Improved direction estimation for Di Zenzo's multichannel image gradient operator

Abstract

Gradient estimation is one of the most important tasks in image/video processing. For multichannel images, a classical and widely-used gradient method is Di Zenzo's gradient operator, which is based on the measure of squared local contrast variation of multichannel images. However, up to now, the indetermination of Di Zenzo's gradient direction has not been well solved, which results in errors occurring in most of the subsequent studies in which Di Zenzo's vector gradient is used. In this paper, this problem is solved thoroughly. Furthermore, the ranges of the values that the gradient angle should take in various cases are also analyzed. As an application in color image processing, a color version of Canny edge detector is implemented by introducing the new gradient estimator to the traditional grayscale image Canny operator. The experimental results indicate that the improved Di Zenzo's gradient operator is currently one of the best color gradient estimators and outperforms other state-of-the-art color image gradient methods. The improved multichannel gradient operator not only provides accurate gradient estimation but also is efficient and easy to implement.

Introduction

Gradient [1] has been widely used in image processing and computer vision in applications such as edge detection [2], [3], [4], [5], [6], image segmentation [7], [8], corner detection [9], image fusion [10], image recognition [11], face detection [12], and object tracking [8], [13]. For example, edge detection can be implemented by thresholding gradient magnitudes or by locating local maximum values of gradient magnitudes, and object tracking and recognition can be obtained by matching the gradient directions and at the same time using the gradient magnitudes of the pixels on the model object edges and candidate object edges.

The gradient associated with an image pixel is usually defined as a 2-D column vector, in which the vectorial angle denotes the direction of the largest growth of the image function. For grayscale images, numerous gradient estimators have been developed. However, for multichannel (multidimensional) images, which are usually described as vector fields [14], this issue has not received enough attention. From a general point of view, multidimensional gradient estimators can be divided into three major categories. The first type is characterized by a single estimate of the orientation and strength of an edge at a point [15]. The first such method is proposed by Robinson [16], who computed 24 directional derivatives (8 neighbors per color channel) and chose the one with the largest magnitude as the color gradient. Later, Ruzon and Tomasi [15] utilized a color distribution to represent a neighborhood and implemented color edge, junction, and corner detection. Their method first divides the current processing window in half with a line segment and computes a color distribution for each half, and then calculates the distance between the two distributions. This process is repeated using line segments with different orientations and the one with maximum strength is assumed as the orientation of the edge. Thus, the maximum strength and the direction normal to the corresponding orientation are regarded as the gradient magnitude and direction.

The second category of multichannel gradient methods is based on grayscale image gradient estimators. These operators calculate the gradient vectors for individual channels and then combine them to produce the final gradient vectors. According to different combination mechanisms, the resultant gradient can be the vector sum of the gradient vectors of individual channels, or the RMS (root mean square), or the maximum of the channel gradient magnitudes, or other mechanisms. However, these component-wise methods, as pointed out in [17], are unsatisfactory in some cases since in these methods the image channels do not actually cooperate with one another.

The third type of multidimensional gradient estimators is based on finding the maximum changes of image vectors. Among them, the simplest one is to define the resultant gradient as the vector in which the magnitude is the maximum of the Euclidean distances between the central pixel vector and its eight neighboring pixel vectors and the direction is estimated from the direction of the maximum change [18]. Di Zenzo [17] proposed a classical and efficient multichannel gradient operator, which is based on the measure of squared local contrast variation of multichannel images. Scharcanski and Venetsanopoulos developed a local vector statistics based gradient method for color edge detection [19], in which they used the differences between the average color vectors of the samples inside the sub-windows in horizontal and vertical directions to estimate the maximum variation of a color image in each pixel position.

More recently, Nezhadarya and Ward proposed a new color image gradient operator [20]. This method first applies highpass and lowpass vector operators in an appropriate manner in both horizontal and vertical directions, where the highpass and lowpass operators are respectively used as vector difference estimate and noise smoothing. Then, an aggregation operator is performed on each direction to find the corresponding partial derivative.

Among these multidimensional gradient estimators mentioned above, perhaps the most classical and widely-used one is Di Zenzo's multichannel gradient operator [17]. However, Di Zenzo did not solve the problem of indetermination of the gradient direction. Although some researchers [21], [22] have made further studies on Di Zenzo's vector gradient, to date this problem has not been well solved, which results in errors occurring in most of the later studies (see Section 2) in which Di Zenzo's vector gradient is referenced. This paper will solve this problem thoroughly and at the same time analyze the gradient angle ranges in various cases.

As an application in image processing, we apply the new multichannel gradient operator to color image edge detection, since gradient is closely related to edge detection and image segmentation. In color images, edges can be defined as meaningful discontinuities of image functions in vector fields [23], [24]. Color edge detection techniques [23], [24], [25], [26], [27] can be roughly divided into two classes: monochromatic-based techniques that first detect edges in individual color channels separately and then combine the component results to be the color edges, and vector-valued techniques that treat color pixels as color vectors in a vector space to detect the abrupt changes. Vector approaches are generally preferred to component-wise techniques owing to the vector nature of color images and the strong spectral correlation that exists between color channels. Vector approaches mainly include the first- [17] and second- [21], [22] order derivative methods which are based on color vector gradients, the directional vectors based difference methods (or called directional operators) [19], the methods based on vector order statistics [28], [29], the difference vector operators [23], [25], and other methods such as morphological gradient approaches [30], vector entropy methods [31], [32], density estimation methods [33], [34], and methods based on physics models [35] and principal axis analysis and moment-preserving [36].

The remainder of this paper is organized as follows. In Section 2, Di Zenzo's vector gradient and the related work are reviewed, and the proposed mechanism for solving the ambiguity in Di Zenzo's gradient angle is described in detail. Section 3 gives an application for color edge detection by applying the new gradient operator to the traditional grayscale image Canny operator. Finally, conclusions are drawn in Section 4.