Gradient histogram: Thresholding in a region of interest for edge detection Debashis Sen * , Sankar K. Pal Center for Soft Computing Research, Indian Statistical Institute, 203 B.T. Road, Kolkata, West Bengal 700108, India article info Article history: Received 28 September 2007 Received in revised form 1 October 2009 Accepted 26 October 2009 Keywords: Edge detection Gradient operators Random process Gradient histogram Skewness and kurtosis Threshold determination Non-maximum suppression Hysteresis thresholding abstract Selecting a threshold from the gradient histogram, a histogram of gradient magnitudes, of an image plays a crucial role in a gradient based edge detection system. This paper presents a methodology to determine the threshold from a gradient histogram generated using any kind of linear gradient operator on an image. We consider the image as a random process with dependent samples, model the gradient histo- gram using theories of random process and random input to a system, and determine a region of interest in the gradient histogram using certain properties of a probability density function. Standard histogram thresholding techniques are then used within the region of interest to get the threshold value. To obtain the edges, this threshold value is then used as the upper threshold of the hysteresis thresholding tech- nique that follows the non-maximum suppression operation applied on the gradient magnitude image. The proposed methodology of determining a threshold in a gradient histogram is deduced through rigor- ous analysis and hence it helps in achieving consistently appreciable edge detection performance. Exper- imental results using different real-life and benchmark images are shown to demonstrate the effectiveness of the proposed technique. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction Edge detection is a very important low-level image processing operation, which is essential in order to carry out various higher level tasks such as motion and feature analysis, understanding, recognition and retrieval from databases. Over four decades of research work on edge detection have resulted in the development of a plethora of simple to complex techniques. However, as stated in [1–6], the problem of finding the edges has not been solved in its entirety and no universally accepted technique exists. Considering the intensity of an image as a two dimensional function, it is generally accepted that the edges in the image are ‘‘meaningful” discontinuities (changes) of this function [6–8]. ‘‘Meaningful” discontinuities refer to the edges due to various re- gions in an intensity image, whereas, the ‘‘non-meaningful” dis- continuities are those due to inherent texture and various noise. It may be noted that we shall henceforth refer ‘intensity image’ as ‘image’ for simplicity. The problem of edge detection has been tackled using various different paradigms such as surface fitting [9,10], optimization of a criterion [11–18], statistical testing [19,20] and soft computing [21,22]. Among the various methods in all the paradigms, the pop- ular are the ones based on finding the ‘‘amount” of edge at each pixel of the image. Authors in [11,15,23] have, respectively, sug- gested the use of first-, second- and fourth-order derivatives to ob- tain the ‘‘amount” of edge at a pixel location. As the use of higher- order derivatives makes the system susceptible to high frequency noise and also results in poorer localization of the edges [6,24], we shall consider the first-order derivative, which is used to calcu- late the gradient at each pixel in the image. Authors in [11–14] have optimized various criteria in order to find operators for computing the gradient, which represents the edge at every pixel in the image. It is interesting to note that all these gradient operators may be considered as a smoothing system followed by the first-order derivative operation. In addition to an optimal gradient operator, Canny [11,25] proposed a technique called the non-maximum suppression (NMS), where the fact that an edge at a pixel is legitimate only when the gradient magnitude at that pixel assumes a maximum in the direction of the gradient. He suggested that the NMS be used as a post-processing operation along with the gradient operator for edge detection in order to ob- tain edges of single pixel width. Most of the optimal gradient operators reported in the litera- ture are based on the three performance criteria, namely, good detection, good localization and only one response to a single edge, given by Canny [11]. As discussed in [11,13], the optimizations of these three criteria are contradictory and hence a trade-off is required. Such a trade-off implies that the smoothing operation cannot remove the ‘‘false” edges due to the ‘‘non-meaningful” dis- continuities completely. Hence, after all the above mentioned pro- cesses of smoothing, first-order derivative and NMS, we still are at a stage where we need a decision making system in order to distin- 0262-8856/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.imavis.2009.10.010 * Corresponding author. Tel.: +91 33 25752048. E-mail addresses: dsen_t@isical.ac.in (D. Sen), sankar@isical.ac.in (S.K. Pal). Image and Vision Computing 28 (2010) 677–695 Contents lists available at ScienceDirect Image and Vision Computing journal homepage: www.elsevier.com/locate/imavis