˆx) := l

j

j

(

ˆx) ⇡ ejTPGFej , (10)

with T the transpose operator, and this can be used as a measure of resolution [9]. It can also be shown [3] that thecovariance matrix for PF-ML reconstruction [7, 8] ran until convergence can be approximated as

Cov(

ˆx) ⇡ PGFGPT

, (11)

whose (j, j) element, the variance, provides a measure of the noise in the reconstruction

V ar

j

(

ˆx) := Cov

j

j

(

ˆx) ⇡ ejTPGFGPT ej . (12)

Finally, we can combine these two measures in the contrast-to-noise ratio (CNR)

CNR

j

=

CRC

j

p

V ar

j

, (13)

Mean projecFon data vector Fisher InformaFon Matrix

y(k)= t

(k)H(k)x (40)


F =

K

X

k=1

t

(k)F(k), F(k)

= H(k)Tdiag

1

�

H(k)x�

i

!

H(k) (41)

with

F(k)= H(k)T

diag

1

�

H(k)x�

i

!

H(k) (42)


i

=

P

k

t

(k)�

F

(k)

i

(�F

(k)

i


(j)).

y(k)= t

(k)Hx(k) (43)

Hx(k)= HT(k)x(1)

= H(k)x (44)

x(k)= T(k)x(1) (45)

H(k):= HT(k), x := x(1) (46)

Hx(k) ) H(k)x (47)


CRC

j

⇡ ejTPGFej , V ar

j


j

=

CRC

j

p

V ar

j

(48)

V ar

j


CNR

j

=

CRC

j

p

V ar

j

(50)

F(k) ⇡ circ

n

F(k)ejo

= QTdiagh

�

F

(k)

i

i

Q (51)⇣

F(k)ej⌘

i

! W

i

·⇣

F(k)ej⌘

i

(52)

�

F

(k)

i

! max

⇣

0,<⇣

�

F

(k)

i

⌘⌘

(53)

�

F

i

=

X

k

t

(k)�

F

(k)

i

, �

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

(54)

�

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

(55)

F ⇡ QTdiag⇥

�

F

i

⇤

Q, �

F

i

=

X

k

t

(k)�

F

(k)

i

(56)

G ⇡ QTdiag⇥

�

G

i

⇤

Q, �

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

(57)

@V ar

j

@c

(k)⇡ ejTP

n


PT ej (58)

@CRC

j

@c

(k)⇡ ejTP

n

GF(k) �GF(k)GFo

ej (59)

@CNR

j

@t

(k)/ ejTPQT

diag

h

�

F

(k)

i

·�

�

G

i

�2i

Qej (60)

y(k)= t

(k)H(k)x (40)


F =

K

X

k=1

t

(k)F(k), F(k)

= H(k)Tdiag

1

�

H(k)x�

i

!

H(k) (41)

with

F(k)= H(k)T

diag

1

�

H(k)x�

i

!

H(k) (42)


i

=

P

k

t

(k)�

F

(k)

i

(�F

(k)

i


(j)).

y(k)= t

(k)Hx(k) (43)

y = Hx (44)

Hx(k)= HT(k)x(1)

= H(k)x (45)

x(k)= T(k)x(1) (46)

H(k):= HT(k)

, x := x(1) (47)

Hx(k) ) H(k)x (48)


CRC

j

⇡ ejTPGFej , V ar

j


j

=

CRC

j

p

V ar

j

(49)

V ar

j


CNR

j

=

CRC

j

p

V ar

j

(51)

F(k) ⇡ circ

n

F(k)ejo

= QTdiagh

�

F

(k)

i

i

Q (52)⇣

F(k)ej⌘

i

! W

i

·⇣

F(k)ej⌘

i

(53)

�

F

(k)

i

! max

⇣

0,<⇣

�

F

(k)

i

⌘⌘

(54)

�

F

i

=

X

k

t

(k)�

F

(k)

i

, �

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

(55)

�

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

(56)

F ⇡ QTdiag⇥

�

F

i

⇤

Q, �

F

i

=

X

k

t

(k)�

F

(k)

i

(57)

G ⇡ QTdiag⇥

�

G

i

⇤

Q, �

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

(58)

@V ar

j

@c

(k)⇡ ejTP

n


PT ej (59)

@CRC

j

@c

(k)⇡ ejTP

n

GF(k) �GF(k)GFo

ej (60)


IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 58, NO. 5, OCTOBER 2011 2205

