Graphics 2. Revision lecture. A list of module sections. 3D graphics pipeline

Transcription

1 Graphics Level 3 10 credits in Semester 2 Revision lecture Make a list of module sections Discuss topics under each heading Professor Aleš Leonardis Office: 235; (Office hours: Fri 2-4pm) a.leonardis@cs.bham.ac.uk A list of module sections 3D graphics pipeline (overview) Object construction Object rendering Colour Viewing and projections Animation Raster conversion Splines and spline surfaces Scan line area-fill Hidden surface removal Texture mapping Ray tracing 3D graphics pipeline Modelling coordinates: - world coordinate system, - object coordinate system Camera coordinates Screen/Window coordinates Device coordinates WORLD/SCENE/OBJECT/CONCEPT 3D MODELLING VIEWING 3D CLIPPING PROJECTION RASTERIZATION 2D PIXELMAP DISPLAY 1

2 Object construction understand 3D coordinate systems (right, le; handed) explain 3D transforma>ons (transla>on, scaling, rota>on) describe the benefits of the homogeneous coordinates know how to represent surfaces (polygon tables: vertex and surface tables (order in which we traverse the ver>ces)) know how to represent volumes (construc>ve solid geometry, oct- tree, sweep func>ons) explain the height maps and parametric surfaces Rotation in the right-handed coordinate system Positive angle of rotation is counter-clockwise when the axis about which it occurs points toward the observer Positive angle of rotation for Z axis Matrix representation Homogeneous coordinates Z axis points at the observer Y Common notation for ALL transformations Common computational mechanism for ALL transformations Simple mechanism for combining a number of transformations => computational efficiency X 2

3 Point transformation in homogeneous coordinates Implemented by matrix multiplication P = M P Volumetric Modules Constructive Solid Geometry x' y' z' 1 = a d g j b e h k c f i l x y z 1 Constructive solid geometry Octrees Source: 3

4 Define sweep path Sweep representations Sweep functions: implementation Translational sweep Define a shape as a polygon vertex table Define a sweep path as a sequence of translation vectors T1 T2 Translate the shape, continue building a vertex table Define shape Define a surface table Spherical coordinates Superquadrics Superellipsoid x = r cosφ cosθ y = r cosφ sinθ z = rsinφ s1 x = r x cos φ cos s y = r cos 1 y φ sin z = r sin s1 z φ s2 s2 θ θ π / 2 φ π / 2 π θ π π / 2 φ π / 2 π θ π S1=S2=0.5 S1=S2=3.0 4

5 Rendering understand the role of objects (geometry, colour, microstructure (reflectance)) lights a camera in rendering characterize different light sources (ambient, directional: diffuse, directional: point source, divergent) explain camera parameters (a pinhole camera) know the shading models (ambient, diffuse, specular) explain the differences among the three approaches for computing shading for polygonal surfaces (flat shading, Gouraud shading, Phong shading) Rendering: setting up the scene Given Object surfaces Light sources Camera Compute Colour of each pixel on the screen This is colour that bounces off the surface point and goes in the direction of the camera (viewer) Normal vectors Normal vectors Computing normal vectors A cross-product of two vectors is a vector perpendicular (orthogonal, normal) to both input vectors if n = a x b, n a and n b Cross product is NOT commutative: a x b b x a although both cross-products are orthogonal to a and b Flat surface patch Curved surfaces 5

