Interactive Computer Graphics Implementation of a Renderer James Gain and Edwin Blake Department of Computer Science University of Cape Town July 2002 jgain@cs.uct.ac.za Collaborative Visual Computing Laboratory

Interactive Computer Graphics Contents Map of the Lecture Introduction and Discussion of Practicals Line Segment and Polygon Clipping Cohen-Sutherland and Liang-Barsky Clipping Hidden Surface Removal z-Buffer and Painter’s Algorithm Scan Conversion DDA and Bresenham Line Drawing Flood Fill and Scan line Polygon Drawing Aliasing and Antialiasing 17/02/2019 Interactive Computer Graphics Contents

Practical 1: Software Rendering Overview: Emulate hardware rendering functionality by implementing it in software Due Date: Wednesday 11 September Requirements: Polygon Scan Conversion using an Edge Table - Ch. 7 Phong and Gouraud Shading - Ch. 6 Coordinate Transformations - Ch. 4 z-Buffer and Backface Removal - Ch. 7 Extra details appear in handout from Foley, Van Dam, Feiner and Hughes: “Computer Graphics: Principles and Practice” 17/02/2019 Interactive Computer Graphics Contents

Practical 2: Scenegraph Construction Overview: Design of Scenegraph nodes, allowing interactive hierarchical scene construction and visualization Due Date: Monday 7 October Requirements: (from Ch. 8 - Hierarchical and Object-Oriented Graphics & Ch. 3 - Input and Interaction) Separator Node, Lighting and Texturing Node, Transform and Material Node, Model Node Insert, update and traverse the scenegraph with mouse and keyboard input Must display a visualization of the scenegraph 17/02/2019 Interactive Computer Graphics Contents

Practical 3: Simple Scenegraph Animation Overview: Extend Scenegraph prac to include an animation node. Create a creature animation Due Date: Friday 17 October Requirements: VRML Node - parse and display an object in VRML format Animating Transform Node - allow animated rotation, translation and scaling, linked to a timer Create an animating model Opportunity to used a modelling package such as Teddy or Blender 17/02/2019 Interactive Computer Graphics Contents

Major Tasks of a Renderer Modelling: Produce sets of vertices that define geometric objects Geometric Processing: Determine which objects should appear and in what colour Normalization  Clipping  Hidden Surface Removal  Shading Vertex-based: floating point processing in 3-D Rasterization: Generate frame buffer pixels by Scan Conversion Pixel-based: integer processing in 2-D Display: Pixels are read from the frame buffer onto the CRT Must overcome display artefacts through Antialiasing Modelling must balance number of vertices against an accurate depiction of objects 17/02/2019 Interactive Computer Graphics Contents

Implementation Strategies Must loop over pixels, geometric primitives, light sources. Which is the outer loop? Image-Oriented (Sort-First): for(each_pixel) assign_a_colour(pixel); Need a data structure to indicate which objects affect which pixels Requires only limited display memory but is computation intensive Object-Oriented (Sort-Last): for(each_object) render(object); Supports a pipelined approach but is memory intensive An Object-Oriented Pipeline 17/02/2019 Interactive Computer Graphics Contents

Line-Segment Clipping Task: Decide which primitives will appear on the display Clipping relationship between primitive and view volume: Accept if fits entirely inside Reject (cull) if falls entirely outside Clip if crosses the boundary Placing Clipping in the pipeline: Many possibilities - during modelling, in 3-D before projection, in 2-D after projection OpenGL clips in 3-D Normalized Device Coordinates before final Orthographic Projection 17/02/2019 Interactive Computer Graphics Contents

Cohen-Sutherland Clipping I Motivation: Many line segments can be trivially accepted/rejected. Want to avoid intersection calculation in these cases Seek to replace FP division with FP subs and bit ops Create algorithm extensible to 3-D Outcode: Extend clipping edges to divide space in and around the clipping rectangle into nine regions Each region has a unique 4-bit binary number o = b0b1b2b3 If P = (x,y) then Likewise b1 is y < ymin, b2 is x > xmax, b3 is x < xmin 17/02/2019 Interactive Computer Graphics Contents

Cohen-Sutherland Clipping II Exercise: What are the outcodes for A-J? A = 0000; B = 0000 C = 0000; D = 0010 E = 0010; F = 0010 G = 1001; H = 1000 I = 0001; J = 1000 Cases (line segment with endpoint outcodes o1, o2): (o1 = o2 = 0) Both endpoints inside (o1 ≠ 0, o2 = 0; or vice versa) One endpoint inside, the other out. Nonzero outcode indicates endpoint for shortening (o1 & o2 ≠ 0) Do bitwise AND. If nonzero, both endpoints lie outside the same side  discard (o1 & o2 = 0) Both endpoints outside but on different clipping edges. Cannot tell from outcodes alone whether to shorten or discard 17/02/2019 Interactive Computer Graphics Contents

