The Viewing Pipeline Coordinate Systems

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1 Overview Interactive Graphics System Model Graphics Pipeline Coordinate Systems Modeling Transforms Cameras and Viewing Transform Lighting and Shading Color Rendering Visible Surface Algorithms Rasterization Hardware 3D Model Model coords The Viewing Pipeline Display 3D World World Coords Rendering Lighting Specification Camera coords Viewing Transform Camera Specification Projection to 2D Screen coords 3D Clipping Clipping coords ITCS 6140/ Review: Computer Graphics ITCS 6140/ Review: Computer Graphics The Viewing Pipeline Coordinate Systems Model Coordinates World Coordinates Viewing (Camera, Eye) Coordinates Perspective/Image Coordinates Screen/Display Coordinates ITCS 6140/ Review: Computer Graphics ITCS 6140/ Review: Computer Graphics

2 Graphics Primitives Point Line, Polyline, Polygon, Rectangle Cubic Curves and Surfaces (Bezier, B-Spline, URBS) Circle, Ellipse Marker, Polymarker Text Texture (2D, 3D), Opacity Graphics Primitive Attributes Lines: style, width Markers: style, size Color (R,G,B or index into lookup table) Filled Primitives: style, pattern Text: style (font), size, origin ITCS 6140/ Review: Computer Graphics ITCS 6140/ Review: Computer Graphics Definition In 3D, point P (P x, P y, P z ) is transformed into Q(Q x, Q y, Q z ) as follows: Affine Transforms Most common in computer graphics. Permits a simple matrix representation Can be composed Q x Q y Q z = ap x + dp y + gp z + t x = bp x + ep y + hp z + t y = cp x + fp y + kp z + t z [ ] Qx Q y Q z = a d g b e h P x P y + t x ty c f k P z t z Q = M P + T Primitive Affine Transforms: Scaling, Rotation and Translation. It would be more convenient to absorb the translation into the matrix. ITCS 6140/ Review: Computer Graphics ITCS 6140/ Review: Computer Graphics

3 Homogeneous Coordinates A mathematical convenience that allows all 3 primitive transforms to use a matrix representation. Add a fourth coordinate, w, to each point and a column and row to each affine transform matrix. To convert from P H to P 2D, divide each coordinate by the w coordinate and discard the 3rd coordinate. w = 0 represents points at infinity. P H = (P x, P y, P z, P w ) P 3d = (P x /P w, P y /P w, P z /P w ) Translation Scale Modeling Transformations t x t T (t x, t y, t z ) = y t z s x s S(s x, s y, s z ) = y s z 0 ITCS 6140/ Review: Computer Graphics ITCS 6140/ Review: Computer Graphics Rotation R z (θ) = R x (θ) = R y (θ) = cosθ sinθ 0 0 sinθ cosθ cosθ sinθ 0 0 sinθ cosθ 0 cosθ 0 sinθ sinθ 0 cosθ 0 Cameras and 3D Viewing A mechanism for specifying the the viewer location, orientation and volume of the world that is of interest. A pin-hole camera model is assumed. Z Y V X U VIEWER (CAMERA) ITCS 6140/ Review: Computer Graphics ITCS 6140/ Review: Computer Graphics

4 Camera Parameters View Reference Point, Look At vector,. Up Vector, V. Vector U = V Window Camera (viewer) position. V RP. ear and Far clipping planes. These parameters specify a view volume (frustum) that is a truncated pyramid. Viewer View Volume Frontt Projection Planet The volume of 3D space that is projected and displayed. Back Volume defined by a 2D window on the projection plane and front and back clipping planes (hither/yon, near/far). View volume is specified in camera coordinates. View volume is a truncated frustum. ITCS 6140/ Review: Computer Graphics ITCS 6140/ Review: Computer Graphics Viewing Transform Y Viewing Transform Z P (x, y, z) (0, 0, 0) World Coordinates (Right Handed) Q (a, b, c) V X U (rx, ry, rz) Camera Coordinates (Left Handed) Object points in world coordinates are transformed into the camera coordinate system with The viewer at the origin. Look-at vector along the negative Z axis. The Up vector along the Y axis. The viewing transform is an affine transform, consisting of a rotation and translation, as follows: u x u y u z r x M wv = v x v y v z r y n x n y n z r z with r = ( r. U, r. V, r. ) ITCS 6140/ Review: Computer Graphics ITCS 6140/ Review: Computer Graphics

