in single photon emission computed tomography ( spect ) , 
 acquired planar projection images or scintigram data can be arranged as 
 sinograms . a spect sinogram consists of radial intensity profiles of the tracer 
 distribution in the body or object as a function of the projection angle . images reconstructed from the acquired sinogram data represent two - dimensional 
 ( 2d ) transaxial slices of the three - dimensional ( 3d ) radioactive distribution 
 in the object . since the image reconstruction process is an ill - posed problem , 
 inherent noise in the projection data resulting from the random nature of 
 radioactivity tends to degrade the quality of the reconstructed images . numerous different methods for spect data filtering have been reported to 
 alleviate the problem . statistical image reconstruction , such as the 
 maximum - likelihood expectation - maximization ( mlem ) algorithm , utilizing the probability 
 distribution of the detected photon counts , provides a sophisticated way to 
 deal with the noise problem . in mlem reconstruction , however , noise and edge 
 artifacts become visible after a large number of itera - tions . to avoid these 
 unwanted artifacts , regularization methods such as early - stopping   or late stopping followed by 
 postfiltering the iterations are 
 stopped before the full convergence in order to avoid  over fitting . this 
 introduces nonuniform resolution and contrast into the image . a large number of 
 iterations followed by after 
 the late - stopping reconstruction , a proper modeling of the noise for filter 
 design can be challenging due to the reconstruction artifacts . a different approach to the quantum noise reduction is 
 smoothing or filtering of the raw projection or sinogram data before image 
 reconstruction . then , in principle , simple and linear filtered back - projection 
 ( fbp ) reconstruction can result in sufficient images , in terms of uniform 
 resolution and contrast . traditionally , the sinogram ( see figure 1 ) data are 
 filtered only in the radial direction ( i.e. , along the projections ) . well - known 
 examples of radial data filtering are , for example , the hanning and butterworth 
 lowpass filters , 
 which are routinely employed in fbp reconstruction . , across the projections of 
 different angular views ) is usually avoided since it introduces tangentially 
 varying blurring into the reconstructed image . recently , we have introduced an approach for sinogram 
 data filtering that allows the use of the angular information of the projection 
 data in a novel way . the proposed new approach is based on the stackgram domain 
 [ 6 , 7 ] . the stackgram 
 representation of the sinogram data decomposes the signals along the sinusoidal 
 trajectories of the sinogram into separate signals . these signals , denoted as 
 locus - signals , can then be processed or filtered without affecting the other 
 trajectory signals in the 3d stackgram ( see figure 2 ) . in the 2d sinogram 
 domain , in contrast , these signals intersect each other . the stackgram 
 filtering technique offers a different and potential alternative to radial sinogram 
 filtering [ 8 , 9 ] . one - dimensional ( 1d ) 
 angular filtering in the stackgram domain does not introduce observable 
 tangential distortion into the reconstructed images . moreover , an interesting 
 aspect of the behavior of the stackgram approach is that , in comparison with 
 the radial sinogram filtering technique , it provides a more  powdery  or 
 natural noise structure in the reconstructed images at a matched resolution . our findings [ 8 , 9 ] were obtained with numerical phantom data simulating acquisitions in positron emission tomography ( pet ) , in which the 
 imaging geometry differs from spect . our previous studies [ 8 , 9 ] gave experimental 
 quantitative evaluations of the stackgram filtering technique by employing 20 
 different filters . in those studies [ 8 , 9 ] , we used ten different noise levels with a hundred 
 noise realizations ( for each level ) to evaluate the noise reduction methods . in this study , we compare the radial sinogram 
 filtering technique with the angular stackgram approach with linear and 
 nonlinear filters , similarly as in [ 8 , 9 ] . in this experimental comparison , two physical 
 spect phantoms were applied for quantitative and visual assessment of the 
 two methods . we employed linear lowpass gaussian filters and nonlinear 
 l - filters . with these filters , the performances of the radial sinogram and the 
 angular stackgram filtering techniques differ substantially [ 79 ] . the employed filters do not 
 exploit any a priori knowledge nor provide optimal noise reduction of spect 
 data . this investigation aims to give an insight into the performance of the 
 novel stackgram approach and its possible tradeoffs in comparison with the 
 commonly accepted radial sinogram filtering technique . the radon transform   maps a function f(x , y ) to a sinogram g(l ,  ) . the 3d stackgram ( i.e. , a stack of back - projections ) 
 is a decomposition of different curves consisting of the values along 
 sinusoidal trajectories of the sinogram [ 6 , 7 ] . the stack - operator s maps the 
 sinogram g(l ,  ) to a stackgram h(x , y ,  ) as 
 ( 1)h  ( x , y, )  = ^sg(l, )  = g(x  cos+y  sin  , ) , 
 where x cos  + y sin  forms the 
 sinusoidal curves on the sinogram for each ( x , y ) coordinate . the 
 stack - operator simply reorganizes the sinogram values into the 3d domain . in 
 some contexts , the 2d layers of the stackgram are referred to as ridge 
 functions   ( which 
 are not associated with the 3d stackgram in ) . the key function of the stackgram approach is to 
 offer an environment to independently process each signal consisting of the 
 values along the different sinusoidal trajectories . in the stackgram domain , these signals are referred to as locus - signals hx , y( ) that can be 
 written as 
 ( 2)hx , y()=h(x , y, ) ,  (x , y){x2+y2c2}2 , 
 where c is a radius 
 defining a support region and  denotes the real 
 set . the stack - operator s , as defined , is unbounded and the ( x , y ) layers of the 
 stackgram are constant along each  angle . 
