Arnold Transform in MATLAB with Inverse Transform and Period Calculation

Arnold transform is a pixel scrambling technique commonly used in image encryption. It achieves encryption effects by altering pixel positions through periodic rearrangement. The core principle lies in the fact that after a certain number of transformations, the image returns to its original state, with the number of transformations required for recovery being called the Arnold period. Arnold Transform Principle The Arnold transform is based on two-dimensional matrix operations that rearrange pixel positions in an N×N image. For a pixel at position (x, y) in the image, the new position (x', y') after transformation is determined by the following formulas: Forward transform: x' = (x + y) mod N y' = (x + 2y) mod N Inverse transform: x = (2x' - y') mod N y = (-x' + y') mod N In MATLAB implementation, the forward transform can be coded using matrix operations with modulo functions to handle boundary conditions, while the inverse transform requires solving the inverse matrix equation to recover original positions. Period Calculation The Arnold transform exhibits periodicity, meaning the image returns to its original state after a certain number of transformations. The period size depends on the image dimension N, with different N values corresponding to different periods. Calculating the Arnold period typically requires iterating through the transformation process until the image matches its initial state. MATLAB implementation approach for period calculation involves creating a loop that applies forward transformations while comparing each result with the original image using matrix comparison functions until identity is achieved. MATLAB Implementation Strategy Forward transform: Iterate through each pixel of the image, recalculate positions according to the transformation formulas, and generate the scrambled image using MATLAB's matrix indexing capabilities. Inverse transform: Utilize the inverse transformation formulas with proper modulo operations to restore the encrypted image to its original form, ensuring correct boundary handling through MATLAB's mod function. Period calculation: Implement a loop that repeatedly applies forward transformations, comparing the current image with the initial image after each iteration using isequal() function, and record the number of iterations when the images match as the period. This technology finds widespread applications in digital watermarking and image encryption fields. Through its periodicity and reversibility properties, it can achieve encryption effects while ensuring the feasibility of decryption and recovery. The MATLAB implementation should include error checking for image dimensions and efficient memory handling for large images.