Adaptive Angular Sampling for SPECT ImagingNan Li and Ling-Jian Meng

Abstract—This paper presents an analytical approach forperforming adaptive angular sampling in single photon emis-sion computed tomography (SPECT) imaging. It allows for arapid determination of the optimum sampling strategy thatminimizes image variance in regions-of-interest (ROIs). Theproposed method consists of three key components: (a) a set ofclose-form equations for evaluating image variance and resolutionattainable with a given sampling strategy, (b) a gradient-basedalgorithm for searching through the parameter space to find theoptimum sampling strategy and (c) an efficient computation ap-proach for speeding up the search process. In this paper, we havedemonstrated the use of the proposed analytical approach with asingle-head SPECT system for finding the optimum distributionof imaging time across all possible sampling angles. Compared tothe conventional uniform angular sampling approach, adaptiveangular sampling allows the camera to spend larger fractions ofimaging time at angles that are more efficient in acquiring usefulimaging information. This leads to a significantly lowered imagevariance. In general, the analytical approach developed in thisstudy could be used with many nuclear imaging systems (such asSPECT, PET and X-ray CT) equipped with adaptive hardware.This strategy could provide an optimized sampling efficiency andtherefore an improved image quality.

Index Terms—Adaptive angular sampling, non-uniform object-space pixelation (NUOP) approach, single photon emission com-puted tomography (SPECT).

I. INTRODUCTION

S INGLE photon emission computed tomography (SPECT)is a commonly used nuclear imagingmodality for small an-

imal studies [1], [2]. One of the recent emphases in SPECT in-strumentation is to push for higher spatial resolution. Examplesof recent developments include the SemiSPECT reported byKastis et al. [4], the SiliSPECT under development by Petersonet al. [5], the MediSPECT proposed (and evaluated) by Accorsiet al. [6] and the U-SPECT-III proposed by Beekman et al. [7],a low-cost ultra-high resolution imager based on the second-generation image intensifier [8] and the use of a pre-existingSPECT camera, arranged in an extreme focusing geometry forultra-high resolution small animal SPECT imaging applications[9]. We have recently developed a prototype ultra-high reso-lution single photon emission microscope (SPEM) system formouse brain studies [10], [11]. This system delivers an ultra-high imaging resolution of around 100 in phantom studies.It was demonstrated that the current dual-headed SPEM systemis capable of visualizing a very small number of radiolabeled

Manuscript received April 03, 2011; revised July 26, 2011; accepted August05, 2011. Date of current version October 12, 2011.The authors are with the Department of Nuclear, Plasma, and Radiological

Engineering, University of Illinois at Urbana Champaign, Urbana-Champaign,IL 61801 USA (e-mail: [email protected]; [email protected]).Digital Object Identifier 10.1109/TNS.2011.2164935

cells in mouse brain [12]. Despite these promising results, theperformance of the SPEM system is limited by the common bot-tleneck for ultra-high resolution SPECT instrumentations—thelimited detection efficiency. Even with the large number of pin-holes in the entire system, the overall detection efficiency forthe SPEM system is typically or lower. The long imagingtime required for ultra-high resolution studies could precludemany interesting applications.This problem could be partially alleviated by using the adap-

tive imaging concept proposed by Barrett et al. [13], Clarksonet al. [14] and Freed et al. [15]. In an adaptive SPECT system,the system hardware could vary in real-time to maximize theefficiency for collecting useful imaging information regardinga given task, and therefore provide an optimum imaging perfor-mance. In [12] (Fig. 12), we have proposed the use of a vari-able aperture system with four sets of apertures that can be in-terchanged during an imaging study. Therefore, apertures withlarger pinholes can be used for localizing the target-region andhighly focusing ultra-high resolution apertures can then be usedfor a closer examination of the target region.In this paper, we propose an analytical approach for adaptive

angular sampling in SPECT imaging. This approach assigns anon-uniform distribution of imaging times across all possiblesampling angles, and the actual sampling strategy will be deter-mined based on the relative importance of each angle for col-lecting useful imaging information regarding a given imagingtask. With the adaptive angular sampling approach, an imagingstudy could start with a uniform time distribution across all pos-sible angles. During the study, the imaging information beingacquired and the input from the user (e.g., the target-region tobe examined) will be used to determine the optimum time dis-tribution based on the expected system performance measuredwith certain analytical performance indices.The adaptive angular sampling approach requires an efficient

computation method for searching through the parameter spaceto find the optimum time distribution in real time. For thispurpose, we have proposed a search algorithm that utilizes thegradient function of certain system performance indices, suchas image variance, with respect to imaging times at individualangles. To allow for a rapid optimization process, we have alsoincorporated a non-uniform object-space pixelation (NUOP)scheme that uses different pixel sizes adaptively according tothe characteristics of the object and the input from the user[3]. By combining the gradient-based search algorithm and theNUOP approach, one can refine the angular sampling strategyadaptively during an imaging study in near real time to achievean improved image quality.Although we have focused on the problem of adaptive

angular sampling in SPECT, the basic approach developed in

0018-9499/$26.00 © 2011 IEEE


Evalua+on of Image Quality •  AnalyFcal formulas for FOM at POI, based on Fisher InformaFon Matrix F •  To increase computaFonal efficiency, F can be simplified:

•  [1]: Non-‐Uniform Object-‐Space PixelaFon approach[3] •  Our method: Local-‐shi] invariance approximaFon[4,5]

Op+miza+on •  Fast opFmizaFon based on the gradient of the FOM

Applica+on •  Adapt Fme per angle in single-‐head single-‐pinhole rotaFng SPECT system[1]

•  Adapt Fme per bed posiFon in staFonary mulF-‐pinhole SPECT system (MILabs U-‐SPECT-‐II)

[1] N. Li and L.-‐J. Meng, IEEE Trans. Nucl. Sci. 58 (2011) [3] L.-‐J. Meng and N. Li, IEEE Trans. Nucl. Sci. 56 (2009)

[4] J. Qi and R. M. Leahy, IEEE Trans. Med. Imag. 19 (2000) [5] K. Vunckx et al., IEEE Trans. Med. Imag. 27 (2008)

Same phantom and system and protocol as in Li and Meng’s paper

In this study we apply these to find the optimum time spent:/We tested our approximations for image quality and the optimization in 2 cases:


We extend the formalism to the case of a stationary…, more complex system and more realistic phantom

@CRCj

@c(k)⇡ ejTP

n

GF(k) �GF(k)GFo

ej (60)

@CNRj

@t(k)/ ejTPQT diag

h

�F

(k)

i

·�

�G

i

�2i

Qej (61)

Grad(k)j

=

@CNRj

@t(k)/ ejTPQT diag

h

�F

(k)

i

·�

�G

i

�2i

QPT ej (62)

Grad(k)j

=

@(FOM)

@t(k)/ ejTPQT diag

h

�F

(k)

i

·�

�G

i

�2i

QPT ej (63)

CRCj

⇡ ejTPQT diag⇥

�G

i

�F

i

⇤

Qej (64)V ar

j

⇡ ejTPQT diag⇥

�G

i

⇤

QPT ej (65)

To keep the validity of these expressions, we used the LSI approximation only on the F(k) matrices, and then insertedthat in (??) to get F̃. Equations (15) and (16) remain valid for F̃(j), with �

i

=

P

k

t(k)�F

(k)

i

(�F

(k)

i

the eigenvalues ofF̃(k)

(j)).—–where �F

i

denotes the ith eigenvalue of ˜F(j), i 2 {1, . . . , N}, and ˜F1 is the first column of ˜F(j). Due to the block-circulant nature of ˜F(j), ˜F1 is obtained by a 3D circulant shift of Fj such that the value in position j is shifted to position1. This approximation greatly simplifies the calculation of 10 and 12, since in Fourier space the FIM and its pseudoinverseare diagonal and their elements can be computed from Fj using 3D discrete fast Fourier transforms [5]. More specifically,we chose to write the pseudoinverse G̃(j) as

˜G(j) = ˜F(j)+ = QTdiag⇥

�G

i

⇤

Q, �G

i

=

(

0, �F

i

= 0

1�

F

i

, �F

i

6= 0

, (66)

which can be shown to be the Moore-Penrose pseudoinverse of a (block) circulant matrix. Thus, following from the factthat QTQ = Id, we can write 10 and 12 as

CRCj

(

ˆx) ⇡ ejTPQTQG̃(j)QTQF̃(j)QTQej , (67)⇡ ejTPQT diag

⇥

�G

i

�F

i

⇤

Qej , (68)

V arj

(

ˆx) ⇡ ejTPQTQG̃(j)QTQF̃(j)QTQG̃(j)QTQej ,

⇡ ejTPQT diag⇥

�G

i

⇤

Qej . (69)

3 Simulations

To validate our approach we used similar settings to the ones used in [2], as those has been shown to provide betterimage quality using the non-uniform time sampling approach(/as those have been shown to benefit from the non-uniformsampling approach). We had a 64 ⇥ 64 ⇥ 64 image with 0.2mm ⇥ 0.2mm ⇥ 0.2mm voxels; within this image there wasa large sphere with a uniform low activity, a high activity ellipsoid in the center, and finally an off-center small spherewith intermediate activity levels, which constituted the region of interest (ROI)/ the phantom consisted of a large spherewith a uniform low activity (background), a high activity ellipsoid in the center (25:1), and an off-center small spherewith intermediate activity levels (5:1), which was the region of interest (ROI). We simulated a single-head single-pinholeSPECT system, which scans the object along 32 equally-spaced angles between 0

� and 360

�, and a total imaging time of64 minutes / for a total of 64 minutes. The collimator was made out of Tungsten and the pinhole was 0.3mm in diameter,located at a distance of 15mm from the center of the image. The detector was 64⇥64 with 0.7mm⇥0.7mm pixels, locatedat 45mm from the pinhole / so that we get magnification of the object.