 therefore , it is normally reasonable to bound the range of the stack - operator 
 as { h(x , y ,  ) = 0 |x + y > c and   [ 0 )}. alternatively , a so - called mollifier function 
  could be used for 
 the bounding . it is straightforward to show that the stack - operator 
 is a linear operator . let g1 and g2 be sinogram 
 functions , and let h1 and h2 be the 
 corresponding stackgrams . thus , for each a , b   , 



 ( 3)s(ag1+bg2)(l, )  = ( ag1+bg2)(x  cos  +y  sin  , )  = ag1(x  cos  +y  sin  ,)+bg2(x  cos  +y  sin  , )  = ah1(x , y,)+bh2(x , y, ) . this is due to the fact that the 
 stackgram layers contain redundant information ( see figure 2 ) . thus , an inverse 
 stack - operator s can be formulated 
 by the simple relations x = l cos  and y = lsin  as 
 ( 4)g(l, )  = ^  s1h = h(l  cos  ,l  sin  , ) . the inversion described above , however , is not a 
 feasible formulation for practical stackgram applications . in practice , an 
 operator ( such as filter ) is employed to modify the locus - signals inside the 
 support region ( 2 ) . a more appropriate formulation , denoted as the 
 generalized inverse stack - operator sw1 , can be defined with both 
 the radon transform and a weight function w(x , y ; l ,  ) as 
 ( 5)g(l,)=^sw1h(x , y,)=w(x , y;l,)h(x , y,)(x  cos  +y  sin  l)dx  dy , 
 where ( ) is the dirac 
 delta function . notice that the operator sw1 is not a true 
 mathematical inverse , since it projects a function from the 3d into the 2d 
 domain , which can not be a one - to - one mapping . the formulation ( 5 ) also gives an 
 insight into our discrete implementation of the stackgram . in the discrete case , we implemented the operators 
 ( 1 ) and 
 ( 5 ) as back- 
 and reprojections by using 2d data rotations . for this , we employed the 
 three - pass rotation algorithm , which decomposes 2d rotation ( by angle  ) into three 1d 
 translations as in 
 ( 6)rot()=[cossinsincos]=[1tan201][10sin1][1tan201 ] 
 the 1d translations of ( 6 ) can be implemented 
 by convolution of a sinc - function . a discrete sinc - interpolator   along with the three - pass 
 rotation algorithm provides a reversible and high quality rotation procedure 
 for 2d band - limited data . since the convolution corresponds to multiplication in 
 the frequency or fourier domain , a fast fourier transform algorithm can be used 
 in the implementation . thus , in the discrete frequency domain , the 
 sinc - interpolator with a translation parameter ( s   ) for signals of odd number of samples ( n ) can be expressed 
 as in 
 ( 7)s(w)={exp(i2swn)if  w=0,  ,n12,exp(i2s(wn)n)if w=n12+1,  ,n1 . 
 in the case of an even number of 
 samples , the sinc - kernel of w with the 
 parameter s can be formulated , 
 respectively , as in 
 ( 8)s(w)={exp(i2swn)if   w=0,  ,n121,1if  w = n12,exp(i2s(wn)n)if   w=n12+1,  ,n1 . 
