Abstract-This paper proposes a closed-form weighted least-squares solution for designing variable two-dimensional (2-D) finite-impulse response (FIR) digital filters with continuously variable 2-D fractional delay responses. First, the coefficients of the variable 2-D transfer function are represented by using the polynomials of a pair of fractional delays ( 1 2 ). Then the weighted squared-error function of the variable 2-D frequency response is derived without sampling the two frequencies ( 1 2 ) and two fractional delays ( 1 2 ), which leads to a significant reduction in computational complexity. With the assumption that the overall weighting function is separable and stepwise, the design problem is reduced to the minimization of the weighted squared-error function. Based on the error function, the closed-form optimal solutions for the coefficient matrices of the variable 2-D transfer function can be determined through solving a pair of matrix equations. In addition, Cholesky decomposition is applied to the final closed-form expressions in order to avoid some numerical instability problem. An example is given to illustrate the effectiveness of the proposed design method.Index Terms-Two-dimensional digital filters, variable digital filters, variable fractional delay filters, weighted least-squares design.