We used a 7-ray tracer [10] (chack Vunckx’s reference), taking sensitivity and resolution correction into account, to getthe system matrix, which is needed to computed the F(k) matrices used in the optimization algorithm, and also for thereconstructions which validated the approach. The operator P was chosen as a 3-D Gaussian filter with a 0.8mm FWHM.

To obtain the optimum imaging time / t(k)’s, we used the CNR at the center of the ROI as figure of merit / atthe VOI (center of the ROI). We started from a constant/uniform imaging time, with the time step t0 = 10; during



•  Method


•  OpFmizaFon



•  Conclusion


1.  Fisher InformaFon-‐based approximaFon for Post-‐Filtered MLEM ran to convergence[5,6,7,8], at voxel j:

y(k)= t

(k)H(k)x (40)


F =

K

X

k=1

t

(k)F(k), F(k)

= H(k)Tdiag

1

�

H(k)x�

i

!

H(k) (41)

with

F(k)= H(k)T

diag

1

�

H(k)x�

i

!

H(k) (42)


i

=

P

k

t

(k)�

F

(k)

i

(�F

(k)

i


(j)).

y(k)= t

(k)Hx(k) (43)

Hx(k)= HT(k)x(1)

= H(k)x (44)

x(k)= T(k)x(1) (45)

H(k):= HT(k), x := x(1) (46)

Hx(k) ) H(k)x (47)


CRC

j

⇡ ejTPGFej , V ar

j


j

=

CRC

j

p

V ar

j

(48)

V ar

j


CNR

j

=

CRC

j

p

V ar

j

(50)

F(k) ⇡ circ

n

F(k)ejo

= QTdiagh

�

F

(k)

i

i

Q (51)⇣

F(k)ej⌘

i

! W

i

·⇣

F(k)ej⌘

i

(52)

�

F

(k)

i

! max

⇣

0,<⇣

�

F

(k)

i

⌘⌘

(53)

�

F

i

=

X

k

t

(k)�

F

(k)

i

(54)

�

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

(55)

F ⇡ QTdiag⇥

�

F

i

⇤

Q, �

F

i

=

X

k

t

(k)�

F

(k)

i

(56)

G ⇡ QTdiag⇥

�

G

i

⇤

Q, �

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

(57)

@V ar

j

@c

(k)⇡ ejTP

n


PT ej (58)

@CRC

j

@c

(k)⇡ ejTP

n

GF(k) �GF(k)GFo

ej (59)

@CNR

j

@t

(k)/ ejTPQT

diag

h

�

F

(k)

i

·�

�

G

i

�2i

Qej (60)

Pseudoinverse of F computaFonal challenge

y(k)= t

(k)H(k)x (40)


F =

K

X

k=1

t

(k)F(k), F(k)

= H(k)Tdiag

1

�

H(k)x�

i

!

H(k) (41)

with

F(k)= H(k)T

diag

1

�

H(k)x�

i

!

H(k) (42)


i

=

P

k

t

(k)�

F

(k)

i

(�F

(k)

i


(j)).

y(k)= t

(k)Hx(k) (43)

Hx(k)= HT(k)x(1)

= H(k)x (44)

x(k)= T(k)x(1) (45)

H(k):= HT(k), x := x(1) (46)

Hx(k) ) H(k)x (47)


CRC

j

⇡ ejTPGFej , V ar

j


j

=

CRC

j

p

V ar

j

(48)

V ar

j


CNR

j

=

CRC

j

p

V ar

j

(50)

F(k) ⇡ circ

n

F(k)ejo

= QTdiagh

�

F

(k)

i

i

Q (51)⇣

F(k)ej⌘

i

! W

i

·⇣

F(k)ej⌘

i

(52)

�

F

(k)

i

! max

⇣

0,<⇣

�

F

(k)

i

⌘⌘

(53)

�

F

i

=

X

k

t

(k)�

F

(k)

i

(54)

�

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

(55)

F ⇡ QTdiag⇥

�

F

i

⇤

Q, �

F

i

=

X

k

t

(k)�

F

(k)

i

(56)

G ⇡ QTdiag⇥

�

G

i

⇤

Q, �

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

(57)

@V ar

j

@c

(k)⇡ ejTP

n


PT ej (58)

@CRC

j

@c

(k)⇡ ejTP

n

GF(k) �GF(k)GFo

ej (59)

@CNR

j

@t

(k)/ ejTPQT

diag

h

�

F

(k)

i

·�

�

G

i

�2i

Qej (60)

3D DFT

[4] J. Qi and R. M. Leahy, IEEE Trans. Med. Imag. 19 (2000) [5] K. Vunckx et al., IEEE Trans. Med. Imag. 27 (2008) [6] J. A. Fessler, IEEE Trans. Image Process. 5 (1996)

[7] J. A. Fessler and W. L. Rogers, IEEE Trans. Image Process. 5 (1996) [8] J. Nuyts et al., Methods 48 (2009)