 in this formulation ( 8) , the highest 
 frequency component at w = ( n  1 ) / 2 is treated 
 differently compared to that in , in order to obtain a reversible kernel . the cost of 
 this small difference is aliasing of the highest frequency component in data 
 interpolation . by assuming that the data are properly sampled , however it can be verified that (w )  ( w ) = 1 and  ( w)(w ) = 1.this means that a rotation of 2d data is reversible , 
 when the three - pass algorithm ( 6 ) with the sinc - kernel ( 9 ) or ( 8) for the 
 translations is employed . that is , fixed forward and back rotations of 2d data 
 introduce negligible errors to the data in practice . based on ( 6 ) , ( 9 ) , and ( 8) , discrete stackgrams can be generated with simple 
 linear operations as follows : ( 1 ) each sinogram projection is replicated over ( x , y ) -plane ; ( 2 ) the 
 resulting 2d data are rotated by appropriate angles ; and ( 3 ) the rotated data 
 are stacked as a stackgram . , the 
 discrete stackgrams can be transformed back to sinogram data using the same 
 procedure vice versa : the stackgram layers are back - rotated and then 
 re - projected to 1d projections ( this transformation would be analogous to 
 ( 5 ) if the 
 weight function w were chosen to 
 be  averaging  ) . our implementation ( ( 6 ) , ( 9 ) , and ( 8) ) enables a 
 reversible sinogram - to - stackgram transformation , which is crucial in 
 investigating the performance of stackgram filtering . we used two physical phantoms in our investigation : a 
 hotspot phantom and the hoffman brain phantom . the hotspot phantom is composed 
 of two cylinders with diameters of 25 cm and 15 mm . the volume of the larger 
 cylinder is 10 liters , while the smaller hotspot volume is 20 ml . the 
 corresponding doses of technetium-99 m for the water volumes were 36 mbq / l and 
 250 mbq / l , respectively . the phantom was acquired on a dual - head spect system 
 ( siemens nuclear medicine ) in tomographic mode . for the hoffman phantom , a tc-99 m dose of 120 mbq was 
 employed . a dual - head spect system ( marconi medical systems ) was operated in 
 tomographic mode with an acquisition time of 45 s / step . in both acquisitions , we employed the hotspot phantom data mainly for 
 quantitative evaluation of the two filtering methods . data filtering along the radial l 
  -direction of 
 the sinogram has a different effect on the data than stackgram filtering along 
 the angular  -direction . the 
 angular  -axis of the 
 stackgram is orthogonal to the radial l -axis . a fixed 
 filter provides neither a comparable evaluation of the resolution nor the noise 
 reduction of the two methods . thus , effects of the different filtering 
 approaches need to be matched for a fair comparison by employing a range of 
 filters . in our investigation , we employed two different types 
 of 1d filters : conventional linear gaussian filters and nonlinear l - filters 
 with gaussian weights . the shift - invariant gaussian filters were implemented as 
 lowpass finite - impulse - response filters . each output point of the l - filter is obtained as a 
 weighted sum of ordered data values in the moving window of length n as in 


 ( 9)l(x;a)=i=1naix(i ) , 


 where a is a weight 
 vector and x(i ) denotes ordered 
 input data . we employed ten different gaussian weights or kernels for both 
 filter types , in order to match filtering effects ( hereafter called the 
 gaussian filters  and  the l - filters  ) . for the filter weights , we used 
 samples from gaussian distributions (  = 0 ) with the standard deviations (  ) of 0.25 , 0.5 , 0.75 ,  , 2.5 . the same weights were employed for both filter 
 types ( 20 filters in total ) . the corresponding lengths of the filters were 3 , 
 5 , 7 ,  , 21 samples . the narrowest kernel corresponds to an 
 identity filter in the case of gaussian filters , whereas the same weights 
 results in a median filter of length 3 in the case of l - filters ( i.e. , a[0 , 1 , 0 ] in ( 9 ) ) . in figure 3 , a selected locus - signal ( 2 ) before and after 
 filtering is shown . for evaluation of the filtering approaches , the 
 acquired hotspot data were filtered employing the 20 filters in the two ways as 
 follows : ( 1 ) along the radial sinogram direction , and ( 2 ) along the angular 
