A structure from motion inequality

We state an elementary inequality for the structure from motion problem for m cameras and n points. This structure from motion inequality relates space dimension, camera parameter dimension, the number of cameras and number points and global symmetry properties and provides a rigorous criterion for which reconstruction is not possible with probability 1. Mathematically the inequality is based on Frobenius theorem which is a geometric incarnation of the fundamental theorem of linear algebra. The paper also provides a general mathematical formalism for the structure from motion problem. It includes the situation the points can move while the camera takes the pictures.Comment: 15 pages, 22 figure

Dynamic Rigid Motion Estimation From Weak Perspective

“Weak perspective” represents a simplified projection model that approximates the imaging process when the scene is viewed under a small viewing angle and its depth relief is small relative to its distance from the viewer. We study how to generate dynamic models for estimating rigid 3D motion from weak perspective. A crucial feature in dynamic visual motion estimation is to decouple structure from motion in the estimation model. The reasons are both geometric-to achieve global observability of the model-and practical, for a structure independent motion estimator allows us to deal with occlusions and appearance of new features in a principled way. It is also possible to push the decoupling even further, and isolate the motion parameters that are affected by the so called “bas relief ambiguity” from the ones that are not. We present a novel method for reducing the order of the estimator by decoupling portions of the state space from the time evolution of the measurement constraint. We use this method to construct an estimator of full rigid motion (modulo a scaling factor) on a six dimensional state space, an approximate estimator for a four dimensional subset of the motion space, and a reduced filter with only two states. The latter two are immune to the bas relief ambiguity. We compare strengths and weaknesses of each of the schemes on real and synthetic image sequences

Drawing with Complex Numbers

It is not commonly realized that the algebra of complex numbers can be used in an elegant way to represent the images of ordinary 3-dimensional figures, orthographically projected to the plane. We describe these ideas here, both using simple geometry and setting them in a broader context.Comment: 14 pages, 5 figure

Structure from motion systems for architectural heritage. A survey of the internal loggia courtyard of Palazzo dei Capitani, Ascoli Piceno, Italy

We present the results of a point-cloud-based survey deriving from the use of image-based techniques, in particular with multi-image monoscopic digital photogrammetry systems and software, the so-called “structure-from-motion” technique. The aim is to evaluate the advantages and limitations of such procedures in architectural surveying, particularly in conditions that are “at the limit”. A particular case study was chosen: the courtyard of Palazzo dei Capitani del Popolo in Ascoli Piceno, Italy, which can be considered the ideal example due to its notable vertical, rather than horizontal, layout. In this context, by comparing and evaluating the different results, we present experimentation regarding this single case study with the aim of identifying the best workflow to realise a complex, articulated set of representations—using 3D modelling and 2D processing—necessary to correctly document the particular characteristics of such an architectural object

Photometric stereo for strong specular highlights

Photometric stereo (PS) is a fundamental technique in computer vision known to produce 3-D shape with high accuracy. The setting of PS is defined by using several input images of a static scene taken from one and the same camera position but under varying illumination. The vast majority of studies in this 3-D reconstruction method assume orthographic projection for the camera model. In addition, they mainly consider the Lambertian reflectance model as the way that light scatters at surfaces. So, providing reliable PS results from real world objects still remains a challenging task. We address 3-D reconstruction by PS using a more realistic set of assumptions combining for the first time the complete Blinn-Phong reflectance model and perspective projection. To this end, we will compare two different methods of incorporating the perspective projection into our model. Experiments are performed on both synthetic and real world images. Note that our real-world experiments do not benefit from laboratory conditions. The results show the high potential of our method even for complex real world applications such as medical endoscopy images which may include high amounts of specular highlights