6 Computing cross product E1=V2 V1 Computing cross-product E2=V3 V2 V3 E2 Front face E1=V2 V1 E2=V3 V2 N1=E1 x E2 = 1 x 1 y 1 z x 2 x 1 y 2 y 1 z 2 z 1 x 3 x 2 y 3 y 2 z 3 z 2 Unit vectors V1 E1 V2 N1 N1=E1 x E2 1 x (y 2 y 1 )(z 3 z 2 ) (y 3 y 2 )(z 2 z 1 ) + 1 y (x 3 x 2 )(z 2 z 1 ) (x 2 x 1 )(z 3 z 2 ) + 1 z (x 2 x 1 )(y 3 y 2 ) (x 3 x 2 )(y 2 y 1 ) Surface visibility from surface normal E1=V2 V1 E2=V3 V2 N1=E1 x E2 = 1 x 1 y 1 z x 2 x 1 y 2 y 1 z 2 z 1 x 3 x 2 y 3 y 2 z 3 z 2 1 x (y 2 y 1 )(z 3 z 2 ) (y 3 y 2 )(z 2 z 1 ) + Unit vectors Inputs to computation Light sources (emitters) Colour (emission spectrum) Geometry (position and direction) Directional attenuation Surfaces (reflectors) Colour (reflectance and absorption spectrum of the material) Geometry (position, orientation of each surface patch) Micro-structure 1 y (x 3 x 2 )(z 2 z 1 ) (x 2 x 1 )(z 3 z 2 ) + 1 z (x 2 x 1 )(y 3 y 2 ) (x 3 x 2 )(y 2 y 1 ) N z 6

7 Surfaces Computing reflectance: shading model Micro-structure Defines reflectance properties Reflectance Diffuse: Matte surfaces Specular: Shiny surfaces + Requires Surface geometry, microstructure and colour Positions and type of light sources Position of the viewer (camera) Combines the three contributions: Ambient light Diffuse reflectance Specular reflectance Pixel colour: Ambient + Diffuse + Specular A complete shading model Shading model Combines all the terms: Pixel colour: Ambient + Diffuse + Specular I = A + D + S I = K a I a + K d I d cos θ d + K s I s (cos δ) n I r = K r I ar + K r I dr cos θ d + K s I sr (cos δ) n I g = K g I ag + K g I dg cos θ d + K s I sg (cos δ) n I b = K b I ab + K b I db cos θ d + K s I sb (cos δ) n Diffuse term: Reflectance Object surface: Lambertian (matte) Light reflected equally in every direction The amount of light reflected depends on the angle between the direction of light and a surface normal at each point Defined by the Cosine Law I = I d cos(θ) π/2 < θ < π/2 I d L K d θ N I 7

8 Shading model Algorithms for shading of surfaces Specular term: Reflectance Object surface: glossy Extends the ideal (mirror) case The reflected light forms a cone around the ideal (Snell-law) reflectance vector Shading model so far showed how to compute reflectance for individual points on a surface Shading varies across surfaces Point-by-point computation very expensive Three approaches for computing shading for polygonal surfaces Flat shading Gouraud shading Phong shading Colour Colour mapping functions know the origins of colour (spectral characteristics, human visual perception) know different colour spaces and how to convert between them (RGB, CIE XYZ, HSV, CMY) explain the role of Colour Lookup Table (CLUT) describe the Colour Mapping Functions 8

9 Viewing and projections Creating a view of the scene an outline define a virtual camera (view reference point (VRP), direction of gaze, view-up direction, viewing distance) understand how to implement a virtual snapshoot show understanding of viewing projections (parallel, perspective) define and explain a pinhole camera model 1. Create vertex tables (3D) for an object in the World coordinate system. 2. Define the (3D) Viewing (camera) coordinate system. 3. Change the 3D coordinates of the object from the World system to the Viewing system. 4. Create (2D) perspective projection of the object. 5. Plot the 2D vertices, edges and surfaces. Animation In-betweening - parametric equations explain how anima>on works know how to create the anima>on sequences explain the no>on of key frames describe in- betweening know about different animated models (rigid, ar>culated, dynamic, par>cle- based, behavior based) explain double buffering Parametric equations a flexible tool for interpolation Example for line segment between two points, (x n,y n ) and (x n+1,y n+1 ) calculate points in between the two given points x i = x n + t (x n+1 - x n ) y i = y n + t (y n+1 - y n ) t is the parameter which always changes between 0 and 1 when t = 0, we get x n when t = 1 we get x n+1 for 0 < t < 1 we get the points in between 9