 direction in the stackgram domain . the filtered data were reconstructed with the fbp algorithm using the ramp 
 filter and a fixed attenuation correction ( hermes nuclear diagnostic ) . with the 
 reconstructed data sets , we determined resolution - noise tradeoff curves for the 
 two compared methods and for both filter types . this was accomplished by 
 quantifying contrast recoveries ( crs ) and coefficients of variation ( covs ) of 
 the reconstructed fbp images . cov versus cr plots provide a straightforward 
 evaluation method for the resolution - noise tradeoff . the symbols h and b represent average intensities of a high - count region 
 ( h ) covering the  hotspot  in the data , and a background region ( b ) round the 
 hotspot . the hot region h was a circle with a radius of two pixels , whereas the 
 background b was a ring - shaped region with thickness of two pixels . in this 
 way the cov is defined as the 
 standard deviation of the data over the mean of the data . after data filtering , 
 the covs for the compared methods were quantified using a large uniform region 
 in the reconstructed hotspot data . stackgram filtering tends to leave  powdery  or high 
 frequency noise in the reconstructed images . an evaluation method like the cov favors images with 
 non - high - frequency noise , such as that resulting from radial ( lowpass ) 
 filtering of the sinogram data . therefore , we also measured 
 full - area - at - half - maximum ( fahm ) values for the hotspot data without any noise 
 measurements . the fahm corresponds to full - width - at - half - maximum , but is a more 
 convenient measurement in 2d data . the fahms were quantified for both filtering 
 approach and for both filter types as follows : ( 1 ) a maximum count value of the 
 hotspot was measured from the fbp reconstruction ; ( 2 ) a cross - sectional area 
 ( in pixels ) of the hotspot was determined at the half - maximum . all the quantitative results ( cr , cov , and fahm ) were 
 averaged over eight data values measured from eight similar transaxial slices 
 of the fbp reconstructions of the hotspot data . for the compared filtering techniques , the determined 
 tradeoff curves ( i.e. , cov versus cr ) were also used to find comparable 
 gaussian filters as well as l - filters , in terms of noise reduction and 
 resolution . the filtered data aim to provide a comparable visual assessment of 
 the two different filtering approaches with both filter types at the matched 
 resolutions . the data were reconstructed with both the fbp ( ramp filter ) and 
 mlem algorithms without attenuation correction . in mlem reconstruction ( hermes 
 nuclear diagnostics ) , our assumption is that the mlem algorithm ( in addition to fbp ) could utilize 
 the stackgram - filtered sinograms due to the natural noise structure of the data 
 [ 8 , 9 ] , resulting in pleasant and 
 reliable images . in theory , data filtering prior to mlem reconstruction would 
 require additional data processing , such as nec scaling , since the noise is 
 presumably no longer poisson - distributed . in this study , resolution - noise tradeoff curves of the compared 
 methods are shown in figures 4 and 5 for the gaussian filters and the 
 l - filters , respectively . with the linear gaussian filters , radial sinogram 
 filtering provides a better tradeoff than angular stackgram filtering at 
 suitable resolution levels ( see figure 4 ) . overall , taking into consideration 
 both filter types ( gaussian and l - filters ) , stackgram filtering with the 
 l - filters provides the best tradeoff in terms of noise reduction at the 
 appropriate resolution , although the differences in the tradeoffs are not so 
 significant ( compare figure 4 to figure 5 ) . note that in figure 5 the measured 
 data values for stackgram filtering below a cr of 0.5 are somewhat biased , since 
 the curve suggests that noise or cov increases . in this case , however , the 
 filters below this cr value are insignificant in our investigation ; because 
 such filters are impractical for noise reduction ( i.e. , they provide a too 
 narrow contrast ) . in figures 4 and 5 , the chosen resolution levels are 
 shown with dashed lines . in the case of the linear filters , the widths (  ) of the gaussian filter kernels at the 
