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A sparsifying transform for use in Compressed Sensing (CS) is a vital piece of image reconstruction for Magnetic Resonance Imaging (MRI). Previously, Translation Invariant Wavelet Transforms (TIWT) have been shown to perform exceedingly well in CS by reducing repetitive line pattern image artifacts that may be observed when using orthogonal wavelets. To further establish its validity as a good sparsifying transform, the TIWT is comprehensively investigated and compared with Total Variation (TV), using six under-sampling patterns through simulation. Both trajectory and random mask based under-sampling of MRI data are reconstructed to demonstrate a comprehensive coverage of tests. Notably, the TIWT in CS reconstruction performs well for all varieties of under-sampling patterns tested, even for cases where TV does not improve the mean squared error. This improved Image Quality (IQ) gives confidence in applying this transform to more CS applications which will contribute to an even greater speed-up of a CS MRI scan. High vs low resolution time of flight MRI CS re-constructions are also analyzed showing how partial Fourier acquisitions must be carefully addressed in CS to prevent loss of IQ. In the spirit of reproducible research, novel software is introduced here as FastTestCS. It is a helpful tool to quickly develop and perform tests with many CS customizations. Easy integration and testing for the TIWT and TV minimization are exemplified. Simulations of 3D MRI datasets are shown to be efficiently distributed as a scalable solution for large studies. Comparisons in reconstruction computation time are made between the Wavelab toolbox and Gnu Scientific Library in FastTestCS that show a significant time savings factor of 60×. The addition of FastTestCS is proven to be a fast, flexible, portable and reproducible simulation aid for CS research.

Magnetic Resonance Imaging (MRI) is a diagnostic modality used to create in-vivo images of 3-Dimensional (3D) biological tissue utilizing magnetic fields, gradients and receivers. When MR signal k-space data are fully sampled based on the Nyquist sampling criteria, some typical high resolution scans take five minutes, allowing time for patient and biologic movement, which negatively impacts Image Quality (IQ). Therefore, it is of great desire to accelerate the data acquisition and achieve better diagnostic images of the body. A promising theory is Compressed Sensing (CS) to under-sample k-space below what the Nyquist criteria requires without compromising IQ. Candès et al. introduce theory and experiments for the general CS problem [

Under-sampling for CS must be carefully designed so that artifacts and errors can be minimized and incoherent. Parallel imaging is another speed-up technique that combines the signals from multiple coils at the same time allowing some under-sampling of k-space to boost the IQ. Deshmane et al. discuss the effects of under-sampling and advantages of using parallel imaging that allows faster acquisition [

CS MRI reconstruction problems are extremely computationally intensive and would have taken orders of magnitude longer to perform even a couple decades ago. Sparsi- fying transforms that perform well in CS can be a wide variety from orthogonal to redundant to adaptive dictionaries, each with varying degrees of complexity and applicability. Ning et al. investigate the patched-based trained directional wavelets and extend them into the translation invariant domain to enhance MRI image features [

In previous works, the Translation Invariant Wavelet Transform (TIWT) has been shown to perform exceedingly well in CS against other transforms [

The focus here contributes a thorough analysis of the TIWT using under-sampling patterns for use in CS with MRI, not previously provided. The Total Variation (TV) minimization vs. the TIWT sparsifying transform are compared in CS reconstruction for IQ improvements. TV is a calculation of the pixel by pixel finite differences in both horizontal and vertical directions across an image. The TIWT is a redundant circularly shifted wavelet calculation that achieves translational invariance, which is lacking in the standard orthogonal wavelet. Various trajectories that cover a wide range of options and possibilities are used as a testing strategy for proving the TIWT is robust for various noise and under-sampling artifacts. The six different under-sampling patterns analyzed are: uniform random, 1D variable density with large and small center, 2D variable density with and without fully sampled center, uniform radial, and logarithmic spiral. The process of developing and comparing these different measurement matrices also involves processing a number of permutations. Through checking several types of applicable under-sampling patterns in simulation, the goal of proving the robustness of this technique is established. This technique can enhance the reconstruction quality of the CS MRI result and therefore enable the use of less data and acquisition time.

Novel software, FastTestCS, is introduced here for the purpose of reducing computation time in CS reconstruction simulation and testing. It is a tool that can be customized for CS reconstructions using different images, sampling patterns, sparse transforms and optimization techniques. It also is written in the compiled language of C++ that can allow for quick runtime of simulations. Additionally, compute performance and time can be measured more in line with an expected fast product implementation. The FastTestCS executable is designed with a full set of versatile arguments to customize all the needed functionality to run several different simulations.

