Mehmet Akçakaya

Shannon-Theoretic Limits on Noisy Compressive Sampling

In this paper, we study the number of measurements required to recover a sparse signal in CM with L nonzero coefficients from compressed samples in the presence of noise. We consider a number of different recovery criteria, including the exact recovery of the support of the signal, which was previously considered in the literature, as well as new criteria for the recovery of a large fraction of the support of the signal, and the recovery of a large fraction of the energy of the signal. For these recovery criteria, we prove that O(L) (an asymptotically linear multiple of L) measurements are necessary and sufficient for signal recovery, whenever L grows linearly as a function of M. This improves on the existing literature that is mostly focused on variants of a specific recovery algorithm based on convex programming, for which O(L log(M - L)) measurements are required. In contrast, the implementation of our proof method would have a higher complexity. We also show that O(L log(M - L)) measurements are required in the sublinear regime (L - o(M)). For our sufficiency proofs, we introduce a Shannon-theoretic decoder based on joint typicality, which allows error events to be defined in terms of a single random variable in contrast to previous information-theoretic work, where comparison of random variables are required. We also prove concentration results for our error bounds implying that a randomly selected Gaussian matrix will suffice with high probability. For our necessity proofs, we rely on results from channel coding and rate-distortion theory.

Combined saturation/inversion recovery sequences for improved evaluation of scar and diffuse fibrosis in patients with arrhythmia or heart rate variability.

To develop arrhythmia‐insensitive inversion recovery sequences for improved visualization of myocardial scar and quantification of diffuse fibrosis.

Low-dimensional-structure self-learning and thresholding: regularization beyond compressed sensing for MRI reconstruction.

An improved image reconstruction method from undersampled k‐space data, low‐dimensional‐structure self‐learning and thresholding (LOST), which utilizes the structure from the underlying image is presented. A low‐resolution image from the fully sampled k‐space center is reconstructed to learn image patches of similar anatomical characteristics. These patches are arranged into “similarity clusters,” which are subsequently processed for dealiasing and artifact removal, using underlying low‐dimensional properties. The efficacy of the proposed method in scan time reduction was assessed in a pilot coronary MRI study. Initially, in a retrospective study on 10 healthy adult subjects, we evaluated retrospective undersampling and reconstruction using LOST, wavelet‐based l1‐norm minimization, and total variation compressed sensing. Quantitative measures of vessel sharpness and mean square error, and qualitative image scores were used to compare reconstruction for rates of 2, 3, and 4. Subsequently, in a prospective study, coronary MRI data were acquired using these rates, and LOST‐reconstructed images were compared with an accelerated data acquisition using uniform undersampling and sensitivity encoding reconstruction. Subjective image quality and sharpness data indicate that LOST outperforms the alternative techniques for all rates. The prospective LOST yields images with superior quality compared with sensitivity encoding or l1‐minimization compressed sensing. The proposed LOST technique greatly improves image reconstruction for accelerated coronary MRI acquisitions. Magn Reson Med, 2011.

A Frame Construction and a Universal Distortion Bound for Sparse Representations

We consider approximations of signals by the elements of a frame in a complex vector space of dimension N and formulate both the noiseless and the noisy sparse representation problems. The noiseless representation problem is to find sparse representations of a signal r given that such representations exist. In this case, we explicitly construct a frame, referred to as the Vandermonde frame, for which the noiseless sparse representation problem can be solved uniquely using O(N2) operations, as long as the number of non-zero coefficients in the sparse representation of r is isinN for some 0 les isin les 0.5. It is known that isin les 0.5 cannot be relaxed without violating uniqueness. The noisy sparse representation problem is to find sparse representations of a signal r satisfying a distortion criterion. In this case, we establish a lower bound on the tradeoff between the sparsity of the representation, the underlying distortion and the redundancy of any given frame.

Accelerated Late Gadolinium Enhancement Cardiac MR Imaging with Isotropic Spatial Resolution Using Compressed Sensing: Initial Experience

PURPOSE To evaluate the use of low-dimensional-structure self-learning and thresholding (LOST) compressed sensing acquisition and reconstruction in the assessment of left atrial (LA) and left ventricular (LV) scar by using late gadolinium enhancement (LGE) magnetic resonance (MR) imaging with isotropic spatial resolution. MATERIALS AND METHODS The study was approved by the local institutional review board and was compliant with HIPAA. All subjects provided written informed consent. Twenty-eight patients (eight women; mean age, 58.0 years ± 10.1) with a history of atrial fibrillation were recruited for the LA LGE study, and 14 patients (five women; mean age, 54.2 years ± 18.6) were recruited for assessment of LV myocardial infarction. With use of a pseudorandom k-space undersampling pattern, threefold accelerated three-dimensional (3D) LGE data were acquired with isotropic spatial resolution and reconstructed off-line by using LOST. For comparison, subjects were also imaged by using standard 3D LGE protocols with nonisotropic spatial resolution. Images were compared qualitatively by three cardiologists with regard to diagnostic value, presence of enhancement, and image quality. The signed rank test and Wilcoxon unpaired two-sample test were used to test the hypothesis that there would be no significant difference in image quality ratings with different resolutions. RESULTS Interpretable images were obtained in 26 of the 28 patients (93%) in the LA LGE study. LGE was seen in 17 of 30 cases (57%) with nonisotropic resolution and in 18 cases (60%) with isotropic resolution. Diagnostic quality scores of isotropic images were significantly higher than those of nonisotropic images with coronal views (median, 3 vs 2, respectively [25th and 75th percentiles: 3, 3 vs 2, 3]; P < .001) and sagittal views (median, 3 vs 2 [25th and 75th percentiles: 3, 4 vs 2, 3]; P < .001) but lower with axial views (median, 4 vs 3 [25th and 75th percentiles: 3, 4 vs 3, 3]; P < .001). For the LV LGE study, all patients had interpretable images. LGE was seen in six of 14 patients (43%), with 100% agreement between both data sets. Diagnostic quality scores of high-isotropic-resolution LV images were higher than those of nonisotropic images with short-axis views (median, 4 vs 3 [25th and 75th percentiles: 3, 4 vs 2, 3]; P = .014) and two-chamber views (median, 4 vs 3 [25th and 75th percentiles: 3, 4 vs 2, 3]; P = .001). CONCLUSION An accelerated LGE acquisition with LOST enables imaging with high isotropic spatial resolution for improved assessment of LV, LA, and pulmonary vein scar.

Compressed sensing reconstruction for whole-heart imaging with 3D radial trajectories: a graphics processing unit implementation.

A disadvantage of three‐dimensional (3D) isotropic acquisition in whole‐heart coronary MRI is the prolonged data acquisition time. Isotropic 3D radial trajectories allow undersampling of k‐space data in all three spatial dimensions, enabling accelerated acquisition of the volumetric data. Compressed sensing (CS) reconstruction can provide further acceleration in the acquisition by removing the incoherent artifacts due to undersampling and improving the image quality. However, the heavy computational overhead of the CS reconstruction has been a limiting factor for its application. In this article, a parallelized implementation of an iterative CS reconstruction method for 3D radial acquisitions using a commercial graphics processing unit is presented. The execution time of the graphics processing unit‐implemented CS reconstruction was compared with that of the C++ implementation, and the efficacy of the undersampled 3D radial acquisition with CS reconstruction was investigated in both phantom and whole‐heart coronary data sets. Subsequently, the efficacy of CS in suppressing streaking artifacts in 3D whole‐heart coronary MRI with 3D radial imaging and its convergence properties were studied. The CS reconstruction provides improved image quality (in terms of vessel sharpness and suppression of noise‐like artifacts) compared with the conventional 3D gridding algorithm, and the graphics processing unit implementation greatly reduces the execution time of CS reconstruction yielding 34–54 times speed‐up compared with C++ implementation. Magn Reson Med, 2013.

Free-breathing multislice native myocardial T1 mapping using the slice-interleaved T1 (STONE) sequence.

To develop a novel pulse sequence for free‐breathing, multislice, native myocardial T1 mapping.

Accelerated isotropic sub‐millimeter whole‐heart coronary MRI: Compressed sensing versus parallel imaging

To enable accelerated isotropic sub‐millimeter whole‐heart coronary MRI within a 6‐min acquisition and to compare this with a current state‐of‐the‐art accelerated imaging technique at acceleration rates beyond what is used clinically.

Accelerated aortic flow assessment with compressed sensing with and without use of the sparsity of the complex difference image

Phase contrast (PC) cardiac MR is widely used for the clinical assessment of blood flow in cardiovascular disease. One of the challenges of PC cardiac MR is the long scan time which limits both spatial and temporal resolution. Compressed sensing reconstruction with accelerated PC acquisitions is a promising technique to increase the scan efficiency. In this study, we sought to use the sparsity of the complex difference of the two flow‐encoded images as an additional constraint term to improve the compressed sensing reconstruction of the corresponding accelerated PC data acquisition. Using retrospectively under‐sampled data, the proposed reconstruction technique was optimized and validated in vivo on 15 healthy subjects. Then, prospectively under‐sampled data was acquired on 11 healthy subjects and reconstructed with the proposed technique. The results show that there is good agreement between the cardiac output measurements from the fully sampled data and the proposed compressed sensing reconstruction method using complex difference sparsity up to acceleration rate 5. In conclusion, we have developed and evaluated an improved reconstruction technique for accelerated PC cardiac MR that uses the sparsity of the complex difference of the two flow‐encoded images. Magn Reson Med 70:851–858, 2013.

Accelerated noncontrast-enhanced pulmonary vein MRA with distributed compressed sensing.

To investigate the efficacy of distributed compressed sensing (CS) to accelerate free‐breathing, electrocardiogram (ECG)‐triggered noncontrast pulmonary vein (PV) magnetic resonance angiography (MRA).