## 1 Introduction  
Magnetic Resonance Imaging (MRI) exploits the resonant frequency of \({}^{1}\)H protons within water _in vivo_ to generate anatomical images of the human body. Magnetic Resonance Spectroscopic Imaging (MRSI) is a similar imaging technique to MRI; however in lieu of the resonant frequency of \({}^{1}\)H protons in water, the resonant frequencies of \({}^{1}\)H protons in metabolites including, but not limited to, creatine (Cr), choline (Cho), glutamate (Glu), glutamine (Gln), lactate (Lac), aspartate (Asp), and \(N\)-acetyl-aspartate (NAA) are used to generate a metabolic image instead [1,2]. Each of these metabolites has a unique resonance spectrum _in vivo_ caused by their chemical environment and covalent bonding structure that can be used to identify and quantify their concentrations within each voxel [3]. Using this concentration information, the biochemical compositions of healthy and diseased tissues can be determined without the need for invasive biopsies and the altered metabolism of cancers can be detected [4]. The four-dimensional (4D) Echo-Planar Correlated Spectroscopic Imaging (EP-COSI) is an MRSI pulse sequence that allows for the simultaneous acquisition of two spatial and two spectral dimensions, (\(k_{y}\),\(k_{x}\),\(t_{2}\),\(t_{1}\)), in one scan _in vivo_[5]. This 4D sequence provides a 2D spectrum for each voxel in a slice, which differs from other MRSI sequences that only provide a single spectral dimension per spatial voxel [6,7]. The overlap of resonance peaks within a single spectral dimension is a major impediment to identifying individual metabolites and the increased spectral dispersion offered by a second spectral dimension can disentangle complex overlapping spectral peaks [8]. Two spectral dimensions have been shown to increase the specificity and sensitivity of tumor grade classification when used with dynamic contrast enhanced (DCE)-MRI in the breast [9]. However, 4D EP-COSI acquisitions are slow compared with most MRI sequences and can require up to 40 min for a typical scan, which is too long to be used on a routine clinical basis. The 4D EP-COSI data acquisition is a rasterized scan that acquires a 2D spatio-temporal plane, \(k_{x}-t_{2}\), from the 4D volume during each TR. The second spatial (\(k_{y}\)) and spectral (\(t_{1}\)) dimensions are incrementally acquired between rasters until the entire 4D volume is sampled. \(k_{y}\) is acquired using standard phase encoding techniques from MRI and \(t_{1}\) is acquired as a series of phase shifted 1D spectra [5]. The 4D EP-COSI data acquisition can be accelerated _in vivo_ by non-uniformly under-sampling (NUS) the incrementally acquired dimensions, \(k_{y}\) and \(t_{1}\). However, NUS produces artifacts in the spatio-spectral mixed domain that can be removed by reconstructing the missing samples in the \(k_{y}-t_{1}\) plane through non-linear, iterative reconstruction [10,11]. Previous work has accelerated MRI acquisition by under-sampling \(k_{y}\) in k-space and reconstructing the missing samples with Compressed Sensing (CS) reconstruction [12,13]. CS has been applied to under-sampled, spatio-temporal and spatio-spectral mixed domains in dynamic MRI [14] and 3D spatial, 1D spectral MRSI [15]. The feasibility of under-sampling the mixed-domain \(k_{y}-t_{1}\) plane of a 4D MRSI dataset and reconstructing the missing samples with Total Variation (TV) denoising or MaximumEntropy (MaxEnt) reconstruction has been shown previously [16,17]. NUS rates as low as 5\(\times\) were demonstrated _in vivo_ and showed that it is possible to accelerate the acquisition of 4D MRSI data down to a clinically acceptable 5-10 min scan time. The current work derives a solution to the Group Sparse (GS) reconstruction problem for 4D MRSI data within the Split Bregman iterative framework and compares the results with CS, TV, and MaxEnt reconstructions of 4D EP-COSI phantom and _in vivo_ brain data under-sampled at 2\(\times\), 4\(\times\), 6\(\times\), 8\(\times\), and 10\(\times\)[18,19].