In the context of reconstruction of MRI images from Kspace data sets, the following are two desirable properties which are diﬃcult to achieve simultaneously: High spatial resolution and High temporal resolution.
The ﬁrst requires longer acquisition time while the second requires less time be consumed acquiring each image. Hence the inherent conﬂict in achieving both simultaneously. One possible remedy is to undersample image acquisition (along a radial or other trajectories such as Cartesian) which results in speed up of data acquisition, hence improving the temporal resolution. Next, an appropriate image reconstruction method is applied to the acquired data which attempts to compensate for some of the eﬀects of the image undersampling.
Due to undersampling, streaking artifacts will be present in the ﬁnal image. These streaking artifacts become more visible the larger the undersampling. Radial undersampling is a preferred method of acquisition compared to using other trajectories such as Cartesian: ”the aliasing artifacts from radial undersampling usually appear as streaks, which are visually less distracting than the wraparound artifacts obtained with Cartesian undersampling.”[6]
Mathematically, the problem of image reconstruction from undersampled Kspace data is an inverse problem: ”Mathematically, the reconstruction problem from sparse Kspace samples is an illposed inverse problem with inﬁnitely many solutions”[6]
Highly constrained backprojection reconstruction (HYPR) was introduced recently for the reconstruction of radially undersampled MRI images. HYPR is able to reconstruct these images with less visible artifacts while maintaining good SNR[4].
The method starts with the construction of one composite image made up from ﬁltered backprojections of a large number of projections (each of these projections is the inverse Fourier transform of a corresponding radial lines from the Kspace data set. This follows from the central slice theorem).
Since the composite image is made up of individual images collected over longer time period, it posses good spatial characteristics. In addition, its SNR is larger since a larger amount of images data is contained in it. However, the composite image temporal characteristics are poor since it combines images that were generated from varying time instances into a single image.
A weight image is then constructed from the ratio of a small number of unﬁltered backprojections obtained from the original projections and from the composite image. Since the weight image is constructed from images that span a smaller time window than the case is with the composite image, the weight image posses good temporal characteristics. However, its spatial characteristic is poor due to the undersampling eﬀect in the original data.
HYPR now generates a new image by multiplying the weight image with the composite image. This results in a HYPR image which combines the best characteristics found in the composite image and in the weight image, resulting in an image with good SNR, good temporal and spatial characteristics and with limited artifacts. The above process is repeated for the next HYPR image reconstruction until all the Kspace data set is processed. The relationship between the weight image and the composite image and the resulting HYPR image is summarized in the following diagram.
The following HYPR based algorithms were analyzed in terms of their mathematical formulation. In addition, their properties were studied by simulation[7] under diﬀerent conditions. The algorithms are: Original HYPR (OHYPR)[4], WrightHuang variation of HYPR (WHYPR)[2], Iterative HYPR (IHYPR)[1] using original HYPR as its kernel, Iterative HYPR using WrightHuang HYPR (IWHYPR)[7] as its kernel, and HYPRLR[5].
For each algorithm, its mathematical formulation is given, its attributes and the situations in which the algorithm is known to work best and where the algorithm can have diﬃculty in terms of the quality of reconstruction are both outlined.
This new algorithm ﬁrst conceived and implemented during this study. Simulation of this new algorithm conﬁrmed that this algorithm reduces noise ampliﬁcation by a much larger amount than IHYPR could during the iterative process.
HYPR simulation software allows one to execute many scenarios and test cases. Here we show the result of two studies that used a set of images (the phantom clip) supplied to us by GE Healthcare where the images exhibit large degree of temporal and spatial dynamics. The HYPR algorithms were run using this clip as input both under the presence of noise and without noise. Noise was Gaussian with zero mean and standard deviation was set at \(5\%\) of the maximum projection from all the original projections. For all test cases, 8 time frames and 8 projections per time frame was used. For the iterative algorithms, 10 iterations were used. The results below are the average RMSE value, which represent the average error in the reconstruction of HYPR images. The smaller this value, the more accurate the algorithm is considered^{1} .
test  OHYPR  WHYPR  HYPRLR 



