Mathematical Optimization in Graphics and Vision - Luiz Velho - Impa

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12 CHAPTER 1. COMPUTER GRAPHICS Example 3 (Image). A grayscale image (see Section1.2.2) is a function f : U ⊂ R 2 → R, therefore it is a graphical object. More generically, an image is a function f : U ⊂ R 2 → R n , where R n is the representation of a color space. In this way, we see that an image is a graphical object whose geometric support is the subset U of the plane (in general, a rectangle), and the attribute function associates a color to each point of the plane. Example 4 (Circle and vector field). Consider the unit circle S 1 centered at the origin, whose equation is given by x 2 + y 2 = 1. A mapping of the plane N : R 2 → R 2 given by N(x, y) = (x, y), defines a field of unit vectors normal to S 1 . The map T : R 2 → R 2 , given by T (x, y) = (y, −x) defines a field of vectors tangent to the circle. (Figure 1.8). Figure 1.8: Circle with normal and tangent vector fields. The circle is a one-dimensional graphical object of the plane, and the two vector fields are attributes of the circle (they can represent, for example, physical attributes such as tangential and radial acceleration). The attribute function is given by f : S 1 → R 4 = R 2 ⊕ R 2 , f(p) = (T (p), N(p)).
1.4. REVISITING THE DIAGRAM 13 1.4 Revisiting the diagram Now that we have discussed the concept of graphical object, it is time to go back to the diagram of Figure 1.2. Every output graphics device has an internal representation of the class of graphical objects that the device is able to display. The space of graphical objects with this representation is called the representation universe of the graphical display. As an example, the representation universe of a regular personal computer (graphics card + monitor) is a matrix representation of an image. Thus, for each output graphics device, the operation of visualization on the device consists of an operator R: O → O ′ from the space of graphical objects to be displayed, to the representation universe of the graphics display. Besides the generic visualization operation described above, we have the operations of graphical object description, graphical object processing and graphical object analysis. This completes the elements to obtain a generic diagram for graphical objects, the contains, as particular cases, the diagrams in Figures 1.2 and 1.3. 1.5 Description, representation and reconstruction In this section we discuss the three most important problems related with graphical object: • How to describe a, continuous, graphical object? • How to discretize a graphical object? • How to obtain a continuous graphical object from one of its discretizations? 1.5.1 Description Describing a graphical object consists in devising its (continuous) mathematical definition. Therefore, this is a problem posed in the mathematical universe. Depending on the nature of the graphical object this topic an important one in different areas of computer graphics. The description of the geometry and topology of
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210 BIBLIOGRAPHY Bibliography Atall
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212 BIBLIOGRAPHY Hansen, E. 1988. G
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214 BIBLIOGRAPHY Ratschek, H., & Ro
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Index abstraction paradigm, 5 accum
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Mathematical Optimization in Graphics and Vision - Luiz Velho - Impa

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