10 In-betweening In-betweening should use interpolation based on the nature of the path, for example: straight path linear interpolation circular path angular interpolation irregular path linear interpolation spline Raster conversion be able to describe raster conversion algorithms (accuracy, speed) explain DDA (digital differential analyzer) algorithm describe Bresenham s line algorithm For in-betweening use parametric representation of lines and curves, e.g. line segment circle Bezier curve Splines and spline surfaces 3D Bezier patches defined on a regular grid be able to calculate Bezier curves given a set of control points explain the construc>on of Bezier surfaces describe the difference between Bezier curves and B- splines 10

11 Scan line area-fill know the purpose of scan line algorithms be able to outline the scan line algorithm Scan-line algorithm - outline For each scan line (each y-coordinate) Compute x coordinates of the intersections of the current scan line with all edges Sort these edge intersections by increasing x value Group the edge intersections by pairs (vertex intersections require special processing) Fill in the pixels on the scan line between pairs of values Hidden surface removal be able to categorize hidden surface removal methods (object- space, image- space) explain back face removal (polygon culling) describe painter s algorithm know about Z- buffer algorithm (complexity, storage requirements) explain scan- line methods and subdivision methods compare different algorithms for surface removal Recommendations for hidden surface methods Surfaces are distributed in z Surfaces are well separated in y Depth sorting Scan-line or area-subdivision Only a few surfaces present Depth sorting or scan-line Scene with at least a few thousand surfaces Depth-buffer method or area-subdivision 11

12 Texture mapping understand the advantages and disadvantages of texture mapping (simple geometry + texture mapping) know the difference between forwards and backwards texture mapping explain the steps involved in texture mapping based on intermediate surfaces (cylinders, spheres) describe environment mapping describe bump mapping (emulates altering normal vectors during the rendering process) explain the difference between bump mapping and displacement mapping describe the aliasing problem and how to alleviate it (interpola>on, filtering, MIP mapping) Three types of mapping Texture mapping Texture image mapping Uses images to fill inside of polygons Environment ( reflection mapping) Uses a picture of the environment for texture maps Bump mapping Emulates altering normal vectors during the rendering process Texture mapping: environment maps Instead of using the ray from the surface point to the projected texture's centre, we use the direction of the reflected ray to index a texture map 12

13 Displacement mapping and Bump mapping Displacement vs. bump mapping Displacement mapping [A. Watt, 3D computer graphics]: Height field is used to perturb a surface point along the direction of its surface normal. Not convenient to implement since the map must perturb the geometry of the model rather than modulate parameters in the shading equation. Bump mapping [A. Watt, 3D computer graphics]: A perturbation is applied to the surface normal according to the corresponding value in the map. If the surface normal is perturbed then the shading changes and the surface that is rendered looks as if it is textured. Ray tracing be able to explain ray tracing technique (reflection, refraction, or absorption) and argue for increased realism with respect to classical techniques (shadows, transparency, reflections and self-reflections) describe ray tracing pipeline (ray generation, ray traversal, intersection, shading) argue why ray tracing is computationally very demanding and explain means to alleviate the problem compare ray tracing versus rasterisation 13

14 Ray tracing speeding up the calculations Bounding Volumes Enclose groups of objects in sets of hierarchical bounding volumes (Octree) First test for intersection with the bounding volume Then only if there is an intersection, against the objects enclosed by the volume. Ray tracing versus rasterization RT gives a very high degree of visual realism, which can not be achieved with rasterization (correct shadows, reflection, transparent and translucent objects etc.) RT is generally slower than rasterization (due to big amount of generated secondary rays). On the other hand, RT is scalable and very well suited for parallel computing (in fact, each ray can be traced independently, so one can exploit as many processors as he has). Rasterization algorithms are easier to implement. Rasterization algorithms are better suited for implementations on GPUs. Summary Good luck! Handouts On-line exercises Computer Graphics, Hearn D & Baker M & Carithers W R, D Computer Graphics, Watt A, hr examination (100%) 14