Image denoising method based on variable exponential fractional‐integer‐order total variation and tight frame sparse regularization

This study presents a variational image restoration algorithm based on variable exponential fractional‐order total variation (TV), variable exponential integer‐order TV and tight frame sparse regularization. The energy functional of this variational problem is composed of four parts: a fractional‐order TV regularization term with a variable exponent, an integer‐order TV regularization term with a variable exponent, a data fidelity term and a tight frame regularization term. The variable exponent is a function of the gradient information of the image. The first three parts of the energy functional is the smooth part, and the last part is the non‐smooth part. To minimize the smooth part, the optimization problem can be simply transformed into a gradient descent flow using the variational method. For the non‐smooth part, we use the soft‐thresholding method over the sparse framelet coefficients. As the combination of the fractional‐order derivative and integer‐order derivative, the variable exponent and the sparse regularisation, the proposed method can effectively remove noise of the image, protect the image boundary, and keep image texture details well. Experiments with simulated data and real data are provided to validate the effectiveness of proposed method. This method is robust to noise and is of some practical application value.