5 Geometric Projections Projection An operation that transforms points in -dimensional space to another of dimensionality less than n Geometric Projection Application to geometric coordinate systems (Cartesian, Polar, Spherical and Cylindric systems) Types of Projections Planar on-planar Application to Computer Graphics 1. Viewing 3D environments on a 2D screen. 2. Visualization of multi-variate data from scientific applications. ITCS 6140/ Review: Computer Graphics OBLIQUE Planar Projections PARALLEL AXOOMETRIC PLAAR ORTHOGRAPHIC ISOMETRIC DIMETRIC TRIMETRIC MULTI VIEW 1 pt PERSPECTIVE ITCS 6140/ Review: Computer Graphics 2 pt 3 pt Perspective Projection Parallel Projection Projectors are parallel to each other. Perspective Projectors converge to a point. Projectors converge to a point. All points on the same projector project to the same point, destroying depth information. Depth information must be carried along the pipeline for use in visible surface calculations. Perspective transformation effectively involves a division of the coordinates of each 3D point by its depth (from the view point). t P P 1 P(pu, pv, pn) P 2 E(eu, ev, en) Projection Plane at n = 0 ITCS 6140/ Review: Computer Graphics ITCS 6140/ Review: Computer Graphics

6 Perspective Transformation P(pu, pv, pn) E(eu, ev, en) t P P 1 P 2 Assuming p n = p n 1 p n /e n as a pseudo-depth, we can write P = (p u, p v, p n, 1 p n /e n ) Projection Plane at n = 0 If P = (p u, p v, p n ) is an object point, E = (e u, e v, e n ) is the camera position and P = (p u, p v, p n) is the projection point, assuming the projection plane contains the origin of the camera coordinate system, it can be shown that p u p u = 1 p n /e n p v p v = 1 p n /e n with = [p u, p v, p n, 1] M p M p = /e n ITCS 6140/ Review: Computer Graphics ITCS 6140/ Review: Computer Graphics Lighting and Shading The Rendering Equation where [ ] I(x, x ) = g(x, x ) ɛ(x, x ) + ρ(x, x, x )I(x, x )dx S I(x, x ) = Intensity of light passsing from x to x. g(x, x ) = Geometry term ɛ(x, x ) = Emitted light intensity from x to x. ρ(x, x, x ) = Light intensity scattered from x to x by a patch of surface at x. Approximations of the Rendering Equation The local lighting model, Ray Tracing Model, Radiosity and Distributed Ray Tracing can all be shown to be approximations of the rendering equation. ITCS 6140/ Review: Computer Graphics ITCS 6140/ Review: Computer Graphics

7 Local Lighting Model Consists of 3 components, ambient, diffuse and specular Ambient Reflection I a = k a I a I a is the ambient intensity, constant for all objects, and k a, the ambient reflection coefficient. Use to model illumination that arrive at object points through indirect sources, such as nearby objects through multiple reflections or scattering. Diffuse Reflection I d = I p k d cosθ = I p k d ( L) where is the unit normal to the surface and L, the vector (of unit length) to the light source, I p the light source intensity. We approximate diffuse reflection using Lambert s law. Intensity of light reflected is a constant, and independent of viewer position. ITCS 6140/ Review: Computer Graphics ITCS 6140/ Review: Computer Graphics Illumination Model Specular Reflection (Phong Model) Including ambient and diffuse reflections, the local lighting model becomes L θ θ α R V Cos ( α) 4 Cos α 64 Cos α α 90 α 90 α 90 I = I a + I d + I s I = k a I a + I p [k d ( L) + k s ( R V ) n ] Assuming a wavelength based distribution of light, the model becomes I s = k s cos n α = k s ( R V ) n I λ = I aλ k a + f att I pλ [k dλ cos( L) + k s λ( R V ) n ] If there are l light sources, in the environment then I λ = I aλ k a + l f atti I pλi [k d λcos( L i ) + k s λ( R i V ) n ] 1 ITCS 6140/ Review: Computer Graphics ITCS 6140/ Review: Computer Graphics

