This paper presents improved autoregressive modelling (AR) to reduce noise in SPECT images. An AR filter was applied to prefilter projection images and postfilter ordered subset expectation maximisation (OSEM) reconstruction images (AROSEMAR method). The performance of this method was compared with filtered back projection (FBP) preceded by Butterworth filtering (BWFBP method) and the OSEM reconstruction method followed by Butterworth filtering (OSEMBW method). A mathematical cylinder phantom was used for the study. It consisted of hot and cold objects. The tests were performed using three simulated SPECT datasets. Image quality was assessed by means of the percentage contrast resolution (CR%) and the full width at half maximum (FWHM) of the line spread functions of the cylinders. The BWFBP method showed the highest CR% values and the AROSEMAR method gave the lowest CR% values for cold stacks. In the analysis of hot stacks, the BWFBP method had higher CR% values than the OSEMBW method. The BWFBP method exhibited the lowest FWHM values for cold stacks and the AROSEMAR method for hot stacks. In conclusion, the AROSEMAR method is a feasible way to remove noise from SPECT images. It has good spatial resolution for hot objects.
Numerous methods for removing noise from SPECT images have been proposed [
In twodimensional AR modelling, each value of an image is regressed on its neighbourhood pixel values, called the prediction region. An AR model can be regarded as a lowpass filter that divides the image into two additive components, a predictable image and a prediction error image. An AR process
In a typical scintigraphic image, there are large local spatial variations in the count number of the image. Therefore, the same model cannot be applied to the entire image, but the model must be adapted to the variations. In this adaptive method, the image area is divided into smaller blocks and the AR model is then fitted into each block separately by using MATLAB subroutines. Recently, a blockwise denoising method has been introduced also for threedimensional ultrasound images [
Flowchart of the autoregressive denoising process. ARF: autoregressive filtering;
Data were simulated using a 3DMAC phantom [
Transaxial slices were reconstructed using either the filtered back projection method (FBP) [
To obtain a fair comparison of the methods, the same amount of filtering was applied in each method. This was done by drawing a circular regionofinterest (ROI) 150 mm in diameter in the uniform part of the phantom and calculating the percentage coefficient of variation (CoV%) in the ROI, that is, the ratio of the standard deviation to the mean multiplied by 100. This kind of presentation ensures that filtering between each method is equal.
Percentage contrast resolution (CR%) values for the activity in each cylinder and uniform activity were calculated. CR% can be expressed by the following formula [
Spatial resolution was estimated by the full width at half maximum (FWHM) of the line spread functions of the cylinders. One, two, four and sixpixelthick profiles were drawn through the 10, 20, 40 and 60mmwide cylinders, respectively. The FWMH values were calculated using Hermes quality control software (version 2.0).
Skeletal SPECT was performed three hours after an intravenous injection of 925 MBq of
The data were analysed using WinSTAT for Excel (version 2007.1; R. Fitch Software, Staufen, Germany). Pairwise comparisons were performed with the nonparametric Wilcoxon’s ranksum test. Comparisons were made between the AROSEMAR and BWFBP methods, the AROSEMAR and OSEMBW methods, and the BWFBP and OSEMBW methods. Data from the cold stacks and the pooled hot stacks were analysed separately. For each cylinder, the paired difference between the values of a variable was computed. The values of the differences were sorted to get a rank order. Finally, the mean rank of negative differences was compared with that of positive differences. Wilcoxon’s ranksum test determines to what extent the difference in mean rank is significant. A
The MSE of the images improved when a different AR model was used for the summed error term image rather than for the original image. A prediction region of four orthogonal neighbours with a block size of
Effect of changing the block size of the summed error term image with a prediction region of 3 × 3 pixels. For the predictable image a prediction region of four orthogonal neighbours with a block size of 5 × 5 pixels was used.
Total counts  Block size  Mean squared error 

28705  5 × 5  0.87 
28705  6 × 6  0.86 
28705  7 × 7  0.86 


54469  5 × 5  2.12 
54469  6 × 6  2.10 
54469  7 × 7  2.14 


108938  5 × 5  6.61 
108938  6 × 6  6.56 
108938  7 × 7  7.04 
Transaxial slice of the Zubal phantom. (a) Poissonnoisecorrupted transaxial slice. (b) Iteratively filtered predictable image. (c) Filtered summed error term image. (d) The final image. The images are individually scaled to their own maximum. Inverse linear grey scale is used for comparison with original phantom. The total count level is 108791 in the Poissonnoisecorrupted image, 102237 in the iteratively filtered predictable image, and 6205 in the filtered summed error term image.
Butterworth filtering was chosen so that the methods had the same amount of statistical fluctuation in the uniform part of the phantom, as confirmed by the CoV% values (Table
Percentage coefficient of variation for different reconstruction techniques.
Method  Count level  RelAct  CF (cycles/cm)  CoV% 

