# Photoacoustic Imaging with a Finite-Size Circular Integrating Detector
## 1 Introduction  
Photoacoustic imaging (PAI) is an emerging biomedical imaging technology that combines the high contrast of optical imaging with the high resolution of ultrasound imaging. PAI uses short-pulse laser excitation on biological tissues. Optically absorbing regions undergo adiabatic expansion, generating ultrasound waves. This phenomenon is known as the photoacoustic effect [1, 2]. By detecting the resulting photoacoustic signals, the spatial distribution of optical absorption in biological tissues can be reconstructed. Recent advances in PAI have significantly propelled its clinical translation. PAI enables microvascular oxygenation visualization [3, 4] and 3D vasculature reconstruction [5], noninvasive hemodynamic tracking in brain studies [6], and high-resolution cortical mapping [7]. In oncology, PAI is used to monitor tumor neovascularization dynamics [8], and multimodal breast scanners [9], along with molecular characterization [10], improve cancer diagnoses. Sentinel lymph node mapping techniques [11, 12, 13, 14] have further revolutionized surgical navigation. Targeted gold nanorods support theranostics for submillimeter bladder cancer [15], while diversified nanomaterials expand functional imaging capabilities [16]. Despite persistent challenges in in vivo targeting efficiency [17], PAI's clinical value continues to grow, demonstrating broad potential across medical disciplines. Ultrasonic detector arrays can be configured in various geometries as a critical component of PAI systems. Circular integrating detectors are compact and symmetric. Thisdesign helps to capture accurate structural and functional information about tissues. This ability is particularly valuable in the early diagnosis and precise treatment of breast cancer and in cardiovascular disease assessments [18; 19]. However, applying existing image reconstruction algorithms [20; 21] that are designed for ideal point detectors to finite-sized circular integrating detectors degrades image reconstruction quality [22]. To mitigate the blurring effects introduced by circular integrating detectors, researchers have proposed the concept of the spatial impulse response (SIR) and developed algorithms to correct the spatial blurring caused by the detectors. Based on a linear and discrete PAI imaging model, [PERSON] et al. designed a spatiotemporal optimal filter using the minimum mean square error criterion to correct SIR effects caused by defocused regions, thereby improving tangential resolution [23]. [PERSON] et al. included SIR in their imaging model. They focused on far-field imaging to improve 3D tomography for small animals [24]. Although incorporating SIR into the imaging model enhances imaging accuracy, it also increases computational complexity. To address this issue, [PERSON] et al. proposed a sensor-based SIR back-projection (BP) method to quickly correct the limitations caused by the detector size, thereby accelerating image reconstruction [25]. The above approach addresses the issue in the data domain before image reconstruction. In the image domain after reconstruction, blind deconvolution methods have become an alternative strategy for handling PAI image blurring. For example, [PERSON] and [PERSON] employed a blind deconvolution method to process the reconstructed data to restore PAI images [26]. They employed iterative deconvolution and used focal measurements from a photoacoustic microscope as the initial PSF. They combined their method with Lucy-Richardson (LR) deconvolution. Their method significantly enhanced the vascular imaging resolution. [PERSON] et al. developed a spatial variant PSF-based PAI image restoration method. PSF was measured experimentally and estimated with pixel-level precision using multi-Gaussian fitting and interpolation algorithms. Their optimized iterative restoration model substantially enhanced image quality and resolution [27]. Additionally, [PERSON] and colleagues introduced the concept of spatial angular convolution to describe \"spin blur\".