Point target measurement camera is one of the key parts in celestial navigation technology, which is used to measure star's centroid coordinate to assist celestial navigators such as star trackers to obtain precise positions and attitude information. Point target measurement cameras using short-wave infrared focal plane detectors have the advantages of low power consumption and high detection rate. Due to factors such as manufacturing materials and production processes, infrared focal plane has bad pixels. The on-orbit infrared camera usually compensates for the bad pixels measured by ground-based test equipment. However, because the infrared camera is inevitably affected by the space irradiation environment and high-energy particles when the infrared camera is on-orbit, new bad pixels will appear, which seriously affects the camera's imaging quality and the accuracy of extraction result of star's centroid coordinate. Therefore, it is meaningful and important to study an accurate and reliable bad pixel detection and compensation algorithm for the on-orbit infrared camera. Nowadays, there are many researches on the detection and compensation of bad pixels. But these methods have some shortcomings. Some methods have a large amount of calculation in the detection process, which lead to the bad real-time performance of the on-orbit infrared camera detecting and compensating bad pixels. Other methods have low detection accuracy, because the detection and compensation results are affected by neighboring bad pixels and different background. This paper analyzes the characteristics of the gray value of bad pixels and proposes a real-time bad pixel detection and compensation method based on gray gradient and sparse vision. The method uses a $\mathrm{5}\times \mathrm{5}$ processing window and sparse vision to determine whether the gray gradient of the pixel to be processed conforms to the characteristics of the bad pixel and calculates the compensated gray value. The gray gradient can directly reflect the difference of the gray value between the bad pixel and the normal pixel in neighborhood, so that the detection result can not be affected by the change of the background gray value. In addition, when bad pixel appears in the neighborhood of the pixel to be processed, the gray gradient of the pixel to be processed is affected. Sparse vision can effectively avoid the mutual influence of neighboring bad pixels on the detection and compensation results. Experiments proved that the gray gradient and sparse vision used in our method has significant effectiveness and our method has great performance of detecting and compensating bad pixels. First of all, the bad pixels in the background and star target area in the infrared image can be accurately detected and compensated, which verifies that the detection based on the gray gradient of the pixel to be processed can not be affected by the change of the background gray value. The robustness of detection and compensation of bad pixels is improved when the shortwave infrared camera is on-orbit. Secondly, adjacent bad pixels both can be detected and compensated, which verifies that sparse vision can effectively solve the problem of the mutual influence of neighboring bad pixels. Finally, the experiment results show that the detection accuracy of our method is 100% and the error of the extracted centroid coordinate is reduced by 88.8% to within 0.2% after compensation. In conclusion, our method adopts gray gradient and sparse vision to detect and compensate bad pixels and has great detection success rate and compensation performance, which significantly improves the accuracy of star's centroid coordinates extraction of the on-orbit short-wave infrared camera. Moreover, $\mathrm{5}\times \mathrm{5}$ processing window has low requirements for data storage space and small calculations in the detection and compensation process, which means that the on-orbit short-wave infrared camera can have good real-time performance.