2.  Local Shi]-‐Invariance assumpFon on F(k) [4,5,8]:

Eigenvalues – obtained using Q Block-‐Circulant Matrix

Post-‐Filter



4.  F and G also become block-‐circulant, with:

5.  In the OpFmizaFon we use:

Grad

(k)j

=

@CNR

j

@t

(k)/ ejTPQT

diag

h

�

F

(k)

i

·�

�

G

i

�2i

QPT ej (61)


i

=

P

k

t

(k)�

F

(k)

i

(�F

(k)

i


(j)).—–where �

F

i


˜G(j) =

˜F(j)+ = QTdiag⇥

�

G

i

⇤

Q, �

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

, (62)


CRC

j

(

ˆx) ⇡ ejTPQTQG̃(j)QTQF̃(j)QTQej , (63)⇡ ejTPQT

diag

⇥

�

G

i

�

F

i

⇤

Qej , (64)

V ar

j

(


⇡ ejTPQT

diag

⇥

�

G

i

⇤

Qej . (65)

3 Simulations


� and 360



To obtain the optimum imaging time / t(k)’s, we used the CNR at the center of the ROI as figure of merit / atthe VOI (center of the ROI). We started from a constant/uniform imaging time, with the time step t0 = 10; duringthe optimization, if the updated CNR decreased its value, t0 ! 0.99t0; when the CNR values between iterations wouldnot change more than 10

�i, with i = 5, 6, 7..., then t0 ! 2t0 - purely pragmatic approach, just to speed up the processand refine the optimization. ; t0 remained constant in the other cases. As the CNR values were ⇠ 10

�4, we consideredconvergence when the absolute difference between consecutive CNR values was lower than 10

�10.To validate both the optimization and the approximations made to achive the CNR value, we also performed actual

reconstructions, with both he uniform (UT) and the optimum non-uniform time (NUT) sampling. To compute the CNR,the variance was calculated based on 300 (600?) noise realizations of the projection data and the CRC based on areconstruction with and without an extra impulse on voxel j of 200% of its value. The number of reconstruction iterationswas chosen to be 500 (800?), as it was seen that the variance value was more or less stable from that point on.

//To validate our approach, we performed actual reconstructions, with both the uniform time (UT) and optimum

non-uniform time (NUT) distributions. To compute the CNR, the CRC was calculated based on a reconstruction with

H(k) the matrix formed by the rows k to k +

N

K

of H. We can then separate the projection data 1 in K components

y(k)= t

(k)H(k)x (40)


F =

K

X

k=1

t

(k)F(k), F(k)

= H(k)Tdiag

1

�

H(k)x�

i

!

H(k) (41)

with

F(k)= H(k)T

diag

1

�

H(k)x�

i

!

H(k) (42)


i

=

P

k

t

(k)�

F

(k)

i

(�F

(k)

i


(j)).

y(k)= t

(k)Hx(k) (43)

y = Hx (44)

Hx(k)= HT(k)x(1)

= H(k)x (45)

x(k)= T(k)x(1) (46)

H(k):= HT(k)

, x := x(1) (47)

Hx(k) ) H(k)x (48)


CRC

j

⇡ ejTPGFej , V ar

j


j

=

CRC

j

p

V ar

j

(49)

V ar

j


CNR

j

=

CRC

j

p

V ar

j

(51)

F(k) ⇡ circ

n

F(k)ejo

= QTdiagh

�

F

(k)

i

i

Q (52)

⇣

F(k)ej⌘

i

! W

i

·⇣

F(k)ej⌘

i

, W

i

= max

✓

0, 1� |~xi

� ~x

j

|N

◆

(53)

�

F

(k)

i

! max

⇣

0,<⇣

�

F

(k)

i

⌘⌘

(54)

�

F

i

=

X

k

t

(k)�

F

(k)

i

, �

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

(55)

�

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

(56)

F ⇡ QTdiag⇥

�

F

i

⇤

Q, �

F

i

=

X

k

t

(k)�

F

(k)

i

(57)

G ⇡ QTdiag⇥

�

G

i

⇤

Q, �

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

(58)

@V ar

j

@c

(k)⇡ ejTP

n


PT ej (59)

@CRC

j

@c

(k)⇡ ejTP

n

GF(k) �GF(k)GFo

ej (60)

@CNR

j

@t

(k)/ ejTPQT

diag

h

�

F

(k)

i

·�

�

G

i

�2i

Qej (61)

Grad

(k)j

=

@CNR

j

@t

(k)/ ejTPQT

diag

h

�

F

(k)

i

·�

�

G

i

�2i

QPT ej (62)

CRC

j

⇡ ejTPQT

diag

⇥

�

G

i

�

F

i

⇤

Qej (63)V ar

j

⇡ ejTPQT

diag

⇥

�

G

i

⇤

QPT ej (64)


i

=

P

k

t

(k)�

F

(k)

i

(�F

(k)

i



F

i


˜G(j) =

˜F(j)+ = QTdiag⇥

�

G

i

⇤

Q, �

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

, (65)


