An Efficient Solution for a Particular Class of Sylvester Matrix Equations in Image Processing Tasks

The Sylvester equation is a special equation described in the form of AX+XB=Y, which is widely applied in many image processing tasks. However, there are two limitations to solving the general solution of the Sylvester equation: Firstly, the general solution method uses the Kronecker product operator to transform the equation into a column vector form equation with Kronecker product operation, which brings heavy storage and computer burdens. Secondly, when the IN⊗A+BT⊗IM is non-invertible, the Sylvester equation does not have a unique solution. Note that the Sylvester equation in image processing often exhibits certain peculiarities (e.g., matrix A can be diagonalized by unitary matrices, and matrix B is a real symmetric matrix). For this type of special Sylvester equation, this paper first derives an efficient method for solving the Sylvester equation based on the properties of diagonal matrices. Then, we discuss the solving techniques when the Sylvester equation has no unique solution. Finally, we demonstrate the application of the proposed method in image processing. Extensive experimental results show that the proposed fast solver yields results consistent with those from conventional Sylvester solvers, with markedly improved computational efficiency.