Resolution Recovery in Ectomography using Filtered Backproj ection. M . Persson, D. Bone, K. Meissinger

Division of Medical Engineering, Karolinska Institute, Novum, S-141 86 Huddinge, Sweden

Abstract The aim of this study was to implement resolution recovery

(RR) during reconstruction of images acquired with the scintigraphic method of Ectomography. Ectomography is a form of limited view angle tomography. During acquisition the detector is stationary and at a constant distance from the object. Consequently, the point-spread function (PSF) varies between reconstructed section images, but is constant within each section. During reconstruction it is possible to deconvolve projection images with the inverse PSF weighted according to the distance of each section from the detector. Reconstruction was performed using filtered back projection (FBP). A function describing the PSF at any distance from the detector was determined, the inverse calculated, weighted according to signallnoise and used to modify a ramp filter. In real data RR a significant improvement in resolution was obtained without additional artifacts when compared to standard FBP. In conclusion FBP with resolution recovery (RR-FBP) can be implemented in Ectomography.

I. BACKGROUND Ectomography is a limited view angle tomographic

method, which we at have implemented on a mobile gamma camera [ 11 [2], hence bringing scintigraphic tomography to the bedside of the critically ill patient. Tomographic projection data is acquired by rotating a collimator with slanted holes in front of a stationary detector head as shown in figure 1. The system is called Cardiotom, and as the name implies is aimed primarily at myocardial investigations. A pilot study has however also been performed in order to investigate the feasibility of using such a system for acute scintigraphic studies of patients with stroke.

Figure 1. The acquisition geometry of Ectomography.

The acquisition geometry of ectomography leads to missing data in the Fourier domain, as do all limited view angle methods. The missing data causes limited depth resolution. The geometry has however some advantages since it allows the camera to be positioned at a constant distance near the organ imaged. The proximity to the organ of interest leads to increased system sensitivity, and possibly also fewer problems with attenuation and scatter.

A parallel-hole collimator has a point-spread function (PSF) that can be approximated by a 2-D circular Gaussian function. The use of slant hole collimators however yields a

planar PSF that has an elliptic Gaussian shape, with greater full-width at half-maximum (FWHM) in the direction of slant.

Reconstructed section images are parallel to the detector and collimator surface. A consequence of this is that the PSF vanes between the reconstructed section images. However, except for a rotational component, the PSF is constant within each section, as the distance from each point in a specific section to the detector is constant. This implies that it should be possible to deconvolve the projection images during reconstruction to achieve resolution recovery.

AIM OF STUDY

The aim of this study was to measure the PSF of the Cardiotom and find an analytical formula to describe the PSF as a function of distance from the collimator surface. A modification of the filtered back-projection (FBP) algorithm has also been developed. This modification has been evaluated using tomographic data from the system to verify the modified algorithm. Data used were from tomographic studies of point sources at varying distance from the detector, phantom studies of a heart phantom with perhsion defects present. The proposed algorithm was also tested on patient data to estimate the image quality and stability in the presence of noise.

11. MATERIALS AND METHODS

A. System measurements The planar PSF for the system fitted with a 40O-slant hole

collimator was measured using small point sources at varying distances. For each projection of the point source a Gaussian curve was fitted to the central profile and the FWHM was used to describe the PSF, h,(x,y), as a function of distance from the collimator, z, as shown in equation 1.

h,(X,Y) = Ae (1)

A is an amplitude scaling constant, and the relation to the FWHM is as shown in Eqn. 2.

B. Resolution recovery during reconstruction To achieve deconvolution the inverse of the PSF was

calculated in the Fourier domain, the inverse being shown in Eqn 3.

(3)

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This approach only works for noise free data and PSF with small FWHM, but even a modest increase in FWHM will lead to overestimation of numerical errors in calculation and noise, as high frequency components are greatly amplified.

A modified filter based on a Wiener filter, was introduced which reduces of noise and calculation errors by suppressing high frequency components depending on count statistics, c, as shown by Helstrom [3] and Bone [4]

In ectomography the reconstruction filter is applied in the direction tangential to the collimator rotation which in turn is perpendicular to the slant direction. The reconstruction filter, G(u,v), for noise free data is a high-pass ramp filter.

Gtu, VI = Iv/ ( 5 )

As O,(u,v) varies with the distance from the detector, each reconstructed section must be processed by a dedicated filter. In our implementation, the filters for different sections are calculated prior to reconstruction and the high pass ramp is included in the section filters as shown in the following equation.

Projection images are filtered during the back-projection process using the filter appropriate for the distance z. The operation is performed in the Fourier domain and consists of multiplication with the filter function.

