Image degradation is commonly unavoidable during acquisition and noise makes the later processing difficult and inaccurate.Partial differential equation methods

especially low order methods

are efficient for grey-scale images denoising.Instead of the heuristic channel-coupled method

researchers begin to deal with color images under Riemann geometry framework.This framework uses the arc element to measure the rates of change and the eigenvectors of the metric tensor to describe the direction of the change.However

these methods generate low-order partial differential equations and their extension to the higher-order model remains an important challenge.The high-order models have ability to eliminate the undesired staircasing effect that accompanies the use of a model based on first-order derivatives.In this paper

a geometry-driven higher-order model for removing noise from color images is proposed.The proposed method introduces a gradient-based weigh function to improve edge detection and preserving ability while removing noise. Within the Riemann geometry framework

the norm of the arc element is calculated and generates a quadratic form called the first fundamental form.This norm can be interpreted as the distance of the ellipse to its center.The eigenvalues of the metric tensor correspond to the semi-major axis and the semi-minor axis of that ellipse.Therefore

its Jacobian matrix allows the measurement of edges in the vector-valued images.Various matrix norms can be established based on this matrix to characterize reasonable measurement for constructing variational models for color image denoising.However

as a low-order model

its corresponding low-order partial differential equation also suffers from staircasing effect.Inspired by the low-order model

a second-order equation is derived from the area element within the geometrical framework for image processing.Its norm square is decided by a second-order-based matrix and the Frobenius norm for this matrix is obtained.Based on this special norm

a higher-order variational model is proposed and a high-order partial differential equation is derived using variational principal.The gradient information is used to guide the higher-order diffusion to preserve edges during the diffusion process.The gradient is first convoluted by Gaussian kernel to predict edge locations to reduce the effect by additive Gaussian white noise.Analysis on the nonlinear diffusion term shows that the diffusion is controlled by the following information:the guide information based on first-order derivatives

the diffusion information based on second-order derivatives

and the second-order derivatives ration between the single-color channel and three-color channels. Experiments are conducted for various data

including one-dimensional signal

synthetic images

and standard test images.In every experiment

the test data are corrupted by additive white Gaussian noise with different variances.The results obtained by the proposed model are quantitatively and visually compared with the related methods.Peak signal-to-noise ratio(PSNR) and structure similarity index(SSIM) are used for quantitative comparison.Zoomed images and residual images are used for visual comparisons.The ability of recovering piecewise linear is verified for one-dimensional synthesized signal.The proposed method has an obvious improvement in both objective index and visual perception.The PSNR of the recovered signal processed by the proposed method increases from 32.64 dB to 33.16 dB compared with the low-order model when the standard deviation of the noise is 35.The SSIM result of the proposed method increases from 0.969 5 to 0.991 8 compared with the channel-coupled mean curvature method under the same condition.The proposed method is also compared with the decorrelated vectorial total variation model.The PSNR increases by 1.37 dB

and the SSIM increases by 0.005 8.The proposed method has the best objective index compared with related methods.The results are plotted in the cubic RGB color space.The same color is mapped to the same point in this space

and the linear segments in the RGB space correspond to the smoothing change of color.The result of the proposed method gives the best performance in the recovery of both constant area and the linear part of the noisy signal.Experimental results on a set of color image data are also given.Compared with the current related methods

the average improvement is 2.33% for PSNR and 0.4% for SSIM. The proposed model can suppress noise efficiently from a piecewise linear image while avoiding the staircasing effect and giving a better performance at the edge of the color image.The proposed method is efficient for removing noise with different variances.