Vector graphics

Summary

Vector graphics is a form of computer graphics in which visual images are created directly from geometric shapes defined on a Cartesian plane, such as points, lines, curves and polygons. The associated mechanisms may include vector display and printing hardware, vector data models and file formats, as well as the software based on these data models (especially graphic design software, computer-aided design, and geographic information systems). Vector graphics is an alternative to raster or bitmap graphics, with each having advantages and disadvantages in specific situations. While vector hardware has largely disappeared in favor of raster-based monitors and printers, vector data and software continues to be widely used, especially when a high degree of geometric precision is required, and when complex information can be decomposed into simple geometric primitives. Thus, it is the preferred model for domains such as engineering, architecture, surveying, 3D rendering, and typography,

Official source

https://en.wikipedia.org/wiki/Vector_graphics

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Change detection and background extraction by linear algebra

Emrullah Durucan, Touradj Ebrahimi

Change detection plays a very important role in real-time image analysis, e.g., detection of intruders. One key issue is robustness to varying illumination conditions. We propose two techniques for change detection that have been developed to deal with variations in illumination and background, with real-time capabilities. The foundations of these techniques are based on a vector model of images and on the exploitation of the concepts of linear dependence and linear independence. Furthermore, the techniques are compatible with physical photometry. A detailed description of the proposed detector and three state-of-the art change detectors is also provided. For the purposes of comparison, an evaluation procedure is presented consisting of both objective and subjective parts. This evaluation procedure results in a final performance value for each detector analyzed

2001

Improved linear dependence and vector model for illumination-invariant change detection

Emrullah Durucan, Touradj Ebrahimi

Change detection generally is the difference between images. The differences or changes could be due to moving objects or a variation of illumination. In general the goal is to extract only changes due to moving object that occur int eh scene, and to ignore changes due to illumination. A requirement is that the detection has to be performed in real-time. The proposed change detection approach relies on a model assigning a vector to every pixel of the reference and the current image. Based on this model, linear independence is used to describe an operator for change detection. This previously published operator is based on the variance. The improved linear independence model consists in redefining the change detection operator to fulfill the real-time requirement and to improve the object detection performance. This model has been applied to surveillance and compared to the variance based linear independence detector. The operator proved to be robust to background and illumination changes and in the same time it detected object changes.

2001

We address the problem of reconstructing scalar and vector functions from non-uniform data. The reconstruction problem is formulated as a minimization problem where the cost is a weighted sum of two terms. The first data term is the quadratic measure of goodness of fit, whereas the second regularization term is a smoothness functional. We concentrate on the case where the later is a semi-norm involving differential operators. We are interested in a solution that is invariant with respect to scaling and rotation of the input data. We first show that this is achieved whenever the smoothness functional is both scale- and rotation-invariant. In the first part of the thesis, we address the scalar problem. An elegant solution having the above mentioned invariant properties is provided by Duchon's method of thin-plate splines. Unfortunately, the solution involves radial basis functions that are poorly conditioned and becomes impractical when the number of samples is large. We propose a computationally efficient alternative where the minimization is carried out within the space of uniform B-splines. We show how the B-spline coefficients of the solution can be obtained by solving a well-conditioned, sparse linear system of equations. By taking advantage of the refinable nature of B-splines, we devise a fast multiresolution-multigrid algorithm. We demonstrate the effectiveness of this method in the context of image processing. Next, we consider the reconstruction of vector functions from projected samples, meaning that the input data do not contain the full vector values, but only some directional components. We first define the rotational invariance and the scale invariance of a vector smoothness functional, and then characterize the complete family of such functionals. We show that such a functional is composed of a weighted sum of two sub-functionals: (i) Duchon's scalar semi-norm applied on the divergence field; (ii) and the same applied to each component of the rotational field. This forms a three-parameter family, where the first two are the Duchon's order of the above sub-functionals, and the third is their relative weight. Our family is general enough to include all vector spline formulations that have been proposed so far. We provide the analytical solution for this minimization problem and show that the solution can be expressed as a weighted sum of vector basis functions, which we call the generalized vector splines. We construct the linear system of equations that yields the required weights. As in the scalar case, we also provide an alternative B-spline solution for this problem, and propose a fast multigrid algorithm. Finally, we apply our vector field reconstruction method to cardiac motion field recovery from ultrasound pulsed wave Doppler data, and demonstrate its clinical potential.

EPFL2005