C. Evaluation

I ) Tomographic PSF Projection images of point sources with PSF applied to

simulate the system response for sources at varying distance from the collimator were simulated. Simulated pixel size was 3.4 mm in a 128*128-pixel projection image. 128 projections were used for image reconstruction. One data set was used for each collimator-point distance. Projection data was reconstructed using regular FBP with a ramp filter, and with the proposed RR-FBP algorithm. Several reconstructions with varying regularization factors, c in Eqn. 6, were performed with the RR-FBP algorithm in order to evaluate how regularization affects tomographic resolution.

FWHM were measured each reconstructed data set, both in the primary reconstructed sections parallel to the detector, and in the depth direction.

2) Noise performance Projection images of a homogenous cylinder were

simulated with effect of PSF incorporated. From this noise free data set, data sets with different noise levels were created by adding statistical uncertainty based on the count value of each pixel. Noise levels corresponding to the following maximum pixel count were simulated: 10000, 5000, 1000, 500, 100 and 50.

Noisy data sets were reconstructed with regularization, c , matching the maximum pixel count in the projection data. For each noise level a noise free data set was also reconstructed for reference. The resulting noise images were visually assessed and energy spectra of the noise were calculated.

3) Two different numerical phantoms were used to visualize

the performance of RR-FBP compared to regular FBP. Phantoms used were a numerical simulation similar to the heart phantom from Data Spectrum Inc, and the MCAT phantom (University of North Carolina at Chapel Hill, USA). Projections were simulated incorporating the PSF. Noise was added to the projection data corresponding to a maximum count of approximately 150 counts.

Projection data from a clinical patient study was also reconstructed using RR-FBP and compared to the results of regular FBP reconstruction.

Heart phantoms and clinical data

111. RESULTS

A. Planar point-spread function Results form the measurements of planar point spread

functions are shown in figure 2. FWHM, is the direction of the collimator hole slant.

20 , 1 t *I

15 f E 10 5

0 50 100 150 200 Distance from collimator [mm]

Figure 2. Planar point-spread functions. The measured data was fitted to a linear function resulting

in Eqn. 7. The data show the increase in FWHM in the slant direction.

(7) FWHM,(z) = 0.05802 + 6.56

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,', FWHM,(z) = 0 . 0 5 5 1 ~ + 4.88

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As seen in the equations above FWHM was found to be a linear function of z, while the PSF is direction dependent, x is the direction in which the collimator is slanted. The difference due mainly to the fact that the collimator consists of hexagonal slanted holes. As the symmetric hexagonal hole is projected on to the detector surface it fonns an elongated hexagon; therefore the PSF is direction dependent.

5

The main reason for the alteration of the appearance with increasing slice-detector distance is that the noise spectral density changes. This change is caused by deconvolution of increasingly broader PSFs. Figure 11 shows the two examples of how noise spectra is altered.

_. + I

I I I

B. Tomographic PSF The resulting tomographic resolutions in the primary

reconstructed slices are shown in figure 3 for varying point source distance. The depth resolution is shown in figure 4.

Tomographic in-rbo m10CuM

, - - - c=5000

I ..... c=50

Figure 5 . Noise level in projection data 1 %. Slices 15, 30 and 45 (left to right). c = 10000

Figure 6. Noise level in projection data 3%. Slices 15, 30 and 45 (left to right). c = 1000

Figure 7. Noise level in projection data 4.5%. Slices 15, 30 and 45 (left to right). c = 500

Tanslrapht depth msdutian ( z 4 k t m j

-c=10000

. . . . . 'c= 1000 Figure 8. Noise level in projection data 1%. Slices 15, 30 and 45 (left to right). c = 1000

Figure 9. Noise level in projection data 4.5% Slices 15, 30 and 45 (left to right). c = 250 0 50 100 1 50 200 250

Dislmce Imml

Figure 4. Tomographic PSF in the depth direction of the reconstructed volume. Distance is from detector surface, collimator 35 is mm thick. c denotes regularization factor in RR-FBP.

B. Noise properties Figures 5 - 10 shows primary reconstructed section images

of the cylinder phantom, with different noise levels in the projection data. Figures 5-7 were reconstructed using RR-FBP with c matched to the maximum pixel count in projection data.

Figures 8-1 0 were reconstructed with increased regularization, i.e. c was set lower than the maximum pixel count in the projection data.

