Radial invariant of 2D and 3D Racah moments Mostafa El Mallahi 1 & Amal Zouhri 1 & Abderrahim Mesbah 1 & Aissam Berrahou 1 & Imad El Affar 1 & Hassan Qjidaa 1 Received: 7 July 2016 /Revised: 13 January 2017 /Accepted: 6 March 2017 # Springer Science+Business Media New York 2017 Abstract In this paper, we introduce new sets of 2D and 3D rotation, scaling and translation invariants based on orthogonal radial Racah moments. We also provide theoretical mathemat- ics to derive them. Thus, this work proposes in the first case a new 2D radial Racah moments based on polar representation of an object by one-dimensional orthogonal discrete Racah polynomials on non-uniform lattice, and a circular function. In the second case, we present new 3D radial Racah moments using a spherical representation of volumetric image by one- dimensional orthogonal discrete Racah polynomials and a spherical function. Further 2D and 3D invariants are extracted from the proposed 2D and 3D radial Racah moments respectively will appear in the third case. To validate the proposed approach, we have resolved three problems. The 2D/ 3D image reconstruction, the invariance of 2D/3D rotation, scaling and translation, and the pattern recognition. The result of experiments show that the Racah moments have done better than the Krawtchouk moments, with and without noise. Simulta- neously, the mentioned reconstruction converges rapidly to the original image using 2D and 3D radial Racah moments, and the test 2D/3D images are clearly recognized from a set of images that are available in COIL-20 database for 2D image, and PSB database for 3D image. Keywords 2D/3D discrete orthogonal radial Racah moments . Radial Racah polynomials . 2D/3D image reconstruction . 2D/3D scaling, rotation and translation moment invariants, 2D/3D pattern recognition 1 Introduction The 3D image recognition and 3D pattern classification play essential roles and interesting part of image processing, engineering and computer vision. In general, 3D image recognition or classification is obtained by looking for descriptors representing the 3D object without taking Multimed Tools Appl DOI 10.1007/s11042-017-4573-5 * Mostafa El Mallahi mostafa.elmallahi@usmba.ac.ma 1 Faculty of Sciences Dhar el Mahraz, CED-ST Center of Doctoral Studies in Sciences and Technologies, Sidi Mohamed Ben Abdellah University, Fez, Morocco