Concerning the loss of weak edges and texture details of the second-order Partial Differential Equation (PDE) amplification algorithm, an image enlargement algorithm was proposed based on improved complex diffusion adaptively coupled nonlocal transform domain model. By utilizing the advantage of accurate edge location of the complex diffusion model, the improved complex diffusion coupled impulse filter to enhance strong edges better; by modeling the sparse characteristics of the transform coefficients coming from three dimensional transformation of the image group composed of similar image blocks, the nonlocal transform domain model could make good use of the nonlocal information of the similar image blocks and had better processing effects on weak edges and texture details. Finally, the second-order derivation of the image obtained by the complex diffusion was used as the parameter to realize the adaptive coupling of the improved complex diffusion model and the nonlocal transform domain model. Compared with partial differential equation amplification algorithm, nonlocal transformation domain amplification algorithm and partial differential equation coupled space domain nonlocal model amplification algorithm, the proposed algorithm has better amplification effect on strong edges, weak edges and detail textures, the mean structural similarity measures of weak edges and texture detail images are higher than those of improved complex diffusion magnification algorithm and the nonlocal transform domain amplification algorithm. The proposed algorithm also confirms the validity of the coupling between the space domain model and the transform domain model, local model and nonlocal model.

To enhance the weak edges and textures and to avoid the staircase effect, an image enlargement method was proposed which coupled anisotropic forth-order partial differential equation to second-order partial differential equation. In the method, the weak edges and textures were enhanced and staircase was reduced by improved anisotropic forth-order partial differential equation with adaptive diffusion coefficient to threshold value constrained by pixel's local variance, improved total variance and adaptive amplitude shock filters controlled by gradient were coupled with the forth-order differential equation to enhance the edges, and the bi-orthogonal projection was used to realize the constraint of the degradation model. Simulation experiment results validate the proposed method on enhancing the edges, details and textures and reducing staircases. Compared with other existing second-order PDE-based zoom methods, the zoomed images using the proposed method have better visual effect and higher Peak Signal-to-Noise Ratio (PSNR) and Mean Structural Similarity Measure (MSSIM), for example, PSNR of zoomed image with larger smooth part by the proposed method is about 0.1 dB higher than that by improved Total Variance (TV) enlargement method and PSNR of zoomed image with larger weak edges and textures by the proposed method is above 0.5 dB higher than that by improved TV enlargement method. Therefore, the zoomed image of the method looks more natural, and the resolution of the weak edges and textures of the image are enhanced better.