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This paper defines a multiple resolution representation for the two-dimensional gray-scale shapes in an image. This representation is constructed by detecting peaks and ridges in the difference of lowpass (DOLP) transform. Descriptions of shapes which are encoded in this representation may be matched efficiently despite changes in size, orientation, or position. Motivations for a multiple resolution representation are presented first, followed by the definition of the DOLP transform. Techniques are then presented for encoding a symbolic structural description of forms from the DOLP transform. This process involves detecting local peaks and ridges in each bandpass image and in the entire three-dimensional space defined by the DOLP transform. Linking adjacent peaks in different bandpass images gives a multiple resolution tree which describes shape. Peaks which are local maxima in this tree provide landmarks for aligning, manipulating, and matching shapes. Detecting and linking the ridges in each DOLP bandpass image provides a graph which links peaks within a shape in a bandpass image and describes the positions of the boundaries of the shape at multiple resolutions. Detecting and linking the ridges in the DOLP three-space describes elongated forms and links the largest peaks in the tree. The principles for determining the correspondence between symbols in pairs of such descriptions are then described. Such correspondence matching is shown to be simplified by using the correspondence at lower resolutions to constrain the possible correspondence at higher resolutions.