Gaussian Filtering in One Dimension

Gaussian filtering in one, two, or three dimensions is one of the most fundamental operations in signal and image processing. Implementing finite impulse response (FIR) filters in the time domain with Gaussian masks is straightforward in both floating-point and fixed-point arithmetic, as Gaussian kernels are inherently strictly positive and bounded. However, these conventional convolution-based implementations often suffer from computational inefficiency when processing large images or using large kernel sizes. To address this limitation, at least two accelerated approaches exist: recursive infinite impulse response (IIR) filters and Fast Fourier Transform (FFT)-based methods. These advanced techniques, while significantly faster, typically require floating-point hardware support for practical implementation.

This paper focuses on developing a fixed-point arithmetic implementation of recursive Gaussian filtering. The methodology employs a non-orthogonal separation scheme for the Gaussian filter kernel, enabling efficient computation while maintaining precision in fixed-point systems. The approach is comprehensively analyzed and successfully applied to both isotropic and anisotropic image filtering scenarios, demonstrating particular implementation advantages in resource-constrained environments where floating-point operations are limited.