 approximately - matched resolution are 1.0 ( the standard gaussian filter ) and 
 0.75 for radial and stackgram filtering , respectively . similarly , in the case 
 of the l - filters , the chosen resolution level provides gaussian filter weights 
 with widths of 0.5 and 0.25 ( median filter ) for radial and stackgram filtering , 
 respectively . note that we do not try to match the linear and nonlinear 
 filters , but the different filtering approaches ( i.e. , radial sinogram and 
 angular stackgram ) for the two filter types . these four different 1d filters 
 were employed for the hoffman data , to be discussed section 3.4 . fahm versus maximum - intensity - value plots for the 
 gaussian filters are shown in figure 6 . as can be seen , stackgram filtering 
 preserves the thickness ( or fahm ) of the hotspot better than radial sinogram 
 filtering , as the kernel width becomes wider . the l - filters , on the other hand , 
 seem to preserve the fahm in a quite exceptional way in stackgram filtering 
 ( see figure 7 ) . that is , regardless of the employed filter kernel , the 
 thickness of the hotspot remains almost the same . in figure 7 , one point of the 
 stackgram - filtered data seems to be apart from the rest of the quantified data 
 values . a possible reason for this might be that the single data point 
 represents a median filter , whereas the other points in the plot represent 
 l - filters with a more gaussian type of weights ( 9 ) . in both radial and stackgram filtering , the two matched 
 l - filters ( see figure 5 ) perform rather equally also in terms of fahm ( see 
 figure 7 ) . thus , these l - filters should give practically as comparable a visual 
 assessment of the two filtering methods as possible . on the other hand , in the 
 case of gaussian filters , the matched filters do not share ( approximately ) the 
 same coordinates on the fahm plot ( see figure 6 ) . again , the 
 maximum - intensity - values predictably correspond to the measured cr values in 
 the plot . a transaxial slice of the filtered hotspot data is 
 shown in figure 8 ( gaussian filtering ) and figure 9 ( l - filtering ) . figure 9(a ) 
 illustrates what sort of geometrical distortion the nonlinear l - filters , in the 
 case of radial sinogram filtering , can introduce into the reconstructed images 
 ( the ring - shaped hotspots ) . a similar effect can not be observed after stackgram 
 filtering ( see figure 9(b ) ) . figure 8 and 9 aim to support the shown curves 
 ( figures figures 47 ) visually . figure 10(a ) shows fbp images of the hoffman phantom 
 ( fbp with ramp filter ) . in figures 10(b ) and 10(c ) ( top row ) , fbp images of the 
 hoffman data are shown for radial sinogram and angular stackgram filtering with 
 the matched gaussian filters . these fbp images are congruent with the shown 
 curves ( figures 4 and 6 ) ; stackgram filtering tends to leave more noise 
 variation in the images ( or this can be regarded as a powdery noise structure ) . figures 10(b ) and 10(c ) ( bottom row ) show fbp images 
 for radial and stackgram filtering with the matched l - filters . as expected 
 ( figures 5 and figure 7 ) , the images of the different filtering methods appear rather 
 similar . there are , however , observable differences in the images , especially 
 in the structure of noise . this effect ( as in the case of gaussian filters ) can 
 be explained by the fact that stackgram filtering preserves the sinusoidal 
 structure of the filtered sinograms , unlike radial sinogram filtering . as stated , the mlem algorithm could perhaps utilize 
 the stackgram - filtered data due to the  powdery  noise structure resulting in 
 visually pleasing images without annoying reconstruction artifacts , although 
 the noise after filtering would not be exactly poisson - distributed . figure 11(a ) shows late - stopping mlem reconstructions of the hoffman data ( 100 
 iterations ) . in figures 11(b ) and 11(c ) ( top row ) , mlem images of 100 
 iterations for radial sinogram and angular stackgram filtering with the matched 
 gaussian filters are shown . figures 11(b ) and 11(c ) ( bottom row ) show the same 
 transaxial slices after filtering with the matched l - filters . as can be 
 observed the mlem images ( figures 11(b ) and 11(c ) ) do not differ substantially . 
