A statistical framework based on a family of full range autoregressive models for edge extraction

Abstract

In this paper, a novel technique is proposed based on a Family of Full Range Autoregressive (FRAR) models to extract edges in 2D monochrome images. The model parameters are estimated based on Bayesian approach and is used to smooth the input images. At each pixel location, residual value is calculated by differentiating the original image and its smoothed version. Edge magnitudes and its directions are measured based on the residual. The edge magnitudes are squared to enhance the edges whereas the other values are suppressed by using confidence limit is based on the global descriptive statistics. Threshold value is fixed automatically based on the autocorrelation value calculated on the smoothed image. This extracts the thick edges. To obtain thin and continuous edges, the nonmaxima suppression algorithm is applied with the confidence limit based on the local descriptive statistics. Then the performance of the proposed technique is compared with that of the existing standard algorithms including Canny’s algorithm. Since Canny’s algorithm oversmoothes across the edges, it detects the spurious and weak edges. This problem is overcome in the proposed technique because it smoothes minimally across the edges. The extracted edge map is superimposed on its original image to justify that the proposed technique is locally characterize the edges correctly. Also, the proposed technique is experimented on synthetic images such as concentric circle and square images to prove that it detects the edges in all directions and edge junctions.

Introduction

Edge detection is a fundamental step in most of the applications of image analysis. It constitutes a crucial initial step before performing high-level tasks such as object recognition, object classification, pattern matching, image segmentation, image compression and boundary detection in computer vision applications. The edges are broadly classified as roof edges, step edges and ridge edges, etc. An edge is characterized by an abrupt change in the gray value of an image. A pixel that represents an edge is mostly decided only on the basis of local characteristics viz. neighbourhood pixel values.

In the literature, a lot of reviews on edge detection are reported and a large number of algorithms are proposed for edge extraction in the case of gray level images. The edge detection methods are broadly classified into enhancement/threshold type, gradient based operators, edge-fitting edge detection, zero-crossings in second derivatives, optimality criteria, residual analysis based technique, etc. The enhancement/threshold type is a well known one due to its simplicity and low computational complexity. This method is based on gradient values of the image under analysis. The amplitude response of the image is obtained by taking either root mean square or the sum of the absolute gradient values of the image along the two spatial coordinates. An edge is assumed to be present if a single criterion like maximization of amplitude response to the image is satisfied. The gradient values of different orders are calculated by convolving the various gradient operators. The gradient based operators such as Roberts, 1965, Prewitt, 1970, Sobel, 1970 operators are well known and are widely used to find the step edges than the roof and ridge edges. All these operators are first order derivatives. Hueckel (1971) proposed a method, based on edge-fitting edge detection and then it was simplified later by Rosenfield (1981), which is not widely used in computer vision applications when compared to the gradient based operators. The Laplacian, a zero-crossing operator, is a second order derivative, and is used to establish the location of edges present in the image. This operator is not used with its original form for edge detection. Instead it is used with the Gaussian function and is called Laplacian of Gaussian (LoG) function (Marr and Hildreth, 1980). Krishnamoorthi and Bhattacharya (1998) used curve fitting model based on zero-crossing in the second derivatives and claimed that their technique is superior to the vector order statistics and entropy schemes. Edge detectors based on optimality criteria are also reported by many authors. For example, Canny’s operator (Canny, 1986) is very popular among all the edge detectors that uses Gaussian filter for smoothing images before extracting the edges. The noise can be reduced, by smoothing the images and then the actual edges are identified. Though, the noise in the images is reduced by smoothing, the Canny’s algorithm captures the spurious edges (Rakesh et al., 2004). Statistical approaches are also adopted to detect the edges (Qie and Bhandarkar, 1996, Stern and Kurz, 1988, Li, 2000, Rakesh et al., 2004) as they give better results for the images with noise and without noise (Rakesh et al., 2004). The residual analysis based approach is studied by many authors (Chen et al., 1991, Lee et al., 1988, Pavlidis and Lee, 1988). For instance, Chen et al. (1991) considered the distribution of residuals, the difference between the original image and its smoothed version. Then they find autocorrelation for the residual by which they extract the features. Zheng et al. (2004) proposed a hybrid edge detector with the combination of gradient and zero-crossing based on Least Square Support Vector Machine (LS-SVM) with Gaussian filter. It reports that it takes lesser time than the Canny’s detector with similar performance on edge extraction. In the earlier works, the threshold is chosen on heuristic basis. Even in the Canny’s edge detector the default value of upper limit is suggested to be 75th percentile of the gradient strength and also it requires at least two threshold values namely, lowest and highest threshold. Though Rakesh et al. (2004) reported that this problem was overcome in their work, their algorithm needs initial threshold value and parameters as input.

Kim et al. (2004) proposed a procedure to determine the edge magnitude and direction and found the 3 × 3 ideal binary pattern for a pixel as follows. The average value is calculated for the pixels in a 3 × 3 block, which is compared to each pixel in the block. If the pixel is greater than the average value then it is marked as 1. Otherwise it is marked as 0. Then, they proposed fixed weights such as 1, 2, 4, 8, 16, 32, 64 and 128 to the eight-neigbouring pixels depending on their position related to the centre pixel and they determine 8-bit code by summing the weights of the pixels marked as 1. This is not justifiable because the pixel values in a 3 × 3 block generally influence the centre pixel more or less equally. In their approach, they use weight 1 to the pixel in the location (1, 1) and 128 to the pixel in the location (3, 3). This shows the great difference between the influences of the two pixels on its centre pixel. Generally, this is not true since both pixels are in opposite direction and closest neighbouring to the centre pixel, so the influences of those pixels on the centre is more or less equal. With this point of view, giving fixed weights may lead to wrong decision on finding the directionality of edge. This problem is overcome in our proposed scheme and the output of our proposed scheme is compared with that of the other existing techniques such as Zheng et al. and the traditional Prewitt, Sobel, and Canny’s detectors.

