A Low-Error, Memory-Based Fast Binary Logarithmic Converter

Abstract

This brief presents a memory based fast binary logarithmic converter based on a piecewise linear approximation technique. The proposed method is simple and arithmetic operation-less, which achieves 10−4 to 10−3 maximum absolute error (MAE) while maintaining a high speed. The approach partitions the logarithmic curve of the fractional component into $2^{L}$ uniform regions and a block RAM (size $2^{L} \times $ bits) stores the approximate value of each sub-region. For any number, most significant $L$ bits of the fractional component address the memory location of the logarithmic converter. The hardware synthesis result, implemented with 26 bits fractional precision on Virtex-6 field-programmable gate array device, shows 75% improvement in MAE and 27% decrease in critical path delay compared to the current state-of-the-art techniques in the worst-case scenario. In Otsu’s image thresholding algorithm, the proposed logarithmic converter with 3.08 ( $=2^{8} \times 12$ ) kbits memory size adequately meets the accuracy requirement for improved image segmentation.