AROSEMAR  50000  0  —  6.35 
BWFBP  50000  2  0.83  6.37 
OSEMBW  50000  4  0.84  6.36 


AROSEMAR  100000  0  —  4.60 
BWFBP  100000  2  0.80  4.56 
OSEMBW  100000  4  0.84  4.65 


AROSEMAR  150000  0  —  4.43 
BWFBP  150000  2  0.89  4.50 
OSEMBW  150000  4  0.86  4.43 
AROSEMAR: autoregressive filtering before and after ordered subset expectation maximisation algorithm; BWFBP: Butterworth prefiltering and filtered back projection; OSEMBW: ordered subset expectation maximisation algorithm and Butterworth postfiltering; RelAct: activity relative to background activity of 1; CF: cutoff frequency. The order of the filter was 2; —: not definable.
Percentage contrast resolution values for the different methods.
Method  RelAct  10 Ø  20 Ø  40 Ø  60 Ø 

A  
AROSEMAR  0  18.4  56.3  74.6  84.2 
BWFBP  0  28.7  67.2  83.1  91.6 
OSEMBW  0  24.9  59.5  75.7  85.6 


AROSEMAR  2  9.0  86.1  84.3  103.5 
BWFBP  2  16.7  75.5  86.6  97.2 
OSEMBW  2  9.9  65.3  76.9  98.3 


AROSEMAR  4  136.5  253.0  270.4  296.5 
BWFBP  4  84.9  229.2  259.2  289.1 
OSEMBW  4  70.2  204.1  247.1  300.0 


B  
AROSEMAR  0  21.5  58.0  76.1  84.8 
BWFBP  0  20.4  61.6  83.2  91.8 
OSEMBW  0  32.9  61.9  78.0  86.6 


AROSEMAR  2  32.8  81.0  81.9  95.7 
BWFBP  2  35.1  80.2  87.4  102.7 
OSEMBW  2  29.7  75.4  78.8  98.3 


AROSEMAR  4  132.8  230.2  269.8  295.7 
BWFBP  4  94.6  236.9  270.3  316.2 
OSEMBW  4  85.6  224.6  255.1  308.5 


C  
AROSEMAR  0  24.0  55.9  75.0  85.5 
BWFBP  0  32.9  68.7  83.0  93.0 
OSEMBW  0  31.2  60.7  76.8  87.8 


AROSEMAR  2  41.1  76.0  86.9  94.9 
BWFBP  2  38.8  82.4  93.9  103.6 
OSEMBW  2  34.3  83.1  85.5  100.6 


AROSEMAR  4  119.4  237.1  266.3  297.7 
BWFBP  4  112.7  248.5  274.5  318.2 
OSEMBW  4  108.1  260.5  261.0  316.9 
A: low count level; B: intermediate count level; C: high count level; AROSEMAR: autoregressive filtering before and after ordered subset expectation maximisation algorithm; BWFBP: Butterworth prefiltering and filtered back projection; OSEMBW: ordered subset expectation maximisation algorithm and Butterworth postfiltering; RelAct: activity relative to background activity of 1; Ø: diameter.
In the analysis of spatial resolution, without exception, the BWFBP and OSEMBW methods exhibited lower FWHM values for the cold stacks than the AROSEMAR method (
Full width at half maximum values for the different methods.
Method  RelAct  10 Ø  20 Ø  40 Ø  60 Ø 

