Industrial image processing is a major technology in the fourth industrial revolution. In recent years, the spline method has gained more and more attention in image processing. To explore an efficient industrial image processing algorithm, this article proposes novel cubic polishing spline algorithms. The cubic polishing spline is a high precision spline algorithm that brings more degrees of freedom by adding nodes to better balance the local and global aspects. First, the direct and indirect cubic polishing spline transforms are used for the image interpolation filter. Then, we construct the least squares cubic polishing spline and pyramid approximation method with application to the image reconstruction and image scaling (image upscaling or downscaling). Next, the frequency domain cubic polishing spline lowpass filter and highpass filter are proposed, and the proposed schemes are applied to image smoothing and image sharpening, respectively. Finally, the peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) and root mean square error (RMSE) evaluation metrics are used to illustrate the reconstruction effects, and experiments show that the algorithms proposed in this article can achieve better reconstruction results with better fidelity and competitiveness compared with other schemes.

Spline functions have received widespread attention in the fields of image sampling and reconstruction. To enhance the performance of splines in reconstruction and reduce the computational burden of solving large linear equations, we propose a family of generalized cardinal polishing splines (GCP-splines) and provide a system of linear equations to obtain the expressions of GCP-splines. First, we propose a cardinal polishing spline basis function with high-precision. Then, we propose a class of GCP-splines and give a general theory of GCP-splines. To calculate the expressions of GCP-splines, we adopt a system of linear equations to obtain the time shifts operator and the convolutional coefficients based on the search spacing and number of terms. Finally, we propose continuous and discrete interpolation models based on GCP-splines, and demonstrate several valuable properties, such as order of approximation and the Riesz basis. To evaluate the performance of GCP-splines, we conduct several experiments on test images from different modalities. The experimental results demonstrate that the GCP-splines for image interpolation and image denoising have better performance and outperform other methods.