CRC

j

(


diag

⇥

�

G

i

�

F

i

⇤

Qej , (67)

V ar

j

(


⇡ ejTPQT

diag

⇥

�

G

i

⇤

Qej . (68)

3 Simulations


� and 360




�i, with i = 5, 6, 7..., then t0 ! 2t0 - purely pragmatic approach, just to speed up the process

ComputaFons reduced to element by element mulFplicaFons and FFTs


N

K


y(k)= t(k)H(k)x (40)


F =

K

X

k=1

t(k)F(k), F(k)= H(k)T diag

1

�

H(k)x�

i

!

H(k) (41)

with

F(k)= H(k)T diag

1

�

H(k)x�

i

!

H(k) (42)

We applied the local shift-invariant approximation to the F(k) matrices, F̃. F̃(j), with �i

=

P

k

t(k)�F

(k)

i

(�F

(k)

i


(j)).

y(k)= t(k)Hx(k) (43)

y = Hx (44)

Hx(k)= HT(k)x(1)

= H(k)x (45)

x(k)= T(k)x(1) (46)

H(k):= HT(k), x := x(1) (47)

Hx(k) ) H(k)x (48)


CRCj

⇡ ejTPGFej , V arj

⇡ ejTPGFGTPT ej ! CNRj

=

CRCj

p

V arj

(49)

V arj


CNRj

=

CRCj

p

V arj

(51)

F(k) ⇡ circn

F(k)ejo

= QT diagh

�F

(k)

i

i

Q (52)

⇣

F(k)ej⌘

i

! Wi

·⇣

F(k)ej⌘

i

, Wi

= max

✓

0, 1� |~xi

� ~xj

|N

◆

(53)

�F

(k)

i

! max

⇣

0,<⇣

�F

(k)

i

⌘⌘

(54)

�F

i

=

X

k

t(k)�F

(k)

i

, �G

i

=

(

0, �F

i

= 0

1�

F

i

, �F

i

6= 0

(55)

�G

i

=

(

0, �F

i

= 0

1�

F

i

, �F

i

6= 0

(56)

F ⇡ QT diag⇥

�F

i

⇤

Q, �F

i

=

X

k

t(k)�F

(k)

i

(57)

G ⇡ QT diag⇥

�G

i

⇤

Q, �G

i

=

(

0, �F

i

= 0

1�

F

i

, �F

i

6= 0

(58)

@V arj

@c(k)⇡ ejTP

n


PT ej (59)


N

K


y(k)= t(k)H(k)x (40)


F =

K

X

k=1


1

�

H(k)x�

i

!

H(k) (41)

with

F(k)= H(k)T diag

1

�

H(k)x�

i

!

H(k) (42)


=

P

k

t(k)�F

(k)

i

(�F

(k)

i


(j)).

y(k)= t(k)Hx(k) (43)

y = Hx (44)

Hx(k)= HT(k)x(1)

= H(k)x (45)

x(k)= T(k)x(1) (46)

H(k):= HT(k), x := x(1) (47)

Hx(k) ) H(k)x (48)


CRCj



=

CRCj

p

V arj

(49)

V arj


CNRj

=

CRCj

p

V arj

(51)

F(k) ⇡ circn

F(k)ejo

= QT diagh

�F

(k)

i

i

Q (52)

⇣

F(k)ej⌘

i

! Wi

·⇣

F(k)ej⌘

i

, Wi

= max

✓

0, 1� |~xi

� ~xj

|N

◆

(53)

�F

(k)

i

! max

⇣

0,<⇣

�F

(k)

i

⌘⌘

(54)

�F

i

=

X

k

t(k)�F

(k)

i

, �G

i

=

(

0, �F

i

= 0

1�

F

i

, �F

i

6= 0

(55)

�G

i

=

(

0, �F

i

= 0

1�

F

i

, �F

i

6= 0

(56)

F ⇡ QT diag⇥

�F

i

⇤

Q, �F

i

=

X

k

t(k)�F

(k)

i

(57)

G ⇡ QT diag⇥

�G

i

⇤

Q, �G

i

=

(

0, �F

i

= 0

1�

F

i

, �F

i

6= 0

(58)

@V arj

@c(k)⇡ ejTP

n


PT ej (59)


N

K


y(k)= t(k)H(k)x (40)


F =

K

X

k=1


1

�

H(k)x�

i

!

H(k) (41)

with

F(k)= H(k)T diag

1

�

H(k)x�

i

!

