Image magnification refers to increasing the image size or resolution, while maintaining a high quality to get a better visual effect, or highlight some details. Image magnification can usually be performed in two steps: first, the image is spatially transformed; second, the image is gray-scale interpolation and processing.
The traditional linear interpolation algorithms include nearest neighbor method, bilinear interpolation method and cubic spline interpolation method. These methods use some known simple functions to approximate the original image according to certain smoothness requirements. They have their inherent shortcomings, such as blurring and jagged edges of the enlarged image, and the larger the magnification, the more obvious these phenomena. The adaptive interpolation method spatially makes the interpolation coefficients better match the local image structure near the edge, but will cause errors when selecting and estimating the edge of interest. The edge-guided interpolation method uses limited quantization of the direction and amplitude of the edge to fit the subpixel edges of the image, preventing inter-edge interpolation, and thus can produce sharp edges, but the fitting of the edges is too simple and rough, and Some image features will be lost. There are also iterative methods based on convex set projection, methods based on wavelet transform and mathematical morphology, etc. These methods still need to be improved in performance, especially when the image contains noise.
The image magnification method based on partial differential equations (ParTIal DifferenTIal EquaTIons, PDE) is based on interpolated images and iteratively evolves to solve diffusion equations to obtain high-resolution images and remove the effects of noise and artifacts. Because it is easy to introduce a priori knowledge, it has obtained good processing performance and has attracted wide attention. The mainstream is based on the regularized PDE method. The isotropic diffusion PDE model is prone to blurry edges and loss of details when enlarging the image. Although the anisotropically diffused PDE can maintain the details of the enlarged image to a certain extent, as the number of iterative solutions increases, some important information of the image will deviate from the original image, resulting in blurred images. Using the Total Variation (Total V-ariaTIon, TV) model for image enlargement can effectively maintain the abrupt edges and has a fast convergence speed, but block effects will occur in flat areas and gradient areas. The fourth-order PDE model has the advantage of maintaining the smoothness of the flat area. Using it for image enlargement will avoid blocking effects, but it reduces the clarity of important geometric structures such as edges.
In order to make full use of the advantages of the fourth-order PDE model to maintain the smoothness of the gradient area, make up for the lack of blocking effect in the total variation TV model, and also retain the advantage of the TV model to maintain the discontinuous edges in the image, this paper proposes an adaptive coupling total Regularized image enlargement model of variational TV and fourth-order PDE. According to the content of the image, the coupling coefficient is adjusted reasonably. In the gradual and flat areas of the image, the fourth-order model is mainly used for smoothing to eliminate the staircase effect and blocking effect; in the abrupt areas of the image, the TV model is used for smoothing to keep the abrupt edges. Simulation experiments show that the algorithm in this paper can effectively improve the subjective visual quality and objective fidelity of the enlarged image.
1 Adaptive coupling TV and fourth-order PDE regularized image enlargement model
The TV model was originally used for image restoration. Let g and u denote low-resolution images and high-resolution images, respectively. According to the maximum likelihood principle, image enlargement can be reduced to the following minimization problem of the regularized energy equation without constraints:
Where: D is the image resolution degradation model matrix, which depicts the low-pass filtering and downsampling process in image acquisition; the first term in the formula is the approximation term, which represents the difference between the image and the degraded image; the second term is the regularity of the image Normalization function, which depends on the image, the function R (·) constrains the image u, generally taken as the Lp (p> 0) norm of the gradient; λ> O is the Lagrange multiplier, between the approximation term and the regularization function Play a balancing role.
If the L2 norm of the image gradient is selected as the regularization function, because the Laplacian has strong isotropic diffusion characteristics, the edge retention characteristics are poor. The total variation TV model replaces the L2 norm with the L1 norm of the image gradient, and has good edge retention characteristics. The TV definition of the image is:
Where: Ω is the image area. So the problem of image enlargement becomes the following unconstrained minimization problem:
In the formula: the first term at the right end of the equation is the total variation norm (TV norm) of the image, which depends on the variation amplitude of the image.
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