# Efficient Tensor Sensing for RF Tomographic Imaging on GPUs
## 1 Introduction  
Radio frequency (RF) tomographic imaging, also called wireless tomography, is a technique for detecting objects within a specific region by analyzing radio frequency signals transmitted between wireless nodes, as shown in Figure 1. This technique uses radio reflections that bounce off the object body and does not require the object to carry any wireless device. This advantage has made it ideal for security checking, rescue operations, space applications, and smart buildings [1]. In general, wireless signal propagating on a link loses power due to shadowing, distance, and multipath fading. The task of RF tomographic imaging is to estimate a spatial distribution of the shadowing loss in a detecting region from the measured power of wireless signals. Classic radio tomography and vector-based methods have been proposed. The work in [2] presented a linear model for using received signal strength (RSS) measurements to obtain images of moving objects. The work in [3] addressed substantial issues for the practical implementation. A tensor-based compressed sensing method [1] was proposed for RF tomographic imaging in a three and higher dimensionally-structured region, which estimates the distribution by utilizing its low-rank property. However, the method requires computing tensor singular value decomposition (t-SVD) in each algorithmic iteration, which leads to high computational complexity. To address this problem, [PERSON] et al. [4] proposed to use the transform-based tensor model to formulate the RF tomographicimaging as a tensor sensing problem, then used a fast iterative algorithm Alt-Min to solve the tensor sensing problem. Their method fully utilizes the geometric structure of the three-dimensional loss field tensor. Compared to the tensor-based compressed sensing method, [PERSON]'s method achieved a lower error rate and faster computation speed. However, the iterative algorithm Alt-Min proposed by [PERSON] et al. [4] iteratively estimates a pair of tensors, which is a computationally-intensive process, thus not being very practical for RF tomographic imaging for big objects or regions, or real-time applications. In this paper, we address this problem by exploiting graphics processing units (GPUs). Because of massive hardware parallelism and high memory bandwidth, GPUs have been widely used in diverse applications including machine learning [5; 6; 7], graph processing [8; 9; 10], big data analytics [11; 12], image processing [13], and fluid dynamics [14]. In order to reap the power of GPUs, the algorithmic steps need to be mapped delicately onto the architecture of GPUs, especially the thread and memory hierarchy. In this work, we design, implement, and optimize the transform model-based tensor sensing method of [PERSON] et al. [4] on an NVIDIA Tesla V100 GPU with CUDA (Compute Unified Device Architecture) and evaluate it in terms of the running time and recovery error. Experiment results show that the GPU algorithm achieves similar relative error as the CPU counterpart. Moreover, the GPU tensor sensing outperforms the CPU tensor sensing on all tensor sizes with 45.39\(\times\) speedup on average and up to 84.70\(\times\) speedup on bigger tensor sizes such as 120 \(\times\) 120 \(\times\) 6. We encapsulate this GPU tensor sensing algorithm into an open-source library called \"cuTensorSensing\" (CUDA Tensor Sensing), which is available at [15]. Our contributions are summarized as follows. First, we analyze the steps of the transform model-based tensor sensing using the Alt-Min algorithm and discuss how to map them onto the GPU architecture. Second, we design, implement, and optimize the tensor sensing on an NVIDIA Tesla V100 GPU. Third, we evaluate and compare the GPU tensor sensing and the CPU tensor sensing in terms of running time and relative error with both synthetic data and IKEA Model data. Fourth, we encapsulate the GPU tensor sensing implementation into an open-source library such that it can be used in diverse applications. The remainder of the paper is organized as follows.