H(k) (42)

F =

X

k

t(k)F(k) (43)


=

P

k

t(k)�F

(k)

i

(�F

(k)

i


(j)).

y(k)= t(k)Hx(k) (44)

y = Hx (45)

Hx(k)= HT(k)x(1)

= H(k)x (46)

x(k)= T(k)x(1) (47)

H(k):= HT(k), x := x(1) (48)

Hx(k) ) H(k)x (49)


CRCj



=

CRCj

p

V arj

(50)

V arj


CNRj

=

CRCj

p

V arj

(52)

F(k) ⇡ circn

F(k)ejo

= QT diagh

�F

(k)

i

i

Q (53)

⇣

F(k)ej⌘

i

! Wi

·⇣

F(k)ej⌘

i

, Wi

= max

✓

0, 1� |~xi

� ~xj

|N

◆

(54)

�F

(k)

i

! max

⇣

0,<⇣

�F

(k)

i

⌘⌘

(55)

�F

i

=

X

k

t(k)�F

(k)

i

, �G

i

=

(

0, �F

i

= 0

1�

F

i

, �F

i

6= 0

(56)

�G

i

=

(

0, �F

i

= 0

1�

F

i

, �F

i

6= 0

(57)

F ⇡ QT diag⇥

�F

i

⇤

Q, �F

i

=

X

k

t(k)�F

(k)

i

(58)

G ⇡ QT diag⇥

�G

i

⇤

Q, �G

i

=

(

0, �F

i

= 0

1�

F

i

, �F

i

6= 0

(59)




•  Method


•  OpFmizaFon



•  Conclusion


Op+miza+on (maximize CNRj)

Accept t(k)new’s

For each k: Gradj(k)>Ek[Gradj(k)]

?

For each k: Compute Gradj(k)

Compute eigenvalues of F(k)

Choose iniFal value for t(k)’s (e.g. uniform) Choose POI j

Compute CNRjnew & ΔCNR=CNRjnew –CNRjold

t(k)new>t(k)old

t(k)new<t(k)old (if t(k)old>0)

ΔCNR>0?

STOP

Reject t(k)new’s & decrease opt. step

Yes

No

Yes

No


|ΔCNR|<ε?

Yes

No

@CRC

j

@c

(k)⇡ ejTP

n

GF(k) �GF(k)GFo

ej (60)

@CNR

j

@t

(k)/ ejTPQT

diag

h

�

F

(k)

i

·�

�

G

i

�2i

Qej (61)

Grad

(k)j

=

@CNR

j

@t

(k)/ ejTPQT

diag

h

�

F

(k)

i

·�

�

G

i

�2i

QPT ej (62)

CRC

j

⇡ ejTPQT

diag

⇥

�

G

i

�

F

i

⇤

Qej (63)V ar

j

⇡ ejTPQT

diag

⇥

�

G

i

⇤

QPT ej (64)


i

=

P

k

t

(k)�

F

(k)

i

(�F

(k)

i



F

i


˜G(j) =

˜F(j)+ = QTdiag⇥

�

G

i

⇤

Q, �

G

i

=

(

0, �

F

i

= 0

1�

F

i

, �

F

i

6= 0

, (65)


CRC

j

(


diag

⇥

�

G

i

�

F

i

⇤

Qej , (67)

V ar

j

(


⇡ ejTPQT

diag

⇥

�

G

i

⇤

Qej . (68)

3 Simulations


� and 360




�i, with i = 5, 6, 7..., then t0 ! 2t0 - purely pragmatic approach, just to speed up the process




•  Method


•  OpFmizaFon



•  Conclusion


Case 1) Single-‐pinhole rota+ng SPECT system

Original phantom[1]

(transversal)

3 * Act(POI)

POI

0.2 * Act(POI)

[1] N. Li and L.-‐J. Meng, IEEE Trans. Nucl. Sci. 58 (2011) [9] C. Vanhove et al., Eur. J. Nuc. Med. Mol. Imaging 34 (2007)

•  Single-‐head single-‐pinhole SPECT

•  RotaFon: 32 Angles

•  Image Size: 12.8mmx12.8mmx12.8mm

•  Total AcFvity=18.5MBq * 64 minutes

•  System Matrix modeled by ray-‐tracing (7-‐ray pinhole aperture subsampling[9])

•  ReconstrucFon algorithm: PF-‐MLEM, Gaussian filter



Op+miza+on Results

•  32 Angles

•  RelaFve increase in CNRj: 34% è Op+miza+on works and agrees with expecta+ons

•  Very fast: ~1-‐2 minutes

8% total Fme

0

0 20 40 60 80

0.8

1

·10�3

Optimization IterationC

NR

-NoE

xtra

Analytical

Reconstructions

0 20 40 60 80

0.9

1

1.1

1.2

·10�3

Optimization Iteration

CN

R(·1

0

�3)

0 20 40 60 80

0.8

1

1.2

·10�3


CN

R(·1

0

�3)

0 20 40

9

9.5

·10�3


CN

R(·1

0

�3)

0 10 20

9.26

9.265

9.27

·10�3


CN

R(·1

0

�3)

Figure 2: Plot of the CNR versus iteration during the optimization. / Plot of the analytical CNR versus iteration duringthe optimization. / Plot of the CNR versus iteration, using the optimization algorithm with changing time step.

