Stable and invertible invariants description for gray-level images based on Radon transform

In a large number of applications, different types of descriptors have been implemented to identify and recognize textured objects in grayscale images. Their classification must be carried out independently of their position, orientation and scale. The property of completeness of descriptors which guarantees their uniqueness for a given shape is also a sought-after property. It is known in the literature that such a property is difficult to obtain. It was possible to achieve it in rare cases for planar curves using for instance Fourier descriptors or the curvature. In grayscale images, we know at least three cases of complete descriptors: Those based on Zernike moments, those computed from the analytical Fourier–Mellin transform or obtained from the complex moments. To our current knowledge, in the case of curved surfaces and 3D volume images, there are yet no complete descriptors invariant to the 3D rigid motions. The property of invertibility of invariants, introduced recently and which implies completeness, allows the reconstruction of the object shape up to a similarity. The two sets of descriptors that we propose here verify on the one hand the invariance and the invertibility and on the other hand, the notion of stability which was introduced to guarantee the fact that the descriptors vary slightly during small variations in the shape. Their construction based on the Radon transform allows a certain robustness with respect to noise. In this article, we rigorously demonstrate the properties of invariance, invertibility and convergence for the two sets of proposed invariants. To evaluate the stability and robustness with respect to noise, experimental studies are carried out on different well-known datasets, namely Kimia 99 and MPEG7. We introduced our own face dataset, which we named FSTEF for further evaluation. On the other hand, several types and levels of noise were added to test the robustness to noise. Therefore, the effectiveness of the suggested sets of invariants are demonstrated by the different studies proposed in the present work.