A  
AROSEMAR  0  19.7  25.0  45.8  64.8 
BWFBP  0  17.9  23.2  43.5  63.8 
OSEMBW  0  19.4  23.2  43.6  62.8 


AROSEMAR  2 

21.5  35.3  59.1 
BWFBP  2 

23.2  36.6  59.6 
OSEMBW  2 

22.2  35.8  59.1 


AROSEMAR  4  13.4  18.4  37.8  57.8 
BWFBP  4  15.8  20.4  37.9  57.9 
OSEMBW  4  15.2  19.9  37.7  57.9 


B  
AROSEMAR  0  23.7  23.7  43.0  62.3 
BWFBP  0  18.6  22.4  41.2  61.4 
OSEMBW  0  18.8  23.2  41.9  61.7 


AROSEMAR  2  19.4  19.8  38.9  59.0 
BWFBP  2  19.0  21.0  38.5  58.9 
OSEMBW  2  17.7  19.9  38.5  59.0 


AROSEMAR  4  13.0  19.7  37.8  58.2 
BWFBP  4  15.4  21.0  37.7  57.9 
OSEMBW  4  14.2  20.2  37.9  58.1 


C  
AROSEMAR  0  20.2  23.9  40.9  63.2 
BWFBP  0  18.5  21.1  40.6  61.7 
OSEMBW  0  16.8  23.2  40.3  62.3 


AROSEMAR  2  15.8  19.5  39.0  58.6 
BWFBP  2  16.9  20.8  38.6  58.7 
OSEMBW  2  16.5  20.3  39.1  58.6 


AROSEMAR  4  12.5  19.0  37.6  58.2 
BWFBP  4  14.0  19.6  36.8  58.0 
OSEMBW  4  13.8  19.3  37.4  57.9 
A: low count level; B: intermediate count level; C: high count level; AROSEMAR: autoregressive filtering before and after ordered subset expectation maximisation algorithm; BWFBP: Butterworth prefiltering and filtered back projection; OSEMBW: ordered subset expectation maximisation algorithm and Butterworth postfiltering; RelAct: activity relative to background activity of 1; Ø: diameter.
Furthermore, the OSEMBW method had lower FWHM values than the BWFBP method (
Visually, the differences between the images produced by the three methods were small (Figures
Transaxial slices of the phantom at the level of cylinders with a diameter of 20 mm. (a) Images reconstructed from noisefree projection images using ordered subset expectation maximisation reconstruction. (b) Autoregressive filtering before and after ordered subset expectation maximisation reconstruction. (c) Filtered back projection reconstruction method preceded by Butterworth filtering. (d) Ordered subset expectation maximisation reconstruction followed by Butterworth filtering. Intermediate count level. The images are individually scaled to their own maximum.
Reformatted coronal slices of the phantom. (a) Images reconstructed from noisefree projection images using ordered subset expectation maximisation reconstruction. (b) Autoregressive filtering before and after ordered subset expectation maximisation reconstruction. (c) Filtered back projection reconstruction method preceded by Butterworth. (d) Ordered subset expectation maximisation reconstruction followed by Butterworth filtering. Stacks with the highest activity. Intermediate count level. The images are individually scaled to their own maximum.
Onepixelthick profiles drawn through the phantom. (a) Profiles at the level of a cylinder with a diameter of 20 mm. (b) Profiles at the level of cylinders with a diameter of 40 mm. Ideal OSEM: images reconstructed from noisefree projection images using ordered subset expectation maximisation reconstruction; AROSEMAR: autoregressive filtering before and after ordered subset expectation maximisation reconstruction; BWFBP: filtered back projection reconstruction method preceded by Butterworth filtering; OSEMBW: ordered subset expectation maximisation reconstruction followed by Butterworth filtering. Intermediate count level. Profiles were rescaled so that they had the same amount of counts as the profile of the image reconstructed from noisefree projection images using ordered subset expectation maximisation reconstruction.
Transaxial slice of skeletal SPECT. (a) Autoregressive filtering before and after ordered subset expectation maximisation reconstruction. (b) Filtered back projection reconstruction method preceded by Butterworth filtering. (c) Ordered subset expectation maximisation reconstruction followed by Butterworth filtering.
This paper presented an improved twodimensional adaptive AR filter and introduced a threedimensional adaptive AR model for reduction of noise in SPECT images. We demonstrated that the quality of scintigraphic images can be improved when the same AR procedure is not applied to the original image and the summed error term image. We have previously shown that if a prediction region of four orthogonal neighbours of the predicted pixel with a block size of
The goal of filtering in SPECT is to suppress statistical noise and simultaneously preserve contrast and spatial resolution [
The BWFBP method produced better performance than the two other methods in the analysis of the cold stacks. FBP’s good performance with cold features has been noticed before [
The FBP method consists of filtering of the projection data and back projection of the filtered data [
The disadvantage of FBP is that it can produce radial streak artefacts because filtered noisy projection profiles do not cancel each other out in back projection. In the present study, the phenomenon was seen in the clinical data. Iterative reconstruction algorithms also provide some other advantages over FBP. They permit the use of several important corrections, such as scatter, attenuation, and collimator response corrections, which can be included in the image reconstruction procedure. Incorporation of anatomical information derived from magnetic resonance imaging or computerized tomography is possible as well [
The Butterworth filter is defined by two parameters: cutoff frequency and order [
In our opinion, a strength of the AROSEMAR method is its simplicity, but the lack of usercontrolled variables can also be regarded as a limitation. Sometimes, adjustable parameters are needed. No particular filter can emerge as the best filter for any organ system. However, filtering should be performed locally in the spatial domain, not globally in the frequency domain, because the correct tradeoff between resolution and smoothing will vary at different points within the image.
Because the AROSEMAR method was only marginally better than the OSEMBW method, an additional study is needed to find out whether image quality will be even better if the AR method is applied to the intermediate results in between the iterations. Secondly, the AROSEMAR method has not yet been tested with positron emission tomography (PET) data, but the method should also be suitable for PET data. The signaltonoise ratio is considerably higher in PET than in SPECT. Therefore, our model will probably provide a good fit for PET data.
The AROSEMAR method is a feasible denoising method in SPECT. It has good spatial resolution for hot features and it is simple to use. It does not have any adjustable parameters.
The authors declare that there is no conflict of interests regarding the publication of this paper.