rapid constructions of circular-orbit pinhole spect...

National Central University

Department of Optics and Photonics

Rapid Constructions of Circular-OrbitPinhole SPECT Imaging System Matrices by

Gaussian Interpolation Method Combined with Geometric Parameter Estimations (GIMGPE)

Ming-Wei Lee1, Wei-Tso Lin1, Yu-Ching Ni2, Meei-Ling Jan2, Yi-Chun Chen1*

1. Department of Optics and Photonics,National Central University, Taiwan.

2. Institute of Nuclear Energy Research, Taiwan.


Department of Optics and Photonics2

Outline• Interpolation of Imaging System Matrices

• Procedures of GIMGPE– Simplified Grid-Scan Experiment– Image Parameterization– Imaging Properties Database– Projection Centroids Model– Gaussian Coefficients Estimation

• Results– OSEM Reconstructions of a Derenzo Phantom– Detectability of SKE/BKE Tasks

• Conclusion



Imaging System Matrix

Linear Digital-Imaging System:

g = Hf + n.

g ─ Image, Discrete, pixel

H ─ Imaging System Matrix, Discrete, voxel-into-pixel

f ─ Object, Discrete, voxel

n ─ Noise Vector.

H = [Hdetector Hgeom] A

2. Simulation1. Measurement

3. Combination

3



Interpolation of Imaging System Matrices• The interpolation of H matrix was firstly proposed by Rowe

et al. [1].

[1] R. K. Rowe et al., Journal of Nuclear Medicine, vol. 34, no. 3, pp. 474-480, 1993.

Coarse-Grid point-source

pinhole

H = R + BR: Total counts & centroid locations

B: Blur functions on the camera

camera

R is a slowly varying function of point-source locations. B is a function of centroid locations.

A finer-grid H matrix can be interpolated.

4



From Stationary to Circular Orbit

5

Pinhole Diameter: 1.5 mm

Camera # 1 Camera # 2

Spherical FOV (Dia. 36 mm.)

d: 33.5 mmf: 91 mm127.4

mm

X-SPECT System:

1. Dual- Headed Gamma Camera,

2. 58 × 58 Discrete NaI(Tl) Crystal Array.

d: 33.5 mm

Previously, the proposed GIMGPE has tested with FASTSPECT II [2].

[2] M.-W. Lee and Y.-C. Chen, 2011 IEEE NSS/MIC Conference Record, pp. 2664 – 2667, 2011.



Simplified Grid-Scan ExperimentA simplified grid-scan pattern contains

• The boundary of Field-of-View, and

• Three inner planes that are parallel to the camera plane.

Reducing ~75% voxels of a full 3D grid-scan pattern

at 0° projection angle



Image Parameterization

Projection of a point source steppedon a voxel within the FOV of apinhole SPECT system.

2 2

2 2

c c

( ) ( )( , ) exp[ ],2 2

: Amplitude, and : Centroid coordinates,:Standard deviation.

c ci

u u v vAh u v

Au v

Circular Gaussian Function

Photon Counts: ~ 6000 (counts)



Circular Orbit Pinhole SPECT System

x

z: rotation axis

Pinhole

Image Planey 58×58 pixels

Pinhole Projection Image

(uc, vc)

d

f

(xo, yo, zo)

(xp, yp, zp) (uc, vc, zi)θ

Object Space

D = f + d and M = f / d.(mm), )z-(z )v-(y )u-(x

(mm), )z-(z )y-(y )x-(x

2ip

2cp

2cp

2op

2op

2op

f

d



Amplitude (D)m around the same pixel

(15,15)

(43,43)

(29,29)

(10,10)

(10,50)

(29,50)

105 110 115 120 125 130 135 140 1450

0.5

1

1.5

2

2.5x 104

110 115 120 125 130 1352000

3000

4000

5000

6000

7000

8000

9000

10000

11000

12000

110 115 120 125 130 135 1402000

3000

4000

5000

6000

7000

8000

9000

10000

11000

116 118 120 122 124 126 128 1303000

3500

4000

4500

5000

5500

6000

6500

7000

7500

8000

110 115 120 125 130 135 1402000

3000

4000

5000

6000

7000

8000

9000

10000

11000

116 118 120 122 124 126 128 1303000

3500

4000

4500

5000

5500

6000

6500

7000

7500

8000

(x-pixel, y-pixel)Image Plane

)Distance/1(.Amp

D

Amp.



σ(M)m around the same pixel

(15,15)

(43,43)

(29,29)

(10,10)

(10,50)

(29,50)

1.5 2 2.5 3 3.5 4 4.5 5 5.5 60.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2 2.5 3 3.5 4 4.5 50.7

0.8

0.9

1

1.1

1.2

1.3

1.4

2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.20.9

0.95

1

1.05

1.1

1.15

1.2

1.25

2 2.5 3 3.5 4 4.50.7

0.8

0.9

1

1.1

1.2

1.3

1.42 2.5 3 3.5 4 4.5

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.20.9

0.95

1

1.05

1.1

1.15

1.2

1.25

(x-pixel, y-pixel)Image Plane

ionMagnificat

M

σ



Imaging Properties Database

Fitting the imaging property curves:• 4th-order Laurent polynomial for Amplitude(D)m.• Linear function for σ(M)m.

1 1 1 13 2

2 2

1 1 1( ) ,

( ) ,where denotes the -th pixel.

m

m

Amplitude D a b c dD D D

M a M bm m



Imaging Properties Database

100 110 120 130 140 1500

0.5

1

1.5

2

2.5 x 104

Distance (D, d+f)

Am

plitu

de (c

ount

s)

Experiment DataEstimation Data

1 2 3 4 5 60.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Magnification (M, f/d)

Stan

dard

Dev

iatio

n (p

ixel

s)

Experiment DataEstimation Data

A full imaging properties database has [58 × 58 × (4+2)] elements: fitting coefficients of Amp.(D)m and σ(M)m curves.

