<p>The polar wavelet transform (PWT) has been proven to be a powerful mathematical tool for signal and image processing in recent years. However, the PWT has some limitations to fully exploit the intrinsic directional features of high-dimensional signals like images. Focusing on this problem, the polar linear canonical wavelet transform (PLCWT) is proposed in this paper. The main goal of this paper is to define PLCWT and investigate its mathematical properties including the inversion formula as well as derive the convolution and correlation theorems of the PLCWT. Finally, a potential application of the PLCWT in medical edge detection is discussed. The ability to detect subtle or weak edges in medical images is the key for accurate early detection and diagnoses. Our numerical experiment demonstrates that the proposed transform PLCWT successfully detects subtle retinal blood vessels of an eye image that were overlooked by both the traditional Canny algorithm and the PWT.</p>