In summary, a comprehensive analysis of the robustness of the TIWT in CS applications is presented through simulations involving six different under-sampling patterns for MRI. Comparisons in reconstruction compute time are made between the Wavelab toolbox in Matlab and Gnu Scientific Library (GSL) in C++ that show a significant time savings factor of 60×. This transform is computationally feasible as it is shown to be only one order of magnitude more computation than the orthogonal wavelet, yet produces higher fidelity image reconstructions. FastTestCS software is introduced and demonstrated as useful for researchers interested in CS simulations using different images, sparsifying transforms, objective functions and under-sampling patterns in a quick development and test environment that can scale to large studies.

The format consists of five main parts. Section 1 and 2 are introductory and background of theory and pertinent literature on under-sampling in CS for MRI. Section 3 details the methods used for the simulations and development of FastTestCS. This includes under-sampling pattern formula and the CS simulation setup. Section 4 presents results from the simulations of TIWT vs TV CS including images and tables of measurements. Finally, Section 5 discusses these results with the use of FastTestCS.

The three main ingredients of CS are correct under-sampling, sparsity and optimization. Candès et al. and Lustig et al. state a CS requirement of incoherence between the under-sampling domain and the sparse representation domain [

For quick adoption of CS in MRI, patterns that follow existing sequences and trajectories with some under-sampling should be used. Both Cartesian and non-Cartesian under-sampling patterns, such as radial and spiral trajectories, have already been investigated for CS in MRI. Lustig et al. utilize a 1D variable density under-sampling [

Radial and spiral sampling techniques are used for their advantages. For example, radial acquisitions reduce structural and motion artifacts over standard Cartesian acquisition, while keeping the requirement of artifact incoherence. King improves non- Cartesian spiral scanning with an anisotropic field of view where acquistion is fast and reconstruction improves IQ utilizing gridding [

Sparsity is the amount of an image or signal that can be represented with information. When a signal is n-sparse, the signal has p elements, but only has n elements of valuable information, in some cases,

Due to under-sampling, the signal recovery in Equation (1) is under-determined. The signal

The solution is non-linear and exponentially complex, therefore an alternative

W represents the sparsifying transform and x is the signal approximation. In order to practically implement this sparse recovery problem, an unconstrained formulation is utilized as shown in Equation (3).

To prove the effective use of the TIWT as a robust technique for CS MRI, several under-sampling patterns are developed and compared with simulation. The goal of an under-sampling pattern for CS is to provide incoherent and minimal noise like artifacts. Developing under-sampling patterns for MRI trajectories that preserve the data at the center and less at the periphery are of great interest, because the sampling domain is k-space and the majority of the signal power is at the center. The design of these trajectories are described next.

The radial under-sampling technique is calculated in a circular pattern, see

The spiral technique presented here is logarithmic, meaning it will grow from the origin as the distance increases, see

Several random distributions are utilized, a basic uniform random under-sampling is used as in

The resulting random distribution matrix can be thresholded to any desired reduction factor %. Additionally, a fully sampled center can be applied to this technique by setting the probability of the points within the center diameter to 100%, see

A 1D variable density under-sampling technique with a fully sampled center is seen in

An analysis is performed for the question of whether to reconstruct CS at a low vs high resolution for the case of a Time of Flight (TOF) Maximum Intensity Projection (MIP). The input high resolution MRI TOF k-space dataset is 256 × 224. It is centered by the highest peak in a 512 × 512 matrix, see

Additionally, a shifting of the under-sampling pattern to correlate with the center of k-space is also performed. For CS MRI acquisition, the under-sampling needs to be centered at the k-space peak to be effective and not lose important information at the k-space center and periphery.

A new CS testing framework called FastTestCS is proposed here. It is fully written in C++, is modular, with simple methods that can be used similar to functions used in Matlab [

Matlab is a rich scientific programming language, ideal for algorithm development, however, having the best optimized performance of simulations may be limited due to abstraction and additional language internal processing. Wavelab [

The speed of the TIWT was tested by comparing the transform reconstruction wall time difference between the Wavelab Matlab implementation vs. the transform written in C++. Two resolutions of an image were chosen, 256 × 256 and 512 × 512. The test code was developed by isolating the transform setup, data memory transfers, compu- tation and tear down of the method. These timing tests are performed without the overhead of other CS processing. This time measurement includes calculating 10 for- ward and inverse transforms on both real and imaginary parts of the image for a total of 40 calls to the TIWT method.

