A new image reconstruction method fur electrical impedance CT is derived rigorously. The Jacobi Matrix is made sparse by analysing each narrow region which is detrmined by equipotential lines seperately. Both storage and computation can be deduced. The method is realized by using first order square FEM and some computer model is reconstructed. The relaxed version of Hildreth's method for interval convex programing is applied, which can hold the sparsity of the Jacobi Matrix and overcome the ill-condition of the matrix.