Hardware-Based Sobel Gradient Computations for Sharpness Enhancement

The majority of imaging systems are software based; they require some kind of microprocessor or microcontroller for the imaging algorithms to run. As the speed requirements of imaging and communications systems increase, the need for more hardware-based imaging systems arises. These fully hardware systems solve the fundamental problem inherent in software-based solutions, in which the speed of the algorithms depend on the instruction cycle speed of the processor. Once an algorithm is designed directly on hardware, the speed of the algorithm depends on the system clock frequency and the propagation delays of the logic cells (or standard cells) used in the design, usually measured in nanoseconds per cell. Therefore, such systems no longer depend on any instruction cycle delays, as there is no microprocessor involved. Most modern imaging and communications systems rely on digital signal processing (DSP) to compute complex mathematical operations. The emergence of powerful and low-cost field-programmable gate array (FPGA) devices with hundreds of arithmetic multipliers has enabled the development of many such DSP hardware applications, traditionally implemented only as software solutions.

Digital signal processing; Edge detection; Gradient; Sobel; VHDL

Lately, there have been several texts (Li & Chu, 1997; Nelson, 2000; Yasri et al., 2009; Mehra & Verma, 2012; Nosrat & Kavian, 2012; Sanduja & Patial, 2012; Singh et al., 2012; Umar et al., 2012; Bhagat et al., 2015) written on hardware-based Sobel implementations on FPGAs using VHDL (Ashenden, 2008) or Verilog. However, nearly all of these advocate the use of calculating the gradient magnitude by obtaining the sum of the absolute values of the gradient in both the horizontal and vertical directions. Implementing gross approximations of many such nonlinear imaging algorithms (Arce et al., 2000; Aubert & Kornprobst, 2006; Bertalm?o et al., 2001; Chambolle, 1994; Kokkinos, 2013; Kornprobst et al., 1999; Mitra & Sicuranza, 2001; Xu & Mueller, 2010) on hardware has become common practice.Although this approach simplifies the hardware implementation by avoiding the more computationally intensive square root calculations, the resulting gradient magnitude suffers from having more errors than a gradient magnitude calculated using the Pythagorean theorem of square-rooting the sum of squares of the gradients in each horizontal and vertical direction.

Before other algorithms are performed, usually, an image filter is applied. This preprocessing filter helps ease the computation of further downstream algorithms, such as those used in optical character recognition systems (Pangestu et al., 2017), or the K-NN algorithms (Naik & Metkewar, 2015) used in artificial intelligence. Either a spatial filter, such as a Sobel edge detector, or a histogram equalizer frequency domain filter may be used as the prefilter, depending on the type of further processing required.

This paper introduces a computationally efficient technique of preserving the precision of the gradient magnitude by using an efficient and fast square root algorithm in the computation of the gradient magnitude. Although we also introduce a different kernel processing scheme that computes kernels in parallel, this paper focuses its discussion on the use of the fast reciprocal square root (FRSR) algorithm for hardware-based Sobel edge detection. 

The Sobel algorithm, used frequently in many edge detection algorithms, has been shown to be feasibly implemented on digital hardware. However, the gradient magnitude of these implementations used the summation of the absolute values of the g_x  and g_x gradients as its estimate , whereas in our implementation, we used the actual square root operator to compute the gradient magnitude. Using the FRSR algorithm gives a more accurate estimate of the gradient magnitude as computed from the square root of the square of the gradients in both the horizontal and vertical directions , compared with using the summation of the absolute values of the gradients.

The authors are thankful to the Ministry of Higher Education of Malaysia for the award of the Fundamental Research Grant Scheme FRGS/1/2015/TK04/MMU/02/10 to support this project. We are also thankful to Jeannie Lau and Ang Boon Chong for the many technical discussions that helped us complete this project.

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