Amp.(D)mσ(M)m



Projection Centroids Model

Geometry of a pinhole SPECT imaging system [3].

[3] D. Beque et al, IEEE Trans. Med. Imag., vol. 24, pp. 180-190, 2005.

Projection centroids are calculated as ( , ),where is the projection angle.

img imgu v

cos ( , , ) cos( , , )

imgu

m xu f m ed y

sin ( , , ) sin( , , )

imgv

m zv f m ed y

This projection model is proposed by D. Beque et al [3].



Procedure of GIMGPEGeometrical Calibration

Projection Centroids Model

Simplified Grid-Scan Experiment

Image Parameterization

Imaging Properties Database:Amp.(D)m and σ(M)m

Rotate the Object Spaceat θ projection angle

Estimate the Projection Centroids of each voxel

Calculate the Blur Functions withthe corresponding projection centroids,D and M of each voxel

16 projection angles

θ: 0° ~ 337.5°

Hθ

14



Configuration of Derenzo Phantom

15

Derenzo Phantom

pinhole

Camera # 1

Rod diameter: 1.7, 2.3, and 3.4 (mm)

Rod length: 15 mm. Camera # 2

Spherical FOV (Dia. 36 mm.)

d: 33.5 mm

f: 91 mm

127.4 mm

4.6 mm 6.8 mm

3.4 mm



OSEM Reconstruction of a Derenzo Phantom

Full 3D Grid-Scan Experiment GIMGPE

2-mm grid spacing Display scale: [10% max - 90% max] 1st -19th Slice



OSEM Reconstruction of a Derenzo Phantom

GIM[4] GIMGPE1-mm grid spacing Display scale: [0 - 90% max]

[4] Y. C. Chen et al, in Small-Animal SPECT Imaging, M. A. Kupinski and H. H. Barrett eds., Springer New York, pp. 195-201, June 2005.

13th - 27th Slice



Line Profile of One Slice

11th Slice

2-mm grid spacing

0 2 4 6 8 10 12 14 16 18 200

2000

4000

6000

8000

10000

12000

The voxel on the cross line

Am

plitu

de (c

ount

s)

Full 3D Grid-Scan ExperimentGIMGPE

4 mm

6 mm

6 mm

4 mm



Line Profile of One Slice

20th Slice

1-mm grid spacing

0 5 10 15 20 25 30 35 400

500

1000

1500

2000

2500

3000

3500

The voxel on the cross line

Am

plitu

de (c

ount

s)

GIMGIMGPE4 mm 5 mm

5 mm 5 mm

6 mm



Detectability

M

m m

mA

m

mM

mmm

m

mM

mmm

gs

dSNR

gg

gg

gg

gg

SNR

1

222

1

22

112

2

1

2

112

2

2

1

)(

signal,contrast -low aconsider If

.

ln)(21

ln)(

,)]()([ :present) (signalH,)( :absent) (signalH

nsbnrfrfHgnbnrHfg

sb

b

Ideal Observer



SKE/BKE Tasks

z

y

R

R

x

y

R

R

R/2

R+R/4

R+R/22R

1. 2.3.

4.5.

2. 6.7.

R = 18 mm

Detectability Estimation:• 7 Uniform Sphere Phantom Against a Flat Background Noise,• 8 Contrast (AS/AB): [0.01; 0.02; 0.05; 0.10; 0.20; 0.50; 1.00; 2.00].

21



Detectability Performance

22

0 0.02 0.04 0.06 0.08 0.10

10

20

30

40

50

60

70

80

Contrast (AS/AB)

SNR

0 0.5 1 1.5 20

200

400

600

800

1000

1200

1400

Contrast (AS/AB)

SNR

1, 5

4, 72

3, 6

1, 5

4, 72

3, 6

2-mm grid spacing○: Full 3D Grid-Scan Experiment

＊: GIMGPE1.16%100)

SNRSNR-SNR

max(Scan-Grid

Scan-GridGIMGPE



Detectability Performance

23

○: GIM

＊: GIMGPE 1-mm grid spacing

0 0.02 0.04 0.06 0.08 0.10

10

20

30

40

50

60

70

80

Contrast (AS/AB)

SNR

0 0.5 1 1.5 20

200

400

600

800

1000

1200

1400

Contrast (AS/AB)

SNR

1, 5

4, 72

3, 6

1, 5

4, 72

3, 6

4.97%100)SNR

SNR-SNRmax(

GIM

GIMGIMGPE



Conclusion• The preliminary evaluation of GIMGPE method shortens

the measurement time of a full 2.0-mm grid H matrixabout 64 times and a full 1.0-mm grid H matrix about 512times.

• The OSEM reconstructed images of a Derenzo phantomwith the interpolated H matrices show comparableresolution and similar line profiles as that reconstructedwith the full 3D grid-scan H matrix and the GIM finer-gridH matrix.

• The SKE/BKE detection tasks demonstrate theinterpolated H matrices have the same detectability levelas the full 3D grid-scan H matrix and the finer-grid GIM Hmatrix.

• Based on the GIMGPE method, further interpolations ofthe H matrix to much finer spacing and more projectionangles could be easily done.



Acknowledgement• This work was supported in part by National Science Council

Grant NSC 101-2221-E-008-135 and Institute of NuclearEnergy Research, Taiwan.

rapid constructions of circular-orbit pinhole spect...

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