Implementation of Logarithmic Polar Coordinate Transformation for Images

This text presents the implementation of logarithmic polar coordinate transformation for images. This transformation possesses rotational invariance, meaning the transformed image maintains identical features regardless of how the original image is rotated. This characteristic proves particularly valuable in digital watermarking applications, as it provides effective resistance against geometric attacks. By applying logarithmic polar coordinate transformation, we can significantly enhance the robustness of digital watermarks against malicious destruction or tampering by attackers. From an implementation perspective, the transformation typically involves these key steps: First, convert Cartesian coordinates (x,y) to polar coordinates (r,θ) using mathematical relations r = √(x²+y²) and θ = arctan(y/x). Then apply logarithmic scaling to the radial coordinate using r' = log(r). Critical functions in implementation include coordinate mapping, interpolation methods (like bilinear or bicubic interpolation for pixel value calculation), and handling boundary conditions. The transformation can be implemented using image processing libraries like OpenCV with functions such as cv2.logPolar(), which requires parameters for the transformation center and output image size. This transformation plays a crucial role in preserving the integrity and security of digital content by making watermarks resilient to rotational manipulations and scaling attacks.