This number of transform calculations is comparable to the minimum that must be performed in a CS program. However, often the number of transforms are much greater, such is the case for the NLCG with a quadratic line search and Wolfe con- ditions. Many extra iterations must be performed to determine appropriate search dire- ctions and step sizes.

Four different implementations are tested for the standard orthogonal wavelet and the TIWT. The translational invariance is achieved by performing circular shifts of the data. The orthogonal wavelet provided by Matlab as “WAVEREC2” and “WAVEDEC2”, is compared with the GSL C++ implementation “gsl_wavelet2d_nstransform_matrix”. The TIWT, provided by Wavelab as “FWT2_TI” and “IWT2_TI”, is compared with the FastTestCS implementation “TIDWT” and “ITIDWT”.

The FastTestCS approach implements CS and several sparsifying transforms that run faster in C++ than in Matlab. Algorithms can be prototyped for accuracy first in Matlab, and then re-implemented in C++. The simulations are run in C++ in less time and in parallel across multiple cores and machines. Additionally, FastTestCS has image com- parison tools such as Mean Square Error (MSE), as well as other useful image operations like Normalize, ReadImg or Write Data. It is a Microsoft Visual Studio project and can be compiled for Windows or Linux. It allows sharing code for easy algorithm prototyping, comparison and dataset manipulation. It is a portable and customizable simulation tool to benefit CS research.

There are many required permutations of several parameters and reconstruction tests to run in order to prove robustness. It is apparent that efficiency can be improved by parallelizing computations and simulations, compared with running on just one machine with one thread. Since some CS routines could run over a week to a month, it would be impractical to try many different options serially.

In response, FastTestCS executable is designed with very versatile arguments to implement all the needed functionality and parameters to run several different simulations from the command line. Multiple machines are used to easily distribute many simulations. Simple scripts are used to tar up the software and unique calling parameters, send the package to the specified machine, build and run the package, save the results and transfer the images and results back to a single archiving repository for review. With this method, there are no limits to the number of machines that can be used and simulations that can be achieved more quickly in parallel.

Distribution is achieved for these tests with a small cluster of five individual quad core machines, with one being the source and archive, in addition to having its own portion of processing. Meta-data in the image identifies what simulation parameters produced the image for distinction. Once all the resulting images are together, quantitative and qualitative image comparisons are performed with a “golden” fully sampled image. An executable loads the reconstructed images and the golden image, and makes several calculations for each image such as MSE. The program can then output results to a comma separated table with one line for each image that is easily viewed in a spreadsheet.

MRI k-space data is generated by a General Electric (GE) Healthcare MRI scanner and saved as a “P”-file. GE provided Matlab software to read the P-file and make a k-space matrix. Due to this data being on a Cartesian grid, a fully sampled recon- struction is simply an Inverse FFT. This fully sampled dataset is input to the under- sampling process used in FastTestCS reconstruction by way of a Comma Separated Value (CSV) file. The resulting CS reconstruction is compared to the original fully sampled dataset and measured by MSE to verify its fidelity.

The MSE is defined as a measure of pixel intensity error between

Three image types are used in the analysis. A phantom image, see

An important way to analyze different parameters and IQ results of CS recon- structions is to keep all parameters the same in the CS algorithm and just change the transform or sampling pattern. The analysis and results here are done with individual transforms without a combination of other transforms. This is to measure the error of each sampling pattern independently and identify the transform quality.

The steps of the CS reconstruction simulation are as follows and in

1) Start with fully sampled k-space data.

2) Sample the k-space data using a desired under-sampling pattern.

3) Perform the iterative non-linear optimization technique.

4) Compare the error between the fully sampled image and the CS reconstruction to measure the quality of the reconstruction.

The CS reconstruction optimization in FastTestCS is the NLCG with backtracking

line search and Wolfe conditions. The formulation of the objective function

where

This unconstrained formulation is versatile when dealing with non-orthonormal transforms. The calculation of the gradients in the unconstrained approach requires the inverse of the transform. For an orthonormal transform, that is simply a multiplication of the coefficients by a transpose of the wavelet dictionary. However, for a non- orthonormal transform, like the TIWT and many of the other transforms, a separate calculation of the inverse transform is used rather than a transpose of the dictionary. In FastTestCS, a transform method is created that calls the forward or inverse of the transform. All coding for the transform is easily integrated, segregated and tested. Real and imaginary parts of MRI data are treated separately throughout all of the optimization.