Stochastic model based methods, such as, Hidden Markov Random field (Chen and Kundu, 1995, Romberg et al., 2001), Markov Random field (Sarkar et al., 2000, Elia et al., 2003), Autoregressive (Kadaba et al., 1998, Krishnamoorthi and Seetharaman, 2007), Multiresolution Guassian Autoregressive (Comer and Delp, 1999) and Gibbs field (Chalmond, 1989, Geman and Geman, 1984) models have attracted many researchers on image processing and computer vision applications such as pattern recognition, object recognition, feature analysis, edge detection, image classification, segmentation, etc. Markov Random field models have been used by many researchers (Bouman and Sauer, 1993, Li, 2000) to capture edges. These models are adopted to impose the smoothness on the image surface function.

Autoregressive model utilizes the linear dependency of the pixels in an image to estimate its surface. This model, also, takes advantage of the spatial interaction of pixels in local neighbourhood. To estimate the gray tone of a pixel in an image region, it needs the conditional probability density function of that pixel given the gray tone of the neighbourhood pixels in that region. That is

$$P\left\{\left(\text{gray tone of the pixel to be estimated}\right)/\left(\text{gray tone of the neighbourhood pixels in the image region}\right)\right\}$$

Generally, a model with statistical properties, which describes the probability structure of a time series and in general any sequence of observations, is called stochastic process. The image to be analysed can be thought of as one particular realization, produced by the underlying probability mechanism of the image under study. That is, in analysing an image we regard it as a realization of a stochastic process.

Definition

If a stochastic process, that is, a family of time dependent random variables {X(t)} satisfies

$$E(X(t)/X(t-j)\text{;}j=1\text{,}2\text{,}\dots )=E(X(t)/X(t-1)\text{,}\dots \text{,}X(t-p))$$

then {X(t)} is said to satisfy the Markov property.

In the above equation, on the left hand side (LHS) the expectation is conditional on the infinite history of X(t). On the right hand side (RHS) it is conditional only on the part of the history. From the definition, an AR(p) model is seen to satisfy the Markov property. In that sense, time series models, Markov Random models and Stochastic Process, all are interrelated and are associated with each other in the context of image processing. In the next section, the proposed, a Family of Full Range Autoregressive (FRAR) models is introduced.

Most of the aforesaid techniques capture spurious and minor edges, and failed to extract the edge junction, that is, meeting point of the edges. It is observed from the literature that the Gaussian derivative is used in most existing techniques to estimate the smoothed surface of the input images. Generally, Gaussian derivative oversmoothes across the edges in the images. This is the main reason to capture the spurious and minor edges, and missed to extract the edge junction. Another important point is that several of the existing techniques require initial input parameters or threshold values. Mainly these two concepts motivated us to carry out this study. In the proposed technique, the input parameters or threshold values are not required because it automatically computes the threshold or parameters values according to the nature of the image data, which is discussed in detail in Sections 5 Edge magnitude and direction, 8 Results and discussion. The main advantage of the proposed technique is that it extracts fine and correct edges, and it captures the edge junction.

The purpose of this paper is to provide automatic threshold and to maintain high performance for edge extraction. Initially a technique, based on FRAR model is proposed to smooth the original input image. Here, the smooth means, slow change of the gray level which have abrupt change in the original image. The input image is modelled as a Gaussian Makov Random Field (GMRF), viz., assuming that each given pixel X(s) depends statistically on the rest of the image only through a selected group of neighbourhood pixels X_n(s) i.e., P(X(s)/X_n(s)). In fact, this assumption reveals that the group of pixels satisfies the linear dependency criteria. Generally, with this assumption, the MRF model captures the features like edges and boundaries of the complicated images to some extent. In our method, when attempting to predict a pixel that pixel represents the features based on its neighbourhood, the model predicts the actual pixel value to some extent. This indicates that, the features are filtered (smoothed) according to the local characteristics. The filtered features are captured by differentiating the original and smoothed images and that are stored in another image array and it can be called residual image. The ‘residual image’ is segregated into various non-overlapping blocks with equal sizes of 3 × 3. The confidence limit is measured based on the autocorrelation values of each block and the pixel values that fall outside the upper limit are identified. Now, the identified pixel values are squared to enhance the edges and are replaced in the corresponding location itself and the other values (within the limits) are with 0. The value other than 0 represents edges and 0 represents non-edge part in the image.

In the proposed method, the centre pixel value in the small image region (3 × 3) is estimated based on the local neighbourhood values with the use of conditional probability mechanism. The centre pixel value is estimated at each pixel location by considering the image region in raster scan fashion. The estimated value is almost closer to the actual value on the homogeneous region whereas there is a small variation between the actual and estimated pixel values on the inhomogeneous region, that is, the region contains features. So, the proposed method smoothes the image surface minimally across the edges.

Overview of the proposed work: The overall concepts used in the proposed technique are presented in the form of flow chart is given in Fig. 1. One can refer the algorithm discussed in Section 5 for more details of the Flow chart.

The rest of the paper is organised as follows. In Section 2, the proposed model to smooth the image is introduced and the smoothing parameters of the model are estimated in Section 3. Section 4 deals with image smoothing and Section 5 focuses on edge magnitude and edge direction. In Section 6, the performance of the proposed technique is compared with existing standard methods. Section 7 deals with the comparison of edge maps extracted by the proposed technique for different types of synthetic images. The results and conclusion are respectively drawn in Section 8 Results and discussion, 9 Conclusion.