Final Time Distribution

14 IntroducFon Method ApplicaFon Conclusion [1] N. Li and L.-‐J. Meng, IEEE Trans. Nucl. Sci. 58 (2011)


0 20 40 60 80

0.8

1

·10�3

Optimization IterationC

NR

-NoE

xtra

Analytical

Reconstructions

0 20 40 60 80

0.8

1

1.2

·10�3


CN

R(·1

0

�3)

0 20 40

9

9.5

·10�3


CN

R(·1

0

�3)

0 10 20

9.26

9.265

9.27

·10�3


CN

R(·1

0

�3)


Valida+on of Image Quality calcula+on

•  CNRAN: AnalyFcal calculaFon •  CNRREC: Computed from

reconstrucFons: -  800 MLEM steps,

Gaussian Post-‐Filtering -  600 noise realizaFons

•  CNRAN increase=34% •  CNRREC increase=38% •  Rela+ve difference between CNRAN and

CNRREC is ~ 30%

CNRAN

CNRREC

ReconstrucFon realizaFon from Uniform Time DistribuFon




•  Method


•  OpFmizaFon



•  Conclusion


Case 2) U-‐SPECT-‐II system

•  MILabs U-‐SPECT-‐II: •  Cylindrical collimator, 75 pinh. •  Shielding cylinder •  3 staFonary detectors

•  MOBY Phantom[10] scaled to 89%, 99mTc-‐tetrofosmin

•  Total AcFvity=75 MBq * 45 minutes

•  System Matrix modeled by ray-‐tracing[9]

•  ReconstrucFon algorithm: PF-‐MLEM, Gaussian filter

[9] C. Vanhove et al., Eur. J. Nuc. Med. Mol. Imaging 34 (2007) [10] W. Branderhorst et al., Phys. Med. Biol. 57 (2012)

[11] W. P. Segars et al., Mol. Imaging Biol. 6 (2004) [12] B. Vastenhouw and F. Beekman, J. Nucl. Med. 48 (2007)


CFOV

Collimator

Bed

t(1) t(k)

12mm

26mm

7mm


Op+miza+on Results 9 Bed PosiFons

•  POI in the heart •  RelaFve increase in CNRj: 10% •  ~ 5-‐10 minutes

0 10 20 30 40 50 60 70 80

0.9

1

1.1

1.2x 10

−3

Iteration

CN

R

0 20 40 60 80

0.8

1

·10�3


CN

R

Analytical

Reconstructions

0 20 40 60 80

0.8

1

1.2

·10�3


CN

R(·1

0

�3)

0 20 40

9

9.5

·10�3


CN

R(·1

0

�3)


CNRAN

23% total Fme

0

Uniform Time

OpFmal Time



MOBY Phantom 3.4x10-‐3 5.1x10-‐3 8.9x10-‐3 9.3x10-‐3

MOBY Phantom Without extra-‐cardiac acFvity

5.5x10-‐3 7.7x10-‐3 13.5x10-‐3 13.9x10-‐3

Valida+on of Image Quality calcula+on •  Same total Fme (45min)

•  Uniform Fme per bed posiFon

•  Transversal bed shi]s

CNRj values (analy+cal)

Transversal Sagital

POI 7.6 * Act(POI)

Bed Posi+ons

Phantom

19 IntroducFon Method ApplicaFon Conclusion [10] W. Branderhorst et al., Phys. Med. Biol. 57 (2012)



•  Method


•  OpFmizaFon



•  Conclusion


Conclusion

•  We presented an efficient method to: 1.  Evaluate Image Quality at a POI 2.  OpFmize Fme per angle (Case 1)/bed posiFon (Case 2)

•  OpFmizaFon works in both cases studied, more impact in Case 1

•  Case 1: analyFcal CNR values consistent with reconstrucFons

•  Case 2: analyFcal CNR values for the full MOBY phantom seem to agree with literature[10], but further invesFgaFon is needed

21 IntroducFon Method ApplicaFon Conclusion [10] W. Branderhorst et al., Phys. Med. Biol. 57 (2012)


Special Thanks To:

Contact: Lara Pato ( [email protected] )

Medical Imaging and Signal Processing Group (MEDISIP) Ghent University, Belgium

http://medisip.elis.ugent.be/

•  Jinyi Qi •  Johan Nuyts •  Kathleen Vunckx

•  Nan Li •  Ling-‐Jian Meng •  Woutjan Branderhorst


eﬃcientopmizaon basedonlocalshiinvariance( for(adap+ve ...€¦ · larar.$v.$pato$ bert...

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