Performance measurements show a big speed up in computation time with optimized C++ code vs Wavelab. See

Tables 2-8, show a comparison of TIWT vs TV CS with several sampling patterns for IQ analysis on three sample images from

Transform | Function | Language | 256 × 256 | 512 × 512 |
---|---|---|---|---|

O Wavelet | WAVEREC2 & WAVEDEC2 | Matlab toolbox | 0.75 | 2.9 |

O Wavelet | gsl_wavelet2d_nstransform_matrix | C++ GSL FastTestCS | 0.33 | 2.21 |

TIWT | FWT2_TI & IWT2_TI | Matlab Wavelab | 342 | 1380 |

TIWT | TIDWT & ITIDWT | C++ GSL FastTestCS | 4.6 | 22.83 |

Samples | Phantom | T1 MRI | TOF MIP | ||||||
---|---|---|---|---|---|---|---|---|---|

% (itr) | Lin | TI | TV | Lin | TI | TV | Lin | TI | TV |

25% (10) | 355 | 55.2 | 145 | 449 | 65.6 | 362 | 194 | 135 | 140 |

25% (100) | 355 | 4.68 | 3.00 | 449 | 65.3 | 59.5 | 194 | 27.2 | 81.5 |

25% (500) | 355 | 0.450 | 0.033 | 449 | 64.0 | 49.7 | 194 | 25.3 | 67.9 |

38% (10) | 197 | 22.6 | 25.7 | 256 | 12.1 | 50.0 | 132 | 36.0 | 73.4 |

38% (100) | 197 | 1.34 | 0.0244 | 256 | 13.0 | 34.8 | 132 | 10.9 | 38.5 |

38% (500) | 197 | 0.004 | 0.02 | 256 | 14.5 | 21.2 | 132 | 9.52 | 27.2 |

50% (10) | 132 | 10.6 | 2.95 | 86.2 | 13.0 | 29.0 | 37.8 | 6.00 | 20.6 |

50% (100) | 132 | 0.163 | 0.0112 | 86.2 | 15.6 | 14.6 | 37.8 | 3.45 | 15.9 |

50% (500) | 132 | 0.0018 | 0.0122 | 86.2 | 14.0 | 23.1 | 37.8 | 3.69 | 14.4 |

75% (10) | 53.6 | 3.11 | 0.021 | 4.6 | 1.94 | 2.51 | 3.67 | 0.92 | 2.66 |

75% (100) | 53.6 | 0.033 | 0.0064 | 4.6 | 1.96 | 2.65 | 3.67 | 0.73 | 2.52 |

75% (500) | 53.6 | 0.0007 | 0.0052 | 4.6 | 2.34 | 2.60 | 3.67 | 0.77 | 2.48 |

Samples | Phantom | T1 MRI | TOF MIP | ||||||
---|---|---|---|---|---|---|---|---|---|

% | Lin | TI | TV | Lin | TI | TV | Lin | TI | TV |

25% | 216 | 28.9 | 1.91 | 354 | 18.8 | 25.3 | 138 | 39.6 | 85.8 |

38% | 164 | 3.2 | 0.0221 | 80 | 12.0 | 14.0 | 70.2 | 10.4 | 33.0 |

50% | 118 | 0.492 | 0.0194 | 34.6 | 6.75 | 19.6 | 30.9 | 4.81 | 14.8 |

75% | 54.9 | 0.061 | 0.0077 | 2.98 | 2.0 | 2.92 | 4.12 | 0.93 | 2.23 |

demonstrates low noise and simple structures. The T1 and TOF MRI brain images portray practical medical images which have differences in noise and fine structure throughout. The 3D TOF additionally represents a sparse image domain which should

Samples | Phantom | T1 MRI | TOF MIP | ||||||
---|---|---|---|---|---|---|---|---|---|

% | Lin | TI | TV | Lin | TI | TV | Lin | TI | TV |

25% | 2629 | 1029 | 1128 | 3709 | 2077 | 3125 | 389 | 304 | 2684 |

38% | 1910 | 837 | 1033 | 3251 | 2121 | 2823 | 281 | 196 | 310 |

50% | 1518 | 789 | 922 | 1346 | 519 | 837 | 154 | 95.5 | 145 |

75% | 1099 | 603 | 824 | 409 | 146 | 269 | 33.2 | 10.1 | 33.9 |

Samples | Phantom | T1 MRI | TOF MIP | ||||||
---|---|---|---|---|---|---|---|---|---|

