there  
 has been large interest in speeding the acquisition   of mri data by acquiring 
 fewer samples in k - space and resolving the artifacts .
recently , there have been 
 significant advances in applying inverse problem techniques to 
 reconstruct images from undersampled k - space mri data [ 16 ] .
the methods use nonuniform undersampling and 
 a nonlinear recovery scheme in which a constraint , such as a spatial total 
 variation ( tv ) constraint , is applied on the estimated solution , while preserving 
 fidelity to the acquired data in k - space .
it has been shown   that using an l1 
  norm or a tv norm as constraint exploits the implicit sparsity in the data , and 
 can be used in both space and time dimensions .
the method is best known to 
 reconstruct piecewise constant or smoothly varying data from its undersampled 
 fourier samples ; but the application of the method to mr imaging techniques 
 like dynamic contrast enhanced myocardial perfusion imaging ( with respiratory 
 motion in the data ) and diffusion tensor imaging ( dti ) can be limited as these 
 images are often not piecewise constant . in 
 this paper , we propose a   technique to improve the reconstruction of 
 general signals that may not fit the tv constraint well .
the technique uses 
 preprocessing of the measured undersampled data to determine an improved 
 ordering of the pixel intensities of the image estimate . if an ordering that 
 improves the match of the estimated images and the constraint being used within 
 the reconstruction can be 
 found , an improved reconstruction can result .
the image estimates are 
 reordered solely to be used with the constraint or regularization term in the 
 iterative reconstruction .
the reordering approach is general in its 
 applicability and can be used in contexts which are based on regularization 
 techniques and in which ordering of the image intensities can be determined a priori . in the next sections ,
we 
 give a brief overview of the compressed sampling or constrained reconstruction 
 method for mri from a regularization point of view and then present the theory 
 and applications of the reordering method .
the 
 compressed sampling method is described rigorously from a mathematical 
 standpoint recently in a series of papers [ 2 , 810 ] .
the method 
 is used to reconstruct a signal from a set of random fourier samples below the 
 nyquist rate by solving a convex optimization problem , in which fidelity to the 
 measured data is preserved at sample locations , while applying an l1 
  or a tv constraint on the estimated solution .
the method exploits the implicit 
 sparsity in the estimated solution or a transform of the estimated solution . 

one useful transform for signals or images is finite differences as the signals 
 that are not directly sparse can be sparse in terms of finite differences 
 ( especially piecewise constant or smoothly varying signals or images ) and hence 
 an l1 
  norm of finite differences ( tv norm ) is used .
although 
 the l1 norm of the signal or image estimate is not a direct measure 
 of sparsity in the data , it has been shown that for a wide variety of data 
 using an l1 norm is equivalent to using the l0 
 norm ( the 
 number of nonzero samples ) , which is a direct measure of sparsity .
solving 
 the optimization problem with an l1 
 norm is generally easier than 
 solving the problem with an l0 
 norm .
the 
 compressed sampling method when applied to mr image reconstruction can be 
 thought as a constrained reconstruction method in an inverse problem framework 
 [ 3 , 1114 ] . for the 1d case
, the relation between fourier data and the signal 
 space estimate can be represented as fm = d ,   where m is the signal of interest , d is the 
 fully sampled k - space data , and f represents the fourier transform ; but it often takes a long time to acquire 
 full k - space data and results in tradeoffs in image quality , resolution , and 
 coverage of the organ . to accelerate the data acquisitions , when full data are 
 not acquired in k - space , and only undersampled data are acquired , the relation 
 between the artifact - free signal estimate m and acquired data is given by ( 1)wfm  =  d , where w implements   a binary undersampling pattern with ones ( where data are 
 acquired ) and zeros ( where data are missing ) , and d is the undersampled fourier data . 