No noise  \(6.83\)  \(6.77\)  \(6.7\) 



With noise  \(10.76\)  \(9.55\)  \(13.7\) 



Handling of noise by each algorithm was analyzed as follows. A copy of the noise signal being added to each projection was used on its own as the input to each HYPR algorithm. In other words, each noise signal was treated as a projection on its own. When each test starts, two separate computations are started: one which process the original projection with the noise signal added to it (in quadrature), and another which process the noise signal only. At the end of the above two separate computations, 2 sets of HYPR images would result. The mean \(\mu \) and the standard deviation \(\sigma \) of each HYPR image generated from the noise computation was then computed and the \(rmse=\sqrt{\mu ^{2}+\sigma ^{2}}\)for each HYPR image was found. The following table contains the average of the above rmse values over all the HYPR images that was generated.
OHYPR  WHYPR  HYPRLR 
\(0.1004\)  \(0.0004\)  \(0.0938\) 
The ﬁrst table above shows that without noise, OHYPR, WHYPR and HYPRLR performed equally well. When iterative HYPR was run, we observe that IHYPR and IWHYPR performed equally well.
When noise was added, WHYPR was more accurate than OHYPR. HYPRLR did not perform as well. But we must note that HYPRLR can be used with diﬀerent low pass ﬁlters and the size of each ﬁlter can be altered as well. Hence it is possible that there exist diﬀerent low pass ﬁlter which can do better than the one used in this particular test. We notice also that when iterative HYPR was run with noise present, the error became larger with more iterations. This is because noise was being ampliﬁed in the process. Notice however that IWHYPR had less noise ampliﬁcation than IHYPR. The second table above shows how each algorithm responded to the noise signal only. We observe that WHYPR had a much smaller rmse. This correlates well with the ﬁndings of using IWHYPR vs. IHYPR given in the ﬁrst table above.
Five HYPR based MRI image reconstruction algorithms were analyzed and simulated. Each algorithm has diﬀerent attributes that need to be examined based on the type of data and the type of acquisition before selecting which algorithm to use. Therefore, the choice of which algorithm to select needs to be examined on a case by case basis. However, there are general guidelines that we can propose in selecting an algorithm. When noise is present, maintaining a good SNR is a requirement that leads one to select the WHYPR. When the images are less sparse and the temporal characteristics are more dynamic, one can choose the HYPRLR algorithm. When the images are more sparse and motion of objects is less prevalent and noise is limited, then one can select the OHYPR.
Finally, when improvement of the temporal characteristics of the generated HYPR images are needed, IHYPR can be used. If noise is present, IWHYPR is the preferred method since it can suppress noise ampliﬁcation more than IHYPR.
[1] Iterative projection reconstruction of timeresolved images using highlyconstrained backprojection (HYPR) by Rafael L. O’Halloran, Zhifei Wen, James H. Holmes, Sean B. Fain
[2] TimeResolved MR Angiography With Limited Projections Yuexi Huang and Graham A. Wright
[3] Principles of computerized Tomographic imaging by Kak and Staney
[4] Highly Constrained Backprojection for TimeResolved MRI by C. A. Mistretta, Wieben,z J, Velikina,W. Block,J. Perry,Y. Wu. K. ohnson and Y. Wu.
[5] Improved Waveform Fidelity Using Local HYPR Reconstruction (HYPR LR). Kevin M. Johnson,Julia Velikina, Yijing Wu, Steve Kecskemeti, Oliver Wieben Charles A. Mistretta
[6] Projection Reconstruction MR Imaging Using FOCUSS. Jong Chul Ye, Sungho Tak, Yeji Han,and Hyun Wook Park
[7] HYPR reports by Nasser M. Abbasi HTML