Computation of 1D and 2D Convolution

Convolution is a fundamental operation in signal and image processing, where 1D convolution is commonly used for signal analysis, while 2D convolution finds extensive applications in image processing. Manual implementation of convolution calculations eliminates dependency on MATLAB and facilitates portability to C-language environments such as embedded systems.

The core of 1D convolution involves sliding-window dot product accumulation. Given an input signal and a convolution kernel, each element of the output signal equals the sum of products between the kernel and corresponding positions of the input signal. Boundary handling typically employs zero-padding. Implementation requires attention to kernel flipping or direct cross-correlation computation (in which case the kernel must be pre-flipped during storage). Code implementation would involve nested loops for index management and element-wise multiplication with accumulation.

The essence of 2D convolution implementation lies in double-nested loops. For image processing scenarios, each output pixel is obtained by element-wise multiplication between the convolution kernel and local image patches followed by summation. To enhance computational efficiency, techniques like image tiling or row-column separation (for separable kernels) can be pre-implemented. Memory access sequence optimization significantly impacts performance, particularly when porting to C where cache-friendly data structures are crucial. Algorithm optimization may involve loop unrolling and spatial locality enhancement through contiguous memory access patterns.

Compared to MATLAB's built-in functions, self-implemented convolution requires no installation of large mathematical software and offers greater flexibility through adjustable loop structures and boundary conditions for specific scenarios. For C language porting, fixed-size kernel configuration enables compiler optimization, while SIMD instructions can accelerate computation-intensive sections through parallel processing of multiple data elements.