reconstructing the signal m directly using ( 1 ) is not feasible as w does not exist in general and hence 
 the solution is not unique .
the existence of the 
 solution is imposed by considering least - square solutions which minimize the functional wfm    d22 , 
 where ||||2 represents an l2 
 norm .
uniqueness 
 of the solution is imposed by using one or more constraints on the solution .
a 
 popular constraint used in the field of compressed sampling is the total 
 variation constraint given by m2  +  1 , 
 where  is the gradient of the estimated signal ,  is a small positive constant to avoid 
 singularities in the derivative of the functional , and ||||1 represents an l1 
  norm .
reconstruction 
 is performed by minimizing a convex cost function ( c )   ( 2)c  =  wfm    d22  +  m2  +  1 .
hence , a 
 solution which preserves fidelity to the acquired data and which has the 
 minimum total variation is chosen as the final solution . in ( 2 )
,  is the regularization parameter which controls 
 the tradeoff between the fidelity and the constraint terms .
the total variation 
 constraint helps to resolve the artifacts while not penalizing the edges 
 heavily .
the 
 method can be extended to 2d and multi - image dimensions and it works very well 
 when the k - space data are undersampled in an irregular fashion and the 
 underlying complex images are smoothly varying or are piecewise constant . 

when the images are not piecewise constant ( which is the case for most mr 
 images ) , the performance of the method can be affected .
we describe below a 
 reordering method to improve the performance of the constrained reconstruction 
 method when the data do not match the constraints well .
the method preprocesses 
 the signal to select a monotonic ordering of the estimated solution in space 
 and/or time and incorporates the reordering in the constraints to obtain better 
 reconstructions . for 
 clarity ,
the reordering method is first described for the 1d case and then the 
 method is extended to 2d and multidimension cases .
applications of the 
 reconstruction method with reordering for dynamic myocardial perfusion imaging 
 with respiratory motion and for brain dti data are presented .
when 
 the signal of interest is varying rapidly and is not smooth or the data are not 
 piecewise constant , the total variation of the signal is already high and hence 
 reconstruction from undersampled fourier domain samples can be inaccurate . 

consider , for example , a smoothly varying 1d signal and a rapidly varying 
 signal that are labeled  original full data  as shown in figures 1(a ) and 1(b ) , respectively .
when the corresponding fourier samples ( k - space data ) of 
 the curves are undersampled by a factor of two in a pseudorandom fashion ( using 
  rand  function in matlab ( the mathworks , natick , mass , usa ) ) and 
 reconstructed using the inverse fourier transform , the signals labeled  undersampled 
 data r = 2  in figures 1(a ) and 1(b ) are obtained . when these undersampled 
 signals are reconstructed according to ( 2 ) ,
the original curves are overlaid in figures 1(c ) and 1(d ) for 
 reference .
reconstruction 
 using ( 2 ) is better for the smooth curve in figure 1 as compared to that for 
 the rapidly varying curve . to improve the reconstruction in the latter case ,
we 
 first reorder the estimated curve in the signal space according to an optimal 
 order , and then apply the total variation constraint .
the optimal ordering can 
 be determined as the ordering that makes the signal intensities in the curve 
 from the fully sampled dataset monotonic and smoothly varying . reordering the 
 estimated solution helps by reducing sudden variations in the curves and gives 
 a better match to the assumed constraint . in practice
, the curve or images from 
 fully sampled data will not be available to obtain the optimal ordering and 
 some sort of approximate reconstruction must be used to determine the ordering . 
 while this area needs more research , we show here that relatively simple 
 methods for determining reorderings can improve reconstructions of some types 
 of undersampled data .
better 
 reconstruction from the undersampled fourier samples is obtained when the 
 reordered curve is used in the constraint term , as the a priori assumption that the curve has lower variation is better 
 satisfied .
so the new reconstruction from undersampled data is performed 
 according to ( 3 ) in which the only difference is that the tv constraint is 
 applied on the reordered data as opposed to applying the constraint directly on 
 the given data .
reordering the estimated signal can also be thought as 
 multiplication of the signal with a reordering matrix  p.
 this matrix can be a permutation matrix of ones and zeros ( it 
 could also be a diagonal matrix for a 1d signal and it can be generalized for 
 multidimensional signals ) as follows : ( 3)m^  =  minmwfm    d22  +  (pm)2  +  1 . 
 