Liang-Barsky Clipping Use parametric form: Endpoints p1, p2; p(a) = (1 - a) p1 + a p2 a > 1 beyond p2 a < 0 before p1 Algorithm: Find a1-4 where line intersects clipping edges (ymin, xmin, ymax, xmax) Examine values and order of a to determine necessary clipping Example: 1 > a4 > a3 > a2 > a1 > 0, intersections in original line segment, innermost a2, a3 determine clipped segment 1 > a4 > a2 > a3 > a1 > 0, intersections in original segment, reject because intersects top and bottom before intersecting left Advantage: avoids multiple shortenings of segments, extensible to 3-D Can rework a calculation to avoid FP division Can also use slope to help 17/02/2019 Interactive Computer Graphics Contents

Interactive Computer Graphics Contents Polygon Clipping Often want to clip polygons against other polygons: For display, shadow generation, hidden surface removal Apply successive line clipping to a polygon Concave Polygons: Can increase the number of polygons or have edges that overlap So, restrict to convex polygons by tesselating concave polygons Sutherland-Hodgeman: Treat line segment clipper as a black box which clips polygon against a line Takes a pair of endpoints returns clipped endpoints Can be linked into a pipeline Show example of problems with concave polygons on board 17/02/2019 Interactive Computer Graphics Contents

Clipping Other Primitives Complex shapes are often represented by approximating polygons Standard clippers apply but may be inefficient Bounding Boxes: Find the axis-aligned extents of the object, use this to trivially accept/reject object Polynomial Curves and Surfaces: Higher than cubic require numerical intersection tests So, approximate by segments or polygons Text: Stroke - approximate with line segments and clip Raster - scissor the bitmapped character in the frame buffer From here on Tuesday 17/02/2019 Interactive Computer Graphics Contents

Interactive Computer Graphics Contents Clipping in 3-D Clipping against a 3-D bounding volume rather than a 2-D bounding region Clipping algorithms (Cohen-Sutherland, Liang-Barsky and Sutherland-Hodgeman) extend to 3-D Perspective Viewing: Clip against the view frustum BUT, easier to clip against a parallelepiped aligned with the axes OpenGL: project to normalized orthographic view then clip Lesson: in pipelining, always consider the process as a whole 17/02/2019 Interactive Computer Graphics Contents

Hidden-Surface Removal Hidden Surface Removal (Visible Surface Determination) seeks to discover which objects are visible or obscured from the viewpoint Object Space Approach: Consider polygons A and B pair-wise and determine if: (a) B partially obscures A, (b) A partially obscures B, (c) A and B are completely visible, (d) B obscures A (or vice versa) Image Space Approach: Follows ray-casting approach Ray leaves COP, passes through pixel and is intersected with polygon planes. Closest intersection gives visibility 17/02/2019 Interactive Computer Graphics Contents

Exercise: Complexity of HSR Problem: Determine the relative upper bound big-O complexity of object space and image space approaches, given k polygons and an n  m display Solution: Object Space - Pick one of the k polygons and compare it pairwise to the other k-1, remove polygon from consideration and repeat = O(k2) Image Space - For each of the n  m pixels intersect and order the k polygons = O(mnk) = O(k) 17/02/2019 Interactive Computer Graphics Contents

Interactive Computer Graphics Contents Back-Face Removal Streamline Hidden Surface Removal by first eliminating all back-facing polygons Front facing if angle  between polygon normal and view vector is -90 ≤  ≤ 90 cos  ≥ 0  n · v ≥ 0 In normalized device coordinates: Views are orthographic  v = (0,0,1) Need only check the sign of nk 17/02/2019 Interactive Computer Graphics Contents

Interactive Computer Graphics Contents z-Buffer Algorithm I Suited to pipelining and hardware Works in image space but loops over polygons z-buffer: Holds current closest intersection depth Same resolution as frame buffer Number of bits per pixel gives depth resolution Algorithm:  x, y: zbuffer(x, y) = MAX FOR each polygon for each pixel in polygon’s projection pz = polygons z-value at pixel coord (x,y) IF pz <= zbuffer(x, y) THEN zbuffer(x, y) = pz framebuffer(x, y) = polygons colour at pixel coord (x,y) 17/02/2019 Interactive Computer Graphics Contents

z-Buffer Algorithm II Amount of incremental work is small Rasterizing a polygon scan line by scan line Polygon is in a plane ax + by + cz + d = 0 Take two points (x1 , y1 , z1) and (x2 , y2 , z2) Rewrite in differential form a ∆x + b ∆y + c ∆z = 0 ∆x = x2 - x1 , ∆y , ∆z similar Across a scan line ∆y = 0 and ∆x is a constant step  ∆z = - c / a · ∆x No pre-sorting needed or object-object comparisons needed 17/02/2019 Interactive Computer Graphics Contents

Interactive Computer Graphics Contents Painter’s Algorithm Render a scene using oil painter’s approach Order polygons from back to front and render with new polygons overwriting old Painting Order of Polygons: IF no z overlap THEN Use depth sort order ELSEIF either x or y extents do not overlap THEN Use either order