Quaternion Color Curvature

In this paper we propose a novel approach to measuring curvature in color or vector-valued images (up to 4-dimensions) based on quaternion singular value decomposition of a Hessian matrix. This approach generalizes the existing scalar-image curvature approach which makes use of the eigenvalues of the Hessian matrix [1]. In the case of vector-valued images, the Hessian is no longer a 2D matrix but rather a rank 3 tensor. We use quaternion curvature to derive vesselness measure for tubular structures in color or vector-valued images by extending Frangi’s [1] vesselness measure for scalar images. Experimental results show the effectiveness of quaternion color curvature in generating a vesselness map. Introduction Hessian-based methods have been widely used from curvature measures to feature detection [1-10]. The Hessian matrix describes the second-order structure of gray-level variations around each pixel of the image. There are two main categories where a Hessian matrix is used. First, the Hessian and the related second-moment matrix have been applied in several operators (e.g., the Harris [11], Harris-affine [12], and Hessian-affine [10] detectors) to find “interest” points where the local image geometry changes in more than one direction. Hessian-based blob detector in color space is proposed in [5]. Second, since the eigenvalues of the Hessian matrix at a pixel measure the principal curvatures of the image intensity surface, it can be used to detect tubular (linear, vessellike) structures, which is useful in many applications [1,3,6-9]. By smoothing with Gaussian kernels of various sizes, the normalized second-order derivatives indicate the scale and orientation of vessels. Vesselness is measured by a large curvature in the crosssectional direction and a small curvature along the vessel. By eigen-analysis of the Hessian matrix, elongated objects (i.e., vessels) are detected wherever the first eigenvalue is positive (or negative) and prominent. This process generates a single response for both lines and edges, producing a clearer sketch of an image’s structure than is usually provided by the magnitude of gradient. Existing first-derivative point/blob detectors are applied to gray scale images. In the case of color images, the basic approach has been to compute the derivatives of each color channel separately, and then add them to produce the final result [5]. However, the first derivatives of a color edge can be in opposing directions, so the summation can lead to cancellation of the derivatives. The same situation occurs in second-derivative-based Hessian detectors. Existing Hessian-based curvature methods are also based on gray scale images, whether the luminance image, or a single color channel. For example, Hessian-based multi-scale segmentation or enhancement of vessels in retinal images has been extensively studied [1,3,6-7], where only the green channel is used. To make use of the extra information in a color image, we use the quaternion representation of color to extend Hessian curvature measures to the color domain. In particular, we extend Frangi’s [1] vesselness approach by estimating principle curvatures in RGB color space using quaternion operations. Sanqwine [13] introduced the quaternion representation of color. Since quaternions, which are an extension of the complex numbers, consist of one real component and three imaginary components, a color can be represented by a pure quaternion having a real component of zero, and imaginary components R, G and B. With colors encoded in quaternions, the entries of the Hessian matrix become quaternions that combine secondary derivatives from all color channels in their imaginary components. Quaternion singular value decomposition (QSVD) [13,14] can then be applied to the Hessian matrix in order to find the principle curvatures as described by the two nonnegative, real-valued singular values. These singular values can be used to measure vesselness or other features. The remainder of the paper is organized as follows. Section 2 reviews the necessary definitions of the Hessian matrix and its eigen-system for scalar images. Section 3 describes the quaternionbased approach to color curvature, and extends Frangi’s vesselness measure to vector-valued images. Section 4 shows the experimental results, and Section 5 concludes the paper. Curvature and Vesselness Measure Viewing an image as an intensity surface, the local shape characteristics of the surface at a particular point can be described by the Hessian matrix. Lines (i.e., straight or nearly straight curvilinear features) and edges have high curvature in one direction and low curvature in the orthogonal direction, and this characteristic can be measured via the Hessian, H. For a 2D scalar image, H is a 2x2 matrix of the second derivatives of image I