note 
 that the reordering in ( 3 ) is not directly based on the intensity values 
 obtained from the aliased signal from undersampled data in k - space ( or for 
 other applications , whatever domain the measurement data is obtained ) .
that is , p is determined once , and is fixed 
 while minimizing ( 3 ) . ordering the undersampled data according to
the optimal 
 order does not mean that the reordered undersampled image estimates are 
 monotonic , but means that 
 if this ordering is used , the original full data in the signal space will best 
 match the tv constraint .
consider figure 1(e ) which shows the sorted curve of 
 the original curve in figure 1(b ) .
the curve is monotonic and smoothly varying 
 and has lower total variation as compared to the original curve .
the ordering 
 in this case was chosen as the sorting order that made the original full curve 
 monotonic and smoothly varying .
although not shown 
 here , the reconstruction with reordering was comparable to that without 
 reordering for the case of the smooth curve in figure 1(a ) . from 
 the compressed sampling point of view , reordering the data can lead to sparser 
 representations of the data and hence higher acceleration factors . 
 alternatively , better reconstructions for a given acceleration factor can be 
 obtained .
figure 2 illustrates the point for the original full curve shown in 
 figure 1(b ) .
the figure compares the sparsity of the original curve and the 
 sorted curve in terms of finite differences .
the finite difference curve for 
 the sorted signal is sparser ( has fewer nonzero values ) as compared to that of the 
 original signal and hence using an l1 
 norm of the finite differences 
 with reordering leads to better reconstructions . choosing 
 the optimal regularization parameter 
						  
					  is important to obtain good 
 reconstructions .
the l - curve method   is a popular technique for choosing 
 the optimal value .
the fidelity 
 norm is plotted against the constraint norm of the reordered data and the optimal 
 parameter is given by the corner of the l - curve .
figures 3(a ) and 3(b ) show the 
 l - curves obtained for reconstructions from undersampled data for the curve 
 shown in figure 1(b ) without and with reordering , respectively .
the l - curve in 
 figure 3(a ) is also overlaid on figure 3(b ) for direct comparison .
the l - curves 
 and the optimal parameters obtained are different for both cases . the optimal 
 parameter without reordering is higher than that with reordering . as in the iterative methods , the number of 
 iterations also plays the role of regularization parameter ; a fixed maximum 
 number of iterations which gave minimum rms reconstruction error for various 
						  
