In this paper, we present a new framework for the design of adaptive transforms which have overlapping and nonoverlapping basis functions. We provide the condition that the overlapping basis functions have to satisfy, and derive the optimal nonoverlapping basis functions in the energy compaction sense. We show the orientation adaptation example, where each adaptive transform is characterized by the angle of edges. In image coding experiments, the proposed adaptive transform can reduce the blocking effect because of the use of overlapping basis functions. Moreover, the use of orientation adaptation can improve visual quality around edges and lines. The performance of the adaptive transform system is superior to that of the nonadaptive transform in the rate-distortion sense. The proposed framework can be applied to any design method for adaptive transforms based on the Karhunen–Loe`ve transform.