The notion of moment-patterns in the Radon space is introduced. A moment pattern of order k, in the Radon space, is the set of moments (of order k) associated with the projections of an image. Some of the interesting properties of the moment patterns are studied. A method of rendering the patterns invariant to translation, scaling, and rotation, within the Radon space, is developed. The resulting set of 1-D invariant moment patterns constitutes a descriptor of an object invariant to geometric transformations. When the data is available in the Radon space, as in computed tomography, the invariants can be constructed within the Radon space. An extension of the technique to 3-D is presented.