					  values was chosen in computing the l - curves . from 
 the above , it is apparent that correct reordering can help in better 
 reconstruction from undersampled fourier data when the signals are not smoothly 
 varying . in the above experiment , we used full data to determine the optimal 
 ordering . to be able to use the 
 reordering method , we need to have an ordering that makes the original signal best 
 match the constraint . in practice
, it is likely not possible to get the exact 
 ordering of the signal curves or images as that obtained using fully sampled 
 fourier data due to various factors like blurring of the prior signal , noise in 
 the prior signal , and so on . to simulate this case , we randomly perturbed the 
 exact sorting order to see the effect of having inexact ordering on the 
 performance of the algorithm . in figure 4 , the 
 x - axis represents the number of random perturbations , that is , the number of indices of 
 the exact  sorting - order vector  that are randomly perturbed .
consider s to be the sorting - order vector for the original signal . when there is one 
 random perturbation ,
( i ) a random number is generated between 1 and the length 
 of s denoted by r_a , then ( ii ) a second distinct random number between 1 and the 
 length of s denoted by r_b , and finally ( iii ) the value of the s at index r_a is exchanged with that at index r_b .
a value of 10 on the x - axis means that the 
 values of the exact  sorting - order vector  at 10 distinct randomly picked 
 indices ( out of 70 ) are exchanged with those at a different set of 10 distinct 
 randomly picked indices .
the y - axis represents the natural log of the total 
 absolute difference between original full data and the reconstructed signal .
we 
 can see that as the number of random perturbations increases , the total absolute error 
 using reordering gradually increases , but this number is still better or comparable to that 
 without the reordering except for a few perturbations toward the end of the 
 plot where the entire sorting - order vector was randomly perturbed .
the 
 reordering method described above for the 1d case can be extended to 2d and 
 applied in the context of images . as in the 1d case , reordering in 2d for 
 images helps in better reconstruction when the images of interest are nonsmooth 
 or are not piecewise constant . for 
 example , figure 5(a ) shows a simulated piecewise constant heart image with 
 blood pools and with an ischemic region in the myocardium .
when full fourier 
 data of the image are undersampled in a variable density ( vd ) cartesian fashion 
  ( so that 5 lines in the center of k - space are fully sampled and the 
 remaining phase encodes are sampled in a pseudorandom fashion to give a net 
 reduction factor of ~6.5 ) , direct inverse fourier transform reconstruction 
 gives the image shown in figure 5(b ) .
when the constrained reconstruction 
 approach with a tv spatial constraint ( xm2  +  ym2  +  1 ) without reordering is used , figure 5(c ) is 
 obtained .
we can see that figure 5(c ) matches well with figure 5(a ) ; but when the 
 image is not piecewise constant , the performance of the constrained 
 reconstruction can be affected .
figure 5(d ) shows an actual mr heart image from 
 a patient at a single time in a perfusion sequence .
the image was reconstructed 
 using a standard 2d inverse fourier transform of the fully acquired k - space 
 data .
figure 5(e ) shows the standard 2d inverse fourier transform 
 reconstruction of undersampled k - space data for the time frame , with zeros 
 inserted for the missing k - space data points .
the data were undersampled by a factor of three in 
 vd cartesian fashion ( 10 phase encodes fully sampled around the center and the 
 remaining ones in a pseudorandom fashion ) . figure 5(f ) 
 shows the constrained reconstruction from the undersampled data using a spatial 
 tv constraint without any reordering which has a few residual artifacts . for 
 improving the reconstruction in this case , the image is reordered independently 
 in x and y directions before applying the 2d tv constraint ,
that is , reorderings 
 are determined separately for each row and each column . in practice , since the 
 data we deal with in mri are 
 complex , the optimal ordering is determined independently for the real and 
 imaginary components of the image and separately in x and y directions . a row - reordered real part 
 of the image of the original   complex 
 mr image of the heart is shown in figure 5(g ) . the tv spatial constraint with 
 reorderings for a complex image
can be explicitly written as 1 , where   =  x(prxmreal)2  +  x(pixmimag)2  +  y(prymreal)2  +  y(piymimag)2  +   , in which mreal is the real part of the image estimate and mimag is the imaginary part of the image 
 estimate .
prx and pix denote the reordering matrices for ordering the signal in x dimension for the real and imaginary parts , respectively , while pry and piy are the corresponding reordering matrices in y dimension . for simplicity and 
 compactness , the above spatial constraint with reordering is referred to as x(pxm)2  +  y(pym)2  +  1 , where pxm gives the image reordered for each row in the x direction and pym is the image reordered for each column in the y direction .
figure 5(h ) shows the 
 reconstruction from the undersampled data using a spatial tv constraint with 
 reordering .
the ordering of the data was 
 obtained using the image reconstructed from fully sampled data . 

figure 6 shows a plot comparing the 
 reconstruction error with increasing number of perturbations in the exact 
 spatial ordering for the actual mr heart image in figure 5(d ) .
the 
 reconstruction error is calculated as the total absolute difference between the 
 full data reconstruction and the data reconstructed using the tv spatial constraint . 