 however , the noise structure in the images after linear stackgram filtering 
 ( see figure 11(c ) , top row ) appears to be most similar to that in the 
 late - stopping mlem reconstruction ( see figure 11(a ) ) , as compared to the rest 
 of the images ( see figure 11 ) . the results of this investigation are congruent with 
 our previous experience of stackgram filtering in numerical pet phantom data ( 192  192 resolution with 256 views ) [ 8 , 9 ] . there are , however , some 
 minor differences resulting ( perhaps ) from the different imaging geometry ( 360 ) and resolution ( 128  128 ) of spect . first , as regards the resolution - noise 
 tradeoffs ( figures 4 and 5 ) , stackgram filtering seems to provide 
 quantitatively better results with narrower filter kernels than does radial 
 sinogram filtering at the appropriate resolution levels . secondly , the differences in the noise after radial and stackgram 
 filtering of the spect data ( see figure 10 ) seem not to be as evident as we 
 have found with pet data [ 8 , 9 ] . stackgram filtering performs better with the nonlinear 
 l - filters than with the linear gaussian filters ( figures 4 and 5 ) . we believe 
 that this kind of effect holds more generally for linear and nonlinear type of 
 filters in 1d stackgram filtering . this can be explained by the fact that the 
 stackgram ( 1 ) 
 contains redundant information , that is , each layer of the stackgram is 
 constant at angle . this means that several locus - signals to be filtered 
 share the same data points , forming , for example , noise peaks . generally , 
 nonlinear filters remove this kind of redundancy or correlation more accurately 
 than linear filters . stackgram filtering seems to preserve the resolution 
 or fahm in a quite exceptional way as the kernel width increases , whereas 
 radial sinogram filtering performs conventionally ( figures 6 and 7 ) . theoretical considerations would naturally provide a more solid explanation , 
 but these are outside the scope of our investigation . as noticed , stackgram 
 filtering  shrinks  the intensity scale of the data when the filter widths 
 get wider , but do not cause spatial blurring in the same way as radial 
 filtering of the sinogram . this can be understood by the structure of the 
 stackgram , which is like a stack of back - projections . summing up the stackgram 
 along the  -axis results 
 in a back - projected ( bp ) image , which can be regarded as a blurred counterpart 
 of the reconstructed image . the bp image has a similar shrunken intensity scale 
 as the stackgram filtered data with the wide filters ( figures 6 and 7 ) . heuristically , if the stackgram data are filtered with more and more  powerful  
 or averaging 1d filters ( which would be inappropriate for noise reduction ) , 
 then each layer of the stackgram converges to a back - projected image , that is , 
 the average of all the layers . this would result in data similar to those 
 indicated with the wide filter widths ( figures 6 and 7 ) . the stackgram allows us to exploit the angular 
 direction of the projection data without introducing tangential blurring to the 
 reconstructed image , as can be observed , for example , in figures 8 and 9 ( i.e. , 
 the shape of the hotspot remains circular ) . in contrast , filtering of the 
 sinogram data along the angular direction introduces observable nonuniform and 
 tangential spatial blurring to the reconstructions . the tangential distortion 
 is extremely obvious in the case of 1d filters . to avoid this undesirable 
 effect , some techniques for sinogram domain filtering utilizing the angular 
 direction have been published [ 18 , 19 ] . these techniques , however , often restrict the 
 available filters or require adjustments to them . in angular stackgram 
 filtering , any 1d filter or denoising technique can be employed for the noise 
 reduction in locus - signal ( 2 ) filtering without the need for regulation of 
 tangential blurring . this claim or observation is based on our experience in 
 stackgram filtering on conventional emission tomography data . stackgram filtering , on the 
 other hand , requires more computational time and especially computer memory 
 space , as compared to sinogram - domain - based filtering methods . in our resolution - noise evaluations , we have 
 explicitly assumed that the two filtering approaches led to images with 
 spatially uniform resolution if shift invariant filters were employed . in 
 radial sinogram filtering , the assumption of uniform resolution is validated by 
 the central slice theorem . based on our experimental studies , stackgram filtering with linear shift - invariant 
 filters also introduces a spatially uniform resolution and contrast to the 
 reconstructed images ( note that the applied discrete stackgram transformation 
 is linear ) . thus , the resolution - noise tradeoffs , as well as the fahm curves 
 for the linear gaussian filters ( figures 4 and 6 ) , should describe the 
 performance of the compared filtering approaches reliably and predictably . on 
 the other hand , the employed nonlinear l - filters ( 9 ) are based on order statistics , in which case a 
 filtering effect on an isolated impulse ( or on a larger  hotspot  ) is not 
 predictive for more general structures . in figure 9(a ) , this explains the distorted 
 shape of the hotspot after radial filtering with the wide filters . however , the 
 plots ( figures 5 and 7 ) seem to provide appropriate evaluations of the 
 filtering approaches , and highlight the different performances . in this study , our objective was to compare the novel 
 stackgram approach with the conventional radial sinogram filtering technique , 