% | Lin | TI | TV | Lin | TI | TV | Lin | TI | TV |

25% | 594 | 129 | 164 | 354 | 124 | 135 | 1128 | 457 | 631 |

38% | 408 | 15.0 | 29.0 | 277 | 66.6 | 108 | 562 | 85.8 | 266 |

50% | 291 | 1.23 | 13.5 | 252 | 26.7 | 35.6 | 444 | 50.8 | 216 |

75% | 85.3 | 0.033 | 1.97 | 43.4 | 3.41 | 6.87 | 29.2 | 3.01 | 15.8 |

Samples | Phantom | T1 MRI | TOF MIP | ||||||
---|---|---|---|---|---|---|---|---|---|

% | Lin | TI | TV | Lin | TI | TV | Lin | TI | TV |

25% | 336 | 264 | 182 | 169 | 85.3 | 89.0 | 370 | 188 | 276 |

38% | 222 | 60.7 | 75.8 | 145 | 31.7 | 57.9 | 217 | 82.6 | 114 |

50% | 174 | 11.0 | 2.24 | 136 | 27.5 | 76.6 | 134 | 41.4 | 67.1 |

75% | 64.6 | 0.014 | 0.386 | 28.5 | 2.66 | 4.72 | 12.9 | 1.94 | 7.70 |

Samples | Phantom | T1 MRI | TOF MIP | ||||||
---|---|---|---|---|---|---|---|---|---|

% | Lin | TI | TV | Lin | TI | TV | Lin | TI | TV |

25% | 1429 | 770 | 960 | 1956 | 848 | 1696 | 87.8 | 31.9 | 84.9 |

38% | 1092 | 568 | 808 | 1618 | 768 | 1541 | 59.5 | 23.7 | 52.5 |

50% | 910 | 581 | 750 | 1496 | 797 | 1505 | 22.2 | 4.03 | 11.8 |

75% | 59.6 | 0.702 | 0.008 | 3.74 | 1.99 | 2.61 | 2.45 | 1.53 | 1.87 |

lend well to CS due to the use of TV and wavelet

Samples | Phantom | T1 MRI | TOF MIP | ||||||
---|---|---|---|---|---|---|---|---|---|

% | Lin | TI | TV | Lin | TI | TV | Lin | TI | TV |

25% | 286 | 65.4 | 3.42 | 280 | 31.7 | 27.1 | 187 | 32.3 | 56.1 |

38% | 221 | 14.3 | 0.0211 | 186 | 11.1 | 20.4 | 124 | 4.96 | 37.4 |

50% | 167 | 0.0328 | 0.0132 | 101 | 7.50 | 10.3 | 61.4 | 2.87 | 20.1 |

75% | 54.2 | 0.0596 | 0.0035 | 10.3 | 2.32 | 3.25 | 4.37 | 0.84 | 2.69 |

In

In

In

CS reconstruction, the noise is significantly reduced, see

To mitigate this, the MIP simulations shown in the tables are all done by shifting the k-space data so that it is not cropped, but rather has the center of k-space and the sampling center match slightly off center. The MSE difference in this case is 0.11, which is negligible, and verifies the assumption that there is no visible difference in IQ between low and high resolution fully sampled reconstructions. With this accuracy, there is confidence in running these tests at low resolution with a shifted k-space.

Performing all reconstructions at a high resolution would come at a high compu- tation cost of about 5× longer, therefore, lower resolution reconstructions are perfor- med. As a time comparison, a 512 × 512 8 channel 192 slice MIP CS reconstruction with the TIWT for 10 iterations takes 20 hours, whereas, the same reconstruction at 256 × 256 takes 4 hours, single threaded, with an Intel Core i7 computer.

Here it is shown that the choice of good under-sampling patterns clearly has a correlation to the data being gathered. The center of k-space visually has a more intense magnitude of data, therefore, sampling density at the center contributes to a better SNR and a lower MSE. The radial, spiral, 1D and 2D variable density with the center fully sampled performs good, compared to the other patterns that did not sample with a dense center. The balancing of image quality with how the under-sampling pattern dissipates, is a complex analysis that requires visual as well as error analysis.

As seen in the synthetic phantom error tables, some CS reconstructions may perform exceptionately well in comparison to real in-vivo MRI data. The lack of noise and the few distinct edges in the original phantom data contribute to a high success for the TIWT and especially TV, which measures differences between pixels. With that in mind, care must be taken for each MRI imaging application that uses CS due to likely differences in noise statistics and image types affecting the IQ performance of the CS reconstruction. This hightlights the value of having a broad spectrum of tests so that the correct CS parameters can be chosen and used.

CS reconstruction of TOF data used in a MIP can be performed at a lower acquisition resolution and then transformed to a higher resolution because there are no visible differences in smaller vessels. The analysis of IQ with MSE is challenging for MIP because the error measurement may not pick up the changes in small vessels being of little numerical error. Additionally, changes in the background intensity of the image do contribute to larger error. Having a larger MSE may be misleading compared to what is observed with a visual analysis.

Special care must be taken for under-sampling with partially encoded Fourier k-space data. Visually, the image generated without small amounts of data at the periphery does have a negative impact on small vessel details of the MIP CS reconstruction. Therefore, adjusting the k-space center is needed to prevent cropping any data and have more meaningful IQ comparisons.

The stopping criteria is determined by analyzing the IQ after a certain number of iterations. This proves to be a very challenging issue due to the fact that different image types will require a different number of iterations to produce the best reconstructions. Image types with very little noise require many more iterations, however, the recon- struction IQ improves greatly as well. This improvement is probably due to the fundamental aspect of CS reconstruction theory of sparsity. When higher sparsity can be achieved in transforming these images, CS has a greater probability of better recon- structions with less data. However, most real images do have noise.

Despite this dilemma, a solution can be observed where enough iterations can be achieved that produces a result that will not change IQ significantly if more iterations are performed. Then this number can be used for other similar tests. This aspect of CS highlights the importance of performing an analysis for determining the best stopping criteria for specific image types and algorithms.

The choice to develop FastTestCS came out of the necessity to speed up reconstructions using a more complex TIWT in CS with larger 3D data sets.

FastTestCS may be customized to include any image or signal, objective function, sampling pattern and sparsifying transform to test different scenarios and compare reconstructions. Additionally, any libraries or algorithms available in C++ can be incorporated into FastTestCS. Having consistent CS input data and algorithms to test various options provides a clear analysis and simulation tool with unlimited possibilities for future enhancements and tests. The executable and command line interface provides a convenient way to parallelize on different compute cores and machines to quickly test across various inputs and options in an automated way. Contact the author for software availability to obtain reproducible results, usage and advancement, http://people.uwm.edu/gxu4uwm/fasttestcs.

The applicability of using the TIWT for CS MRI is wide spread due to the variety of image types and sampling patterns encountered in MRI. Therefore, there are many applications along those lines that could be researched in the future. Additionaly, different wavelet filters and settings could be investigated to find which applications perform best with which filters to produce better IQ. Also, future expansions to FastTestCS are possible to add functionaltiy for testing other images, sparse transforms, sampling patterns and objective functions.

By simulation of CS reconstructions and measurement of IQ differences, results indicate robustness of the TIWT. Several comprehensive under-sampling patterns are designed to compare the TIWT with TV as sparsifying transforms in CS MRI recon- struction. The TIWT performs consistently well in terms of MSE for all under- sampling patterns. An in-depth look at high and low resolution TOF MIP CS recon- struction is evaluated for IQ differences. Analysis shows that careful under-sampling must be performed at the center of k-space to preserve important data for better recon- struction. With the TIWT, good k-space under-sampling and a reliable CS recon- struction algorithm, improved IQ and greater speed-up for CS MRI can be achieved.

FastTestCS software is introduced and demonstrated as a helpful tool for CS reconstruction research that offers the needed speedup with process distribution to perform experiments with large datasets. In the case of the TIWT, a 60× time improve- ment was achieved. FastTestCS also is easily customized with each under-sampling pattern used in this investigation.

The author wishes to thank Professor Guangwu Xu for guidance, Academic Advisor, Department of Computer Science and Engineering at the University of Wisconsin- Milwaukee, and Dr. Kevin King, Scientist at GE Healthcare for providing the MRI data along with helpful review.

Baker, C. (2016) Comparison of MRI Under-Sampling Techniques for Compressed Sensing with Translation Invariant Wavelets Using FastTestCS: A Flexible Simulation Tool. Journal of Signal and Information Processing, 7, 252-271. http://dx.doi.org/10.4236/jsip.2016.74021