a value of 10 on the x - axis means that the sorting - order vectors for 10% of the 
 total number of rows ( rounded to nearest integer and randomly picked ) and those 
 for 10% of the total number of columns ( rounded to nearest integer and randomly 
 picked ) are randomly perturbed for both the real and imaginary parts of the 
 complex image data .
perturbation for 
 a given row or a column is done independently for the entire length of the 
 sorting - order vector as described for the 1d case in section 2.4 .
a value of 
 100 on the x - axis means that the sorting - order vectors for all the rows and all 
 the columns are completely perturbed ( randomly and independently ) for real and 
 imaginary parts of the image .
the error for the reconstruction with reordering 
 is gradually increasing with increasing perturbations and when the exact 
 sorting orders are severely modified , the error gets higher than that without 
 reordering .
the 
 reordering method described above   can be extended to multi - image mr 
 acquisitions like dynamic myocardial perfusion imaging and brain dti . in 
 perfusion imaging , a series of images of the heart are acquired to track the 
 uptake and washout patterns of the contrast agent in the myocardium .
dti 
 requires the acquisition of multiple images with diffusion weightings in different 
 directions .
reordering can be done in the multi - image dimension  in the time 
 dimension for the case of myocardial perfusion imaging and in the diffusion 
 encoding dimension for the case of dti . as in the 1d case , reordering in the multi - image dimension for the 
 images can give a better reconstruction when the signal changes in the 
 dimension are not smoothly varying which is the case for perfusion imaging with 
 respiratory motion and for dti .
the 
 constraint for the reordering in the multi - image dimension is represented as t(ptm)2  +  1 , where t represents the gradient operator in the 
 multi - image dimension and ptm is the data reordered in the corresponding 
 dimension .
the subscript t is used 
 because the multi - image dimension is analogous to the temporal dimension of 
 dynamic perfusion datasets . for a given image frame in the multi - image dataset , 

reconstruction 
 can then be performed by using tv constraints in both space and multi - image 
 dimensions   and with reordering in the corresponding dimensions 
 as follows : ( 4)c  =  wfm    d22  +  1t(ptm)2  +  1 +  2x(pxm)2  +  y(pym)2  +  1 .
a similar 
 framework to ( 4 ) was proposed 
 in   for reconstructing undersampled radial 
 myocardial perfusion data but without reordering and with a different temporal 
 constraint . as in , ( 4 ) 

the reordering method for 
 multi - image acquisitions ( 4 ) was tested using a dynamic phantom .
gd was slowly 
 injected into a tube running through a water phantom and fully sampled 
 cartesian k - space data were 
 acquired over time using an echoplanar imaging sequence on a siemens 3 t trio 
 scanner .
raw k - space data was then undersampled offline in a variable density 
 ( vd ) pseudorandom fashion , in which 12 central low - resolution k - space lines 
 were sampled for all time frames , and the remaining phase encodes were sampled 
 in a pseudorandom fashion to give a net reduction factor of three . the acquisition 
 matrix for
reconstruction from the undersampled data was then performed according 
 to ( 4 ) in two steps . in the first step
, 
 the information about the reordering was obtained using images obtained using 
 the central low - resolution data from vd undersampling .
the   image 
 estimates were reordered first in the time dimension only and the 
 reconstruction was performed . in this step ,
the real and imaginary parts of the 
 complex low - resolution image space data for each pixel were sorted 
 independently in the time dimension according to their intensity values .
the 
 corresponding sorting orders for the real and imaginary parts were used for 
 reordering the real and imaginary parts of the complex undersampled image space 
 data . after performing an initial reconstruction with only temporal reordering
results of the final reconstruction with reordering were compared to full data 
 reconstructions using the standard inverse fourier transform and to the 
 reconstructions without any reordering .
the reordering method was applied 
 on dynamic myocardial perfusion imaging with respiratory motion and on brain 
 dti data .
full cartesian raw k - space perfusion data were obtained using a siemens 3 t trio 
 scanner using a turboflash saturation recovery sequence .
the parameters for the 
 data acquisition were tr   =   1.8 milliseconds , te   =   1 millisecond , flip angle   =   12 , gd dose   =   0.025 mmol / kg , slice thickness   =   6 mm , and 
 acquisition matrix   =   192      96 .
the data were acquired with 
 informed consent in accordance with the university of utah institutional review board .
brain dti image data were 
 acquired on a ge 3 t scanner and full k - space data were generated from the magnitude image data by 
 applying 2d fourier transforms on each diffusion encoding direction .
full 
 k - space data for both perfusion and brain dti data were undersampled in a variable density 
 pseudorandom fashion outside the center and with the central 18 k - space lines 
 sampled for each time frame , and reconstruction was performed in two steps as 
 described in section 3.1 .
the net r value for the perfusion data was 2.5 while 
 that for the dti data was 3 .
the 
 results of the reordering method on the multi - image phantom data are shown in 
 figure 7 .
figure 7(a ) shows the image of a slice that was reconstructed from 
 full data using ift at single time point .
figure 7(b ) shows the corresponding 
 image reconstructed from r~3 data using stcr without any reordering .
figure 7(c ) 
 shows the image reconstructed from r~3 data using stcr with reordering in both 
 temporal and spatial dimensions . 

