In this paper

a new unitary transform called U transform is constructed with the recursion method.The U transform contains piecewise constant basis vectors

piecewise linear basis vectors and piecewise polynomials of degree 2 basis vectors

and so it is a generalization of Walsh transform and Slant transformThe corresponding fast algorithm based on Kronecher product is deduced.The new orthogonal U transform is applied to digital image compression with quantization table based on Human Visual System(HVS).The experimental results indicate that the compression performance of U transform outperforms the Slant transform and is equivalent to that of DCT.