Figure 10. Noise level in projection data 4.5% Slices 15, 30 and 45 (left to right). c = 100

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Noise spectra

-_ ------- e-. .. _ " . . - . . i \ ' " ~ - -.\

.--- I 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Normaliiea frequency a

Figure 1 1. Noise spectra changes with increasing detector distance. Data with 10% and 4.5% noise in the projection images are shown.

C. Heart phantoms and clinical data Projection data with PSF incorporated generated from the

MCAT phantom and reconstructed with FBP and RR-FBP are shown in figure 12. In figure 13, noise was added corresponding to count statistics of 100 counts/pixel in the myocardium.

The RR-FBP algorithm also worked well when used on tomographic data from a study of a heart phantom in water with a perfusion defect present with 45" extent, as shown in Figure 14. The RR-FBP more accurately resolves the perfusion defect.

Figure 12. MCAT phantom with simulated PSF and noiseless projections, reconstructed with FBP and RR-FBP.

Figure 13. MCAT phantom with simulated PSF and 10% noise level in projection data, reconstructed with FBP and RR-FBP.

To further validate the proposed method, data from typical patient studies has been assessed. A comparison of

reconstructed images using FBP and RR-FBP is shown in Figure 16, together with a projection image from that study (Figure 15). This study was performed with a 4-segment 40"- slant hole collimator, so that each acquired projection image contains 4 projections of the myocardium. For clinical images, Hanning weighted ramp has usually been used for FBP reconstruction in Ectomography. In this data set count statistics were approximately 50 countdpixel in the myocardium.

Figure 14. Heart phantom reconstructed without resolution recovery (upper row) compared to the same slices reconstructed with resolution recovery (lower row)

Figure 15. Example of projection data from the patient study.

IV. DISCUSSION In SPECT, the distance to the detector of a fixed point off

center of rotation will vary in the different projections. In Ectomography however, a fixed point will have a constant distance to the detector in all projections. This should in theory allow for analytic deconvolution of the PSF during the filtered backprojection process in Ectomography.

The proposed algorithm works well for noise free data sets. If perfect deconvolution could be performed (Eqn. 3), the result would be a PSF independent of the distance from the detector. However, even for noise free data it is necessary to regularize the deconvolution function in order to suppress numerical errors in the reconstruction process, yielding the results shown in figure 3 and figure 4. These results are achieved using the deconvolution shown in Eqn. 6, which is based on the Wiener filter, and show a potential for substantial improvement of resolution, especially for the depth resolution.

The performance in the presence of noise shows the problem with this method as the noise undergoes a change in spectral distribution. This results in low frequency granularity

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FBP ramp filter

FBP RR-FBP RR-FBP RR-FBP hanning c=20 c=50 c=100

Figure 16. Comparison of FBP and RR-FBP when applied to projection data from a patient study. Count statistics in projection data was approximately 50 count'pixel. For a clinical setting FBP with Hanning weighted ramp should be compared to RR-FBP with c = 5@in this particular image set.

when deconvolving broader PSFs, i.e. the noise fiequency spectrum becomes a function of slice to detector distance as shown in figures 10-15. However we have seen that the variance of the relative noise level within a reconstructed volume does not increase with deconvolution of broader PSFs. It is primarily the noise spectrum that is modified resulting in granular images for large slice-detector distances (figure 16).

For clinical data sets, and simulations using cardiac phantoms, the algorithm perfoms well, even in the presence of a high noise level as can be seen in figure 16. In order to achieve good results from the RR-FBP algorithm, it is necessary to estimate the parameter c in the filter function. This corresponds to the image statistics and noise level and should be chosen to match the signal-to-noise ratio in the object of interest. In the studies shown above, this constant has been chosen according to the registered counts per pixel in the organ of interest.

Iterative reconstruction techniques can also be implemeEted in Ectomography. As iterative reconstruction algorithms can incorporate restraints on the reconstructed data, depth dependent noise spectra may be avoided. Pilot investigations have also shown that iterative techniques, such as ML-EM, have a beneficial effect on the depth resolution, when applied to the acquisition geometry of Ectomography.

One pilot study performed with an iterative method based on maximum entropy called COMET[5] has yielded promising results, although at the cost of computational time.

v. CONCLUSION For noise fiee and low noise data the proposed method

improves the topographic resolution, especially in the depth direction. The trade-off is an alteration of the spectral distribution of the noise with increasing detector-slice distance. However, in clinical data the modification of the noise spectra is not as obvious as in the simulated cylinder phantom.

The proposed algorithm can be used for Ectomography in scintigraphic imaging, even where noise levels are relatively high, but iterative techniques should offer better control over noise properties at increased computational cost.

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VI. REFERENCES

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