We present a method for organizing a heterogeneous collection of 3D shapes for overview and exploration. Instead of relying on quantitative distances, which may become unreliable between dissimilar shapes, we introduce a qualitative analysis which utilizes multiple distance measures but only in cases where the measures can be reliably compared. Our analysis is based on the notion of quartets, each defined by two pairs of shapes, where the shapes in each pair are close to each other, but far apart from the shapes in the other pair. Combining the information from many quartets computed across a shape collection using several distance measures, we create a hierarchical structure we call categorization tree of the shape collection. This tree satisfies the topological (qualitative) constraints imposed by the quartets creating an effective organization of the shapes. We present categorization trees computed on various collections of shapes and compare them to ground truth data from human categorization. We further introduce the concept of degree of separation chart for every shape in the collection and show the effectiveness of using it for interactive shapes exploration.

The computation of a rigid body transformation which optimally aligns a set of measurement points with a surface and related registration problems are studied from the viewpoint of geometry and optimization.We provide a convergence analysis for widely used registration algorithms such as ICP, using either closest points (Besl and McKay [2]) or tangent planes at closest points (Chen and Medioni [4]), and for a recently developed approach based on quadratic approximants of the squared distance function [24]. ICP based on closest points exhibits local linear convergence only. Its counterpart which minimizes squared distances to the tangent planes at closest points is a Gauss-Newton iteration; it achieves local quadratic convergence for a zero residual problem and - if enhanced by regularization and step size control - comes close to quadratic convergence in many realistic scenarios. Quadratically convergent algorithms are based on the approach in [24]. The theoretical results are supported by a number of experiments; there, we also compare the algorithms with respect to global convergence behavior, stability and running time.

We propose a novel framework for reconstructing lightweight polygonal surfaces from point clouds. Unlike traditional methods that focus on either extracting good geometric primitives or obtaining proper arrangements of primitives, the emphasis of this work lies in intersecting the primitives (planes only) and seeking for an appropriate combination of them to obtain a manifold polygonal surface model without boundary. We show that reconstruction from point clouds can be cast as a binary labeling problem. Our method is based on a hypothesizing and selection strategy. We first generate a reasonably large set of face candidates by intersecting the extracted planar primitives. Then an optimal subset of the candidate faces is selected through optimization. Our optimization is based on a binary linear programming formulation under hard constraints that enforce the final polygonal surface model to be manifold and watertight. Experiments on point clouds from various sources demonstrate that our method can generate lightweight polygonal surface models of arbitrary piecewise planar objects. Besides, our method is capable of recovering sharp features and is robust to noise, outliers, and missing data.

We present an algorithm for shape reconstruction from incomplete 3D scans by fusing together two acquisition modes: 2D photographs and 3D scans. The two modes exhibit complementary characteristics: scans have depth information, but are often sparse and incomplete; photographs, on the other hand, are dense and have high resolution, but lack important depth information. In this work we fuse the two modes, taking advantage of their complementary information, to enhance 3D shape reconstruction from an incomplete scan with a 2D photograph. We compute geometrical and topological shape properties in 2D photographs and use them to reconstruct a shape from an incomplete 3D scan in a principled manner. Our key observation is that shape properties such as boundaries, smooth patches and local connectivity, can be inferred with high confidence from 2D photographs. Thus, we register the 3D scan with the 2D photograph and use scanned points as 3D depth cues for lifting 2D shape structures into 3D. Our contribution is an algorithm which significantly regularizes and enhances the problem of 3D reconstruction from partial scans by lifting 2D shape structures into 3D. We evaluate our algorithm on various shapes which are loosely scanned and photographed from different views, and compare them with state-of-the-art reconstruction methods.

The human eye can perceive a vast array of colors. Whether light or dark, the colors that our eyes can see are supposedly unlimited. However, this is not the case. In reality, every image has two aspects or descriptors to it, a reflectance and an illumination. While the reflectance essentially shows an images true color, the illumination is what causes the colors to seem different to the human eye. This effect, originally discovered by Helmholtz, is known as Color Constancy. Color Constancy ensures that the color the Human Visual System (HVS) receives is the true color of the image, regardless of illumination. As a result of this effect, in 1971, Land and McCann created the Retinex theory. Using the pixels in the image, Land tried to estimate the value of the reflectances and thus reveal the true color of the image. This theory was basically a color constancy algorithm that tried to explain why colors look different when exposed to lighting. By calculating the pixels, Land was able to depict the sameness in a gradient of colors in an image. However, the algorithm is both inefficient and complicated. Following their footsteps, many other people have tried to formulate new algorithms around the Lands original Retinex algorithm. In this paper, different methods such as least squares and discrete cosine transform are explained as well as how to enhance images using both Lands idea and histogram equalization.