 not to develop optimal stackgram - based filters . to develop suitable filters for 
 spect , it would be necessary to study further the noise properties of the spect 
 stackgrams . it is quite obvious , however , that the noise distribution of the 
 locus - signals ( 2 ) in the stackgram domain ( 1 ) follows a poisson 
 distribution , since the spect sinogram data are simply reorganized in the 
 stackgram . furthermore , in general , a sufficiently large subset of data 
 presumably follows the same distribution as the initial data set ( cf . a 
 sinusoidal trajectory signal as the subset and the sinogram as the complete 
 set ) . besides , in our previous studies , we have found that the noise 
 distribution of the locus - signals in discrete pet stackgrams follows 
 approximately a generalized poisson distribution . it is important to note that 
 the implementation of the discrete stackgrams ( ( 6 ) and ( 7 ) or ( 8) ) implicitly introduces 
 minor changes to the noise in the locus - signals , for example , due to the 
 ringing effect associated with sinc - interpolation . it is also worth noticing 
 that many of the locus - signals to be filtered contain rather a constant 
 intensity range ( such as in figure 3 ) . this implies that the ( poisson ) noise 
 has a constant variation in these 1d signals , that is , the noise distribution 
 is actually close to gaussian . in summary , considering the discussion above , 
 the applied nonlinear l - filters with gaussian weights ( 9 ) may turn out to be 
 close to an optimal filter design for noise reduction of spect stackgrams . statistical iterative reconstruction algorithms can 
 take into account the poisson nature of noise in the emission projection data . the iterative methods such as mlem are therefore routinely used nowadays in 
 image reconstruction . it is a well - known fact that radial filtering of the 
 spect projections modifies the poisson distribution of the noise , especially 
 with lowpass filters , thus preventing the use of the mlem algorithm in theory . a similar modification effect on the noise distribution introduced by stackgram 
 filtering is not so obvious , because the angular direction of the stackgram is 
 orthogonal to the radial direction or the acquisition plane . this supposition , 
 and our previous studies [ 8 , 9 ] , gave us the motivation to test the stackgram - filtered 
 data using the mlem reconstruction algorithm , although the noise distribution 
 of the filtered data is unknown and would require further studies . however , 
 since the discrete stackgram transformation is linear , we assumed that 
 ( particularly ) a linear filter could approximately preserve the poisson noise 
 distribution in stackgram - filtered sinograms . thus , the use of the mlem 
 algorithm would lead to visually pleasing or artifact - free images , due to lower 
 noise variation in the sinogram data . stackgram filtering , however , does not 
 seem to provide significant differences in the reconstructed mlem images , as 
 compared to the  unnatural  radial - filtered data ( see figure 11 ) . nevertheless , the stackgram approach incorporated into the mlem algorithm could 
 be a useful , or at least interesting , future direction of research . other 
 authors have reported preliminary results of a similar approach to image reconstruction 
 in . for noise reduction of spect sinograms , angular 
 stackgram domain filtering offers an alternative to conventional radial 
 sinogram domain filtering . we compared these two filtering approaches with both 
 linear gaussian filters and nonlinear l - filters using physical spect phantoms . the chosen filters were mainly used for evaluation purposes of the two 
 filtering domains . in this experimental investigation , stackgram filtering with 
 the nonlinear l - filters provides the best resolution - noise tradeoff . at 
 appropriate resolution levels , however , the differences in the resolution - noise 
 tradeoffs of the compared filtering approaches seem to be quite small . the overall performance of 1d stackgram filtering 
 differs significantly from 1d radial sinogram filtering . that is , angular 
 filtering in the stackgram domain seems not to introduce conventional blurring 
 to the reconstructed images , but rather shrinks the intensity scale of the 
 data . furthermore , the noise in the images after stackgram filtering does not 
 appear to be as smooth as in the case of radial filtering at the same 
 resolution . more ( theoretical ) studies , however , are need - ed to explore 
 the novel stackgram approach in order to develop suitable or optimal 
 stackgram - based filtering techniques for raw spect data .
we discuss data filtering prior to image reconstruction . for this kind of filtering , the radial direction of the sinogram 
 is routinely employed . 
 recently , we have introduced an alternative approach to sinogram data processing , exploiting the angular information in a novel way . 
 this new stackgram representation can be regarded as an intermediate form of the sinogram and image domains . in this experimental study 
 , we compare the radial sinogram and angular stackgram filtering methods using physical spect phantoms . 
 our study is carried out by employing simple linear and nonlinear filters with ten different gaussian kernels , in order to provide a comparable investigation . according to our results , 
 angular stackgram filtering with the nonlinear filters provides the best resolution - noise tradeoff of the compared methods . 
 besides , stackgram filtering with these filters seems to preserve the resolution in an exceptional way . 
 visually , noise in the reconstructed images after stackgram filtering appears more  powdery  in comparison with radial sinogram filtering .