Compressed SensingWhat is compressed sensing and what is it good for?
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Compressed sensing (CS) refers to a group of methods for accelerated MR data acquisition based on semi-random, incomplete sampling of k-space. A final image is created through an iterative optimization process using non-Fourier transformation and thresholding of intermediately reconstructed images. Although still relatively early in development, CS methods appear especially promising for MRA, 3D/4D MRI, dynamic contrast enhanced (DCE) studies, and cardiac MR imaging applications. Their utility has been recently expanded by incorporation of deep learning models that improve reconstruction accuracy and reduce computational time.
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Cine cardiac study using compressed sensing with acceleration factors up to 15-fold. (Adapted from Abascal et al under CC BY)
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Most readers will be aware that file size of photos can be reduced as much as 10:1 while maintaining good quality using a format like jpeg. Such compression is possible because photographic (as well as MR) images contain many pixels lacking unique information content (e.g., those in the background with zero intensity and those with nearly the same values as their neighbors).
This leads directly to the ingenious idea underlying CS MRI: If MR images can be compressed after reconstruction, why not save time up front and only collect only the "essential" components of the MR signal rather than all k-space data? To successfully accomplish this, three factors are required: incoherent undersampling, sparsifying transformation, and iterative reconstruction.
Incoherent Undersampling
The term undersampling means that only a portion of k-space is measured, thus allowing significant reduction in data acquisition times. However, this undersampling must be performed in an incoherent (semi-random) manner. Data undersampling in a coherent (predictable) pattern, such as selecting every 3rd line of k-space, creates discrete aliasing "ghost" artifacts that would wrap over the image. By sampling irregularly, these artifacts are instead distributed as diffuse noise across the entire image that can later be removed. At the same time, data sampling cannot be too random, because the center of k-space contains more "essential" image information than its periphery. Accordingly, most current CS approaches (variable density Poisson disk, Gaussian, and Golden-angle radial) employ semi-random acquisition of data with preferential sampling near the k-space center.
Sparsifying Transformation
Sparsity reflects the extent to which an imaging matrix is filled with meaningful data. Some matrices are intrinsically sparse, such as those associated with MRAs, containing mostly zeros (corresponding to black/nonvascular regions). Images of solid organs may not appear sparse in their conventional display, but can be mathematically transformed/decomposed into representations having relatively sparse components. The application of so-called sparsifying transforms (such as wavelets) are an essential feature of CS reconstruction methods described below.
MRA - an intrinsically sparse image
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Brain MRI - not intrinsically sparse
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Relatively sparse wavelet components
(From Zhao et al under CC BY) |
Iterative Reconstruction
Unlike single-step Fourier reconstruction of conventional MR images, Compressed Sensing involves multiple steps and data transformations, mediated by an iterative optimization algorithm repeated dozens of times, following a scheme something like this:
- Collect initial k-space data using limited, incoherent (semi-random) sampling.
- Fourier transform this data to create an initial image. The initial image will be very crude, suffering from diffuse aliasing artifacts manifest as "noise".
- Apply a sparsifying transform (such as wavelet decomposition) that will concentrate meaningful imaging characteristics into a smaller number of high-intensity pixels. The aliasing "noise" will now be of lower intensity and distributed over many pixels, most in the background.
- Remove aliasing ("denoise") the sparsified data by zeroing out pixels with values below a minimum threshold, digital filtering, and/or subtraction.
- Apply inverse sparsifying and Fourier transforms to reconvert this "denoised" data back into k-space format.
- Create a k-space "difference matrix" by subtracting the original from the denoised k-space data and setting all other points to zero. By Fourier transformation, a "difference image" can then be created. Add the initial and difference images together to create an "updated image".
- Compare the initial and updated images. If significant disparity is present, repeat Steps 3-7. Otherwise Stop.
Commercial Applications of CS
CS methods are now being offered as commercial products: Compressed Sensing (Siemens), Compressed SENSE (Philips), HyperSense (GE), and Compressed Speeder (Canon) These are now being combined with deep learning based reconstruction methods to shorten exam time and improve image quality.
References
Abascal JFPJ, Montesinos P, Marinetto E, Pascau J, Desco M. Comparison of total variation with a motion estimation based compressed sensing approach for self-gated cardiac cine MRI in small animal studies. PLos ONE 2014; 9(10): e110594.
Blasche M, Forman C. Compressed sensing — the flowchart. MAGNETOM Flash 2016; 66:4-7. (Good review of the Siemens iterative construction algorithm).
Donoho DL. Compressed sensing. IEEE Trans Information Theory 2006; 52:1289-1306. [DOI LINK]
(First paper on theory of CS)
Geethanath S, Reddy R, Konar AS, et al. Compressed sensing MRI: a review. Critical Reviews in Biomedical Engineering 2013; 41:183-204.
Jaspan ON, Fleysher R, Lipton ML. Compressed sensing MRI: a review of the clinical literature. Brit J Radiol 2015; 88:20150487
Liang D, Cheng J, Ke Z, Ying L. Deep magnetic resonance image reconstruction: inverse problems meet neural networks. IEEE Signal Process Mag 2020; 37:141-151. [DOI LINK]
Lustig M, Donoho D, Pauly JM. Sparse MRI: the application of compressed sensing for rapid MR imaging. Magn Reson Med 2007; 58:1182-1195.
Zhao D, Du H, Han Y, Mei w. Compressed sensing MR image reconstruction exploiting TGV and wavelet sparsity. Comput Math Methods Med 2014; 958671.
Abascal JFPJ, Montesinos P, Marinetto E, Pascau J, Desco M. Comparison of total variation with a motion estimation based compressed sensing approach for self-gated cardiac cine MRI in small animal studies. PLos ONE 2014; 9(10): e110594.
Blasche M, Forman C. Compressed sensing — the flowchart. MAGNETOM Flash 2016; 66:4-7. (Good review of the Siemens iterative construction algorithm).
Donoho DL. Compressed sensing. IEEE Trans Information Theory 2006; 52:1289-1306. [DOI LINK]
(First paper on theory of CS)
Geethanath S, Reddy R, Konar AS, et al. Compressed sensing MRI: a review. Critical Reviews in Biomedical Engineering 2013; 41:183-204.
Jaspan ON, Fleysher R, Lipton ML. Compressed sensing MRI: a review of the clinical literature. Brit J Radiol 2015; 88:20150487
Liang D, Cheng J, Ke Z, Ying L. Deep magnetic resonance image reconstruction: inverse problems meet neural networks. IEEE Signal Process Mag 2020; 37:141-151. [DOI LINK]
Lustig M, Donoho D, Pauly JM. Sparse MRI: the application of compressed sensing for rapid MR imaging. Magn Reson Med 2007; 58:1182-1195.
Zhao D, Du H, Han Y, Mei w. Compressed sensing MR image reconstruction exploiting TGV and wavelet sparsity. Comput Math Methods Med 2014; 958671.
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