In this thesis, we study the fast transform based operators and their applications in some large-scale image restoration problems. The fast transform based operator maps every arbitrary matrix Aₙ to the optimal transform based preconditioner. This preconditioner is defined to be the minimizer of ‖Vₙ-Aₙ‖F over a set of matrices Vₙ that can be diagonalized by a fast discrete transform matrix. We discuss the algebraic and geometric properties of the operators. Then we consider the spectral properties of the operators. Numerical tests are also given. We then study the solutions of block Toeplitz systems Aₘₙx=b by the preconditioned conjugate gradient (PCG) method. Here Aₘₙ=Aₘ⊗Aₙ and Aᵢ,i=m,n are Toeplitz matrices. By applying the PCG method, Jin [23] introduced a fast algorithm with the optimal circulant preconditioners for solving these systems. This fast algorithm allows a tensor problem to be reduced to a one-dimensional problem. It was proved that if the mn×mn system is well-conditioned, then the PCG method converges superlinearly and only O (mn log mn) operations are required in solving the preconditioned system. However, only well-conditioned systems were considered in [23].Therefore we will apply this fast algorithm with {w}-circulant preconditioner proposed in [31] to manage the ill-conditioned systems. Numerical results are included to illustrate the effectiveness of this fast algorithm for solving the preconditioned systems by using the PCG method. Finally, the image restoration problems are introduced. Actually, many image restoration problems are ill-posed, so it is difficult to find the required solution. Regularizaton is commonly used to coper with ill-posed problems. Numerical experiments with different optimal transform based preconditioners are reported to illustrate the effectiveness of these preconditioners for solving the image restoration problems.