Algorithm Implementation for Converting Images from 2D Cartesian to Polar Coordinate System

<p>Implementing image transformation from Cartesian to polar coordinates in MATLAB is a common image processing task, particularly useful for analyzing circular, annular, or radially symmetric images. The core concept involves converting pixel coordinates (x,y) to polar coordinates (r,θ), where r represents the radial distance from the origin and θ denotes the angle from the x-axis.</p> <p>The transformation process begins by defining the center point of the polar coordinate system, typically selecting the image center as the origin. For each pixel, MATLAB's built-in cart2pol function efficiently computes the radial distance r = sqrt(x² + y²) and angular component θ = atan2(y,x). These computed values form the basis for reconstructing the image in the new polar coordinate grid. The implementation requires careful consideration of coordinate mapping using meshgrid functions to generate coordinate matrices before applying cart2pol.</p> <p>Since the transformed polar coordinates may not align perfectly with the new pixel grid, interpolation methods are essential for estimating pixel values at non-integer positions. MATLAB provides several interpolation algorithms through the imwrap function or custom implementations using interp2: nearest-neighbor interpolation for speed, bilinear interpolation for optimal quality-performance balance, and bicubic interpolation for superior smoothness. The choice depends on specific requirements for image quality versus computational efficiency, with bilinear interpolation serving as the recommended default for most applications.</p> <p>Special attention is required when the transformation extends beyond original image boundaries. Standard practices involve padding these regions with specific colors (e.g., black) or employing extrapolation techniques. Furthermore, due to different scaling characteristics along radial and angular directions, applications may require independent scaling adjustments - for instance, using different resolution parameters for radial (r-axis) and angular (θ-axis) dimensions through appropriate scaling factors in the output image initialization.</p>