The discrete Walsh and Hadamard transforms are often used in image processing tasks such as image coding, pattern recognition, and sequency filtering. A new discrete Walsh transform (DWT) algorithm is derived in which a modified form of the DWT relation is decomposed into smaller-sized transforms using vectorized quantities. A new sequency-ordered discrete Hadamard transform (DHAT) algorithm is also presented. The proposed approach results in more regular algorithms requiring no independent data swapping and fewer array-index updating and bit-reversal operations. An analysis of the computational complexity and the execution time performance are provided. The results are compared with those of the existing algorithms.