figure 7(d ) 
 shows the absolute difference image between figures 7(a ) and 7(b ) and 
 figure 7(e ) shows the absolute difference image between figures 7(a ) and 7(c ) .
the images in figures 7(d ) and 7(e ) are scaled to the 
 same window level to highlight the differences .
the 
 results of the reordering method for dynamic myocardial perfusion imaging are 
 shown in figure 8 .
figure 8(a ) ( column ) shows images at two different time 
 points in a perfusion sequence reconstructed from full perfusion data using 
 standard inverse fourier transforms .
figure 8(b ) ( column ) shows the 
 corresponding stcr reconstructions from r~2.5 data   without any reordering 
 in time or space dimensions .
figure 8(c ) ( column ) shows the corresponding stcr 
 reconstructions with reordering in both time and space dimensions .
the 
 reordering helps in reconstruction when there is a significant respiratory motion in the 
 data .
we previously reported higher acceleration 
 factors ( r~4 with interleaved undersampling and r~5 with variable density 
 sampling ) in   for myocardial perfusion , when there was minimal or no 
 respiratory motion in the data . in the presence of significant respiratory 
 motion , the method in   was not fully able to resolve the artifacts from 
 undersampling .
the current method was better able to reduce the artifacts even 
 in the presence of large respiratory motion .
the 
 results obtained by applying the reordering method on brain dti data are shown 
 in figure 9 . 

figure 9(d ) compares the line intensity 
 profiles for full data reconstruction ( figure 9(a ) ) and the reconstruction 
 without reordering ( figure 9(b ) ) .
figure 9(e ) compares the intensity profiles 
 for full data reconstruction ( figure 9(a ) ) and the reconstruction with 
 reordering ( figure 9(c ) ) .
the signals in figure 9(e ) match better than those in 
 figure 9(d ) .
this 
 paper introduces a modified constraint term for compressed sampling and 
 constrained image reconstruction approaches . in general
, it is possible to 
 choose a regularization or constraint term which is a good model for the image 
 being reconstructed .
the basic idea of the reordering method is that it is 
 possible to tailor these regularization or constraint operators to improve the 
 reconstruction by reordering the signal . from a compressed sampling point of 
 view
reordering can be thought as a new set of data - specific  transforms  that 
 further improve the sparsity .
recently , a new method using a prior image 
 constraint   was proposed to improve the constrained reconstruction of 
 dynamic ct images .
the additional prior image constraint minimizes the l1 
 distance between the estimated solution and the prior image .
the reordering 
 method proposed here is different in the sense that it does not directly use 
 the intensities in the prior image .
the method uses only the ordering 
 information from a prior image or set of images , which can be preserved if the 
 prior images are at a different and unknown intensity scale as compared to the 
 estimated solution .
reordering 
 can be done in multiple dimensions to improve the sparsity when the signals are 
 not smoothly varying . here , we used images from the central low - resolution data 
 to determine orderings initially in the multi - image dimension and then used the 
 resulting images to obtain the spatial reordering for each image separately . 

this is because the central low - resolution data are more faithful in the multi - image dimension 
 than they are in 
 the spatial dimension .
reordering in the multi - image dimension offered more 
 significant improvements as compared to reordering in the spatial dimensions . 

this is because the temporal constraint generally plays a more important role 
 in resolving the artifacts as compared to the spatial constraint for dynamic 
 imaging . to obtain significant improvements just using spatial reordering
improved ways of 
 obtaining reordering , like doing a separate training scan before the actual 
 acquisition , may help to achieve higher accelerations .
the 
 reordering method incorporates the ordering information of the signal to better 
 match the total variation constraint assumption and thus improves the 
 reconstruction from undersampled data .
methods like adaptive regularization 
 [ 18 , 19 ] were proposed to improve the performance of tv regularization - based denoising 
 techniques by using a priori 
 information about the signal . in these methods ,
the regularization parameter is 
 varied based on the a priori 
 knowledge of the locations of edges and smooth regions in a signal , so that 
 less regularization is done where strong edges are present in the signal .
while 
 this type of approach may be extended to tv constrained reconstruction , 
 choosing the optimal amount of variation of the regularization parameter can be 
 complicated for rapidly varying signals .
an alternative method that can use 
 such a priori information was explored here with the reordering method that 
 uses only a single regularization parameter .
the 
 reordering method may not be appropriate when the ordering is incorrect in such 
 a way that the total variation of the reordered full image sets is increased as 
 compared to that of the original full data . in practice
in such cases , l - curves can be 
 used to determine to some extent if reordering is appropriate .
l - curves can be 
 computed for reconstructions with and without reordering and the tv norms 
 corresponding to the optimal regularization parameters can be compared . if the 
 ordering is appropriate , then the tv norm corresponding to the optimal 
 regularization parameter with reordering is lower than that for the 
 corresponding optimal regularization parameter without reordering .
consider 
 figure 10(a ) which shows the l - curve obtained for the 1d randomly varying curve 
 shown in figure 1(b ) , with a large number of random perturbations ( ~65% ) in the 
 exact ordering .
the tv norm corresponding to the optimal  is 36.08 . when the number of random perturbations is decreased ( ~21% ) , the l - curve 
 in figure 10(b ) is obtained and the tv norm corresponding to the optimal  is 5.02 .
the l - curve in figure 10(a ) is 
 overlaid in figure 10(b ) for reference .
the tv norm corresponding to the 
 l - curve obtained without reordering is 14.05 .
the 
 reconstruction time with image reordering was higher than the standard l1  
 norm 
 reconstruction , as in each iteration , the estimated signal is reordered before 
 computing the constraint update .
for the data reordering in 1d case , the 
 reconstruction time was 1.04 times slower , while that for the dynamic case with 
 reordering in both spatial and temporal dimensions was 2.8 times slower .
it took   ~35 seconds 
 per iteration on a linux machine with an amd processor ( 2.5 ghz ) and 6 gb ram 
 for stcr with reordering in multiple dimensions .
the implementation was done in 
 matlab and a host of methods including the use of gpus are available to greatly 
 speed up reconstruction methods .
a 
 method involving reordering in time and space dimensions of the image estimates 
 to better match the chosen constraints of an inverse problem - type 
 reconstruction was presented .
the method uses non - reordered reconstructions to 
 obtain information about the signal to be reconstructed to determine the 
 orderings of the pixel intensities .
the orderings can be estimated from the low - resolution 
 images when a   variable density undersampling scheme is used , and from 
 non - reordered constrained reconstructions .
the method can be forgiving to errors 
 in the images used to choose the orderings as the method does not use the data 
 directly but uses only its ordering information .
the method was shown to have 
 promise for cardiac perfusion imaging and offered some small improvements for 
 dti data .
future improvements in finding more optimal reorderings , perhaps as 
 part of the estimation process , may make the approach useful in a wide array of 
 applications .
 recently , there has been a significant interest in applying reconstruction techniques , like constrained reconstruction or compressed sampling methods , to undersampled k - space data in mri . here 
 , we propose a novel reordering technique to improve these types   of reconstruction methods . in this technique , the intensities of the signal estimate are reordered according to a preprocessing step when applying the constraints on the estimated solution within the iterative reconstruction . 
 the ordering of the intensities is such that it makes the original artifact - free signal monotonic and thus minimizes the finite differences norm if the correct image is estimated ; this ordering can be estimated based on the undersampled measured data . 
 theory and example applications of the method for accelerating myocardial perfusion imaging with respiratory motion and brain diffusion tensor imaging are presented . 
1. INTRODUCTION
2. THEORY
3. APPLICATIONS
4. RESULTS
5. DISCUSSION
6. CONCLUSION