## 9 Multiplexed PET and Variable-Density MRI Sampling with Joint Hankel-Based Reconstruction  
Efficient data acquisition in MR-PET imaging critically hinges on the exploitation of sparse sampling strategies that minimize measurement redundancy while preserving essential anatomical and functional information [8; 27; 29]. The proposed framework leverages structured sparse sampling designs tailored to the joint nature of MR and PET data, thus enabling accelerated acquisition and reduced hardware complexity without compromising reconstruction fidelity. In the PET domain, a novel multiplexed acquisition scheme has been implemented [55], where signals from multiple detector elements are combined through a carefully de -signed sensing matrix \(C_{\text{PET}}\). The multiplexed measurements \(y^{\text{PET}}\) can be mathematically expressed as follows:  
\[y_{n}^{\text{PET}}=\sum_{k=1}^{K}c_{k,n}\,s_{k}^{\text{PET}}+\eta_{n}^{\text{ PET}}, \tag{15}\]  
where \(s_{k}^{\text{PET}}\) denotes the true signal from the \(k\)-th detector, \(c_{k,n}\) are elements of the sensing matrix representing the multiplexing weights, and \(\eta_{n}^{\text{PET}}\) models acquisition noise. The sensing matrix \(C_{\text{PET}}\) is constructed leveraging structured random matrices such as subsampled Toeplitz and Hankel matrices [31], which have been theoretically proven to satisfy the Restricted Isometry Property (RIP) with high probability [56], thereby ensuring stable and accurate sparse recovery. In this formulation, \(y_{n}^{\text{PET}}\) represents the \(n\)-th observed PET measurement resulting from multiplexed acquisition. The variable \(s_{k}^{\text{PET}}\) corresponds to the true underlying signal generated by the \(k\)-th PET detector channel. The coefficients \(c_{k,n}\) are entries of the sensing matrix \(C_{\text{PET}}\), which controls how detector signals are linearly combined during acquisition. This multiplexing process allows for a reduction in the number of acquired measurements while preserving critical information. The noise term \(\eta_{n}^{\text{PET}}\) models random acquisition noise, typically assumed to follow a Gaussian distribution. The sensing matrix \(C_{\text{PET}}\) is carefully designed using structured random matrices--such as subsampled Toeplitz and Hankel matrices [31]--which exhibit favorable properties like the Restricted Isometry Property (RIP) [56], ensuring that the compressed measurements still enable stable and accurate recovery of the underlying PET image through sparse reconstruction algorithms. In parallel, the MR acquisition strategy employs a variable density Poisson-disc sampling pattern in k-space [35], encoded by a sampling matrix \(M_{\text{MRI}}\), which is modeled as a block Toeplitz operator acting on the Fourier domain representation of the image:  
\[y^{\text{MRI}}=M_{\text{MRI}}F(x^{\text{MRI}})+\eta^{\text{MRI}}, \tag{16}\]  
where \(F\) denotes the Fourier transform, and \(\eta^{\text{MRI}}\) accounts for measurement noise. In this equation, \(y^{\text{MRI}}\) denotes the acquired k-space measurements in the MRI acquisition process. The variable \(x^{\text{MRI}}\) represents the underlying spatial-domain MRI image to be reconstructed. The operator \(F\) refers to the discrete Fourier transform, which maps the spatial-domain image into the frequency (k-space) domain, as performed during MRI signal acquisition. The sampling matrix \(M_{\text{MRI}}\) is a binary mask that encodes the variable-density sparse sampling pattern applied in k-space. Specifically, \(M_{\text{MRI}}\) selectively retains a subset of Fourier coefficients, enabling acceleration of the acquisition process by reducing the number of required measurements. The term \(\eta^{\text{MRI}}\) models additive measurement noise present in the acquired k-space data, typically assumed to follow a Gaussian distribution.