8 Rendering Algorithms Phong Shading Gouraud Shading a 4 1 A v y 1 y s y 2 3 I 2 Face ormals are averaged to obtain vertex normals. I a I 1 I p Vertex Intensities (I 1, I 2, I 3 ) are obtained by application of a lighting model. ITCS 6140/ Review: Computer Graphics 2 I b I 3 b Compute ormals at vertices of polygon. B Interpolate each component of normal at any interior point of polygon Apply illumination model. Can easily be incorporated into a scanline algorithm. ITCS 6140/ Review: Computer Graphics C c Visible Surface Algorithms Z (Depth) Buffer Algorithm (Catmull 74) A buffer of Z or depth values is maintained, one for each pixel, in addition to the frame buffer. During scan conversion, if primitive s depth is less (closer) than the existing value in the Z buffer, replace it and corresponding color (intensity) in the frame buffer. Each primitive is scanconverted once, independent of any other primitive and in any order. Z (depth) can be calculated from the polygon plane equation, or interpolated from depth at polygon vertices Algorithm: For each object primitive { For each pixel in projection { Z = primitive s depth value at pixel(x,y). if Z < Z BUF[x][y] { Write ZBUF (x, y, Z) Write Pixel (x, y, pixel color) } } } ITCS 6140/ Review: Computer Graphics ITCS 6140/ Review: Computer Graphics

9 Rasterization Scanline Polygon Fill Algorithm With some modifications and additional parameters, is used to scanconvert polygons along with the Z buffer algorithm Approach Exploit the geometry of the polygon. Use scanline and edge coherence for efficient filling. j E I I 0 1 I 2 C I3 Algorithm for each scanline { Calculate Intersections between scanline and all edges of the polygon. Sort Intersections on X. Fill pixels between pairs of intersection points. } D A B ITCS 6140/ Review: Computer Graphics ITCS 6140/ Review: Computer Graphics Color Models RGB Color Model Blue Cyan CMY Color Model Complement of the RGB model. Magenta Black (0,0,0) White (1, 1, 1) Green Uses subtractive primaries, Cyan, Magenta, Yellow. Application C M Y = R G B Red Yellow HardCopy Devices, such as ink jet color printers. Application Color CRT Monitors. ITCS 6140/ Review: Computer Graphics ITCS 6140/ Review: Computer Graphics

10 HSV Color Model green V yellow Display Architecture cyan white red blue magenta CPU Display Processor Peripheral Devices black H S System Bus Model defined in a cylindrical coordinate system. Colors are defined in a hexcone or six-sided pyramid. Based on the artist s model of tints, shades and tones. Hue is measured around the vertical axis. Saturation ranges from 0 on the centerline to 1 on the triangular sides. Value ranges from 0.0 (black) at the apex of the pyramid to 1.0 (white) at the base. ITCS 6140/ Review: Computer Graphics System Memory Frame Buffer Video Controller Display Video Controller reads the frame buffer through a special port. A single address space system. Memory contention is still a problem, but the flexibility of a single address space (for raster ops, for instance) outweighs this factor. ITCS 6140/ Review: Computer Graphics Graphics Processing Units (GPU) Modern graphics cards (vidia GEForce, ATI Radeon etc) are extremely powerful computing platforms for processing/shading large amounts of geometry Pipelined architecture Support for 3D textures (multitexturing, dependent texturing) May be used for general purpose computing Examples: Accelerated ray tracing, solving linear systems, fluid flow. ITCS 6140/ Review: Computer Graphics

Visualisatie BMT. Rendering. Arjan Kok

Visualisatie BMT. Rendering. Arjan Kok Visualisatie BMT Rendering Arjan Kok a.j.f.kok@tue.nl 1 Lecture overview Color Rendering Illumination 2 Visualization pipeline Raw Data Data Enrichment/Enhancement Derived Data Visualization Mapping Abstract