Guest Editorial Compressive Sensing for Biomedical Imaging

W HILE it is common knowledge that most images can be greatly compressed, compressive sensing (CS) theory has established that such compression can be done during the data acquisition process and then the uncompressed image can be recovered through a computationally tractable optimization procedure such as L1-norm minimization [1]–[4], or greedy algorithms [5]. In the field of biomedical imaging, CS is exciting for several reasons. First, it allows accurate recovery of an image from far fewer measurements than the number of unknowns, and does not require close match between the sampling pattern and characteristic image structures. Also, CS techniques can often efficiently deliver practical results in terms of increased imaging speed, reduced radiation dose, enhanced image quality, and other benefits. Given its transformative potential for system design, algorithm development, and preclinical and clinical applications, CS has recently become a hot topic in biomedical imaging [6]. Some of the key ideas and methods of CS were introduced and applied to biomedical imaging well before the advent of modern CS theory. While not an imaging method per se, an early observation of the ability of regularization to produce sparse signals was in geophysics [7]. Early sparsity-based imaging methods employed variations on reweighted least-squares [8] to solve the associated nonlinear recovery problem. Theoretical results and algorithms for spectrum-blind sampling in imaging [9]–[11] introduced the possibility of perfect image recovery from highly undersampled Fourier data subject to object sparsity. Combining random sampling and an efficient nonlinear sparse reconstruction algorithm led to a compressive sensing method for Fourier imaging, including MRI and tomography [12]. As shown in Fig. 1, CS has seen impressive successes and fast growth over the past ten years, including applications in medical imaging. Applications of CS to MRI have been the earliest, most numerous, and most diverse, owing to the tremendous flexibility in designing the acquisition process and the pressing need that MRI has, as a slow acquisition modality, to reduce the sampling requirements. Representative works, classified by subtopic/application include the following: static MRI [13], dynamic MRI [14]–[16], parallel MRI [17]–[19], MR spectroscopic and parametric imaging [20]–[22], diffusion tensor imaging [23], flow imaging [24], optimized acquisition [25]–[27], CS reconstruction algorithms [28]–[30], and image quality evaluation [31], [32]. Optical imaging modalities rank next in level of activity, covering bioluminescence [33], [34], optoacoustics [35], [36], fluorescence molecular tomography [37], [38], and diffusion optical tomography [39]–[41]. CT is drawing an increasing number of CS applications [42]–[45], while CS in ultrasound [46]–[48] is only beginning to be