# Motion Deblurring for Single-Pixel Spatial Frequency Domain Imaging
## 1 Introduction  
Spatial frequency domain imaging (SFDI) works by projecting the sinusoidal structured light onto the tissue and measuring the light reflected from the tissue surface [1, 2]. It enables fast, non-invasive and wide field mapping of the tissue optical properties, and has been applied to many clinical applications [3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Recently, since the single-pixel imaging (SPI) technique has brought significant performance advantages in band extension and sensitivity enhancement [13, 14], single-pixel SFDI is receiving extensive attention [15, 16, 17, 18, 19, 20].  
Despite its combined merits of both SPI and SFDI, single-pixel SFDI has been restricted to low frame rates because it requires hundreds of measurements to obtain a single image. This adversity can cause severe quality degradations of the measured reflected images when imaging a moving target, such as scenarios of the in vivo imaging of animals in motion or clinical interrogation of patients for breathing and heartbeat, and accordingly reduces the accuracy of the extracted optical properties. Despite the many techniques proposed for motion deblurring in the image restoration regime, they fail in direct translation to single-pixel SFDI because of its essentially different imaging mechanism from the traditional photography.  
Many works have been devoted to improving the dynamic performance of SPI. For instance, the high-speed illumination module [21] and digital micromirror device (DMD) [22, 23] were exploited to achieve a high frame rate. However, these dynamically enhancing methods normally require high illuminating power to ensure a reasonable signal-to-noise ratio (SNR) at a high refresh rate, which, therefore, can cause harm to tissues when applied to single-pixel SFDI. Other methods obtain the high-quality image by successively transforming the illumination patterns according to the target locations or superimposing a sequence of under-sample images, with each image shifted with the targetdisplacement [24; 25; 26; 27; 28; 29; 30]. The motion information required in these methods is estimated by the optimization algorithms or using the cross-correlation or the low-order moments of the under-sample images. However, these methods require the image to possess a uniform background because the target motion is regarded as the translation of the entire image. This requirement prevents these methods from solving the motion blurring in single-pixel SFDI, as the images in single-pixel SFDI have the sinusoidal fringes in the background that remain stationary when the target moves.  
In this work, we present a novel method of motion deblurring for single-pixel SFDI. The method adopts a Fourier single-pixel imaging (FSI) technique to acquire the Fourier coefficients of the reflected image. On this basis, a motion-degradation-model-based compensation, which is derived by the phase-shift and frequency-shift properties of Fourier transform, is adopted to eliminate the effects of target displacements on the measurements. Thereafter, the deblurred image is reconstructed from the processed data. The target displacements required in the method are obtained using a fast motion estimation approach. The novelty of this work is to derive the motion degradation model for single-pixel SFDI that describes the effects of target displacements on both the planar (DC) and spatially modulation (AC) components of the image. The compensation of the measurements based on this model can deblur the target while retaining the background sinusoidal fringes. The method enables the fast deblurring of targets with varying speeds and directions. We performed a series of numerical simulations and phantom experiments to validate the proposed method. The numerical results show that the method can dynamically locate and deblur the target in the reflected image, and the phantom experiments further demonstrate the method to significantly improve the accuracy of the subsequently extracted optical properties.