Research
Research statement
On peut avoir trois principaux objets dans l'étude de la vérité: l'un, de la découvrir quand on la cherche; l'autre, de la démontrer quand on la possède; le dernier, de la discerner d'avec le faux quand on l'examine." Blaise Pascal.
Computer vision consist in endowing a machine with human-like visual abilities. For this purpose, it is used a photographic camera, which senses and capture data, and a processing information unit. The photographic camera is called a projective device, since images are formed by the perspective transformation of the 3D world to the 2D image plane; hence, geometric properties are embedded at the core of computer vision problems. My research focuses in finding such properties, and applying geometric concepts in the design of computer vision algorithms. A distinctive feature of my research, is the use of geometric algebra as a tool for modeling and implementing concepts, principles and algorithms.
My research program is devoted to vindicate the use of geometric principles in the developing of computer vision algorithms, to show how they can be used to improve state-of-art methods, and to exemplify how geometric algebra simplifies the modeling, computing and application of such principles.
Projects
This project proposes a novel algorithm based on classical voting schemes like Hough Transform and Tensor Voting. In particular, it combines the use of perceptual saliency functions and geometric flags with a clustering technique, all within the mathematical framework of the conformal model of Geometric algebra. So that, the voting scheme can represent complex configurations of visual data we want to detect; in addition, this algorithm can be use as building blocks to design hierarchical architectures. Experimental results show its ability to extract pair of points, lines, circles, and symmetry axes from synthetic and real images.
[1] Gerardo Altamirano-Gómez and Eduardo Bayro-Corrochano. “Conformal Geometric Method for Voting”. In: Journal of Intelligent Data Analysis 20 (2016), S69-S82.
[2] Gerardo Altamirano-Gómez and Eduardo Bayro-Corrochano. “Conformal Geometric Algebra Method for Detection of Geometric Primitives”. In: 23nd International Conference on Pattern Recognition (ICPR). 2016.
[3] Gerardo Soria-Garcı́a et. al. “Conformal Geometric Algebra Voting Scheme implemented in reconfigurable devices for geometric entities extraction”. In: IEEE Transactions on Industrial Electronics (2017).
Torque-based geometric perceptual maps.
In this project, geometric algebra is used to re-define the image torque operator, proposed by Nishigaki et. al (2012). Our definition generalize this operator to higher dimensions and permits the use of high order moments; as a result, we obtain an algorithm that implements the grouping by closure principle of the Gestalt theory, to construct saliency maps from n-dimensional data.
Quaternion convolutional neural networks.
This study establishes the theoretical bases and implementation details of quaternion convolutional neural networks. In particular it is proposed: the basic model of quaternion convolution layers and quaternion pooling layers, the back-propagation mechanism, and the connection between quaternion convolution and Quaternion Fourier Transform.
[1] Gerardo Altamirano-Gómez, “Geometric methods of perceptual organisation for computer vision,” Ph.D. dissertation, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional (Cinvestav), México, 2017.
[2] Gerardo Altamirano-Gómez and Carlos Gershenson, “Quaternion Convolutional Neural Networks: Current Advances and Future Directions,” 2023, preprint at https://arxiv.org/abs/2307.08663.
This work proposes a geometric method to calculate the position of a camera within a non-calibrated system. The solution is formalized using Grassmann-Cayley algebra.
[1] Gerardo Altamirano Gómez. “¿Dónde está el fotógrafo? Usando álgebra de Grassmann-Cayley”. Master thesis. Baja California, México: Centro de Investigación Cientı́fica y de Educación Superior de Ensenada, 2012.
[2] Mariana Chan-Ley et.al. “Método no-calibrado para localización automática de una cámara aplicable en robótica”. In: Congreso Mexicano de Robótica (COMROB), 2018.
[3] Mariana Chan-Ley et.al. "Self-localization of an uncalibrated camera through invariant properties and coded target location," Appl. Opt. 59, D239-D245, 2020.
Linux-RTAI Open architecture for the robot Mitsubishi RV-M1.
This work proposes a linux-based real-time open system architecture for an industrial robot of 6DOF. Using this system, it is possible to implement different control algorithms, to integrate different types of sensors to the system (video camera, force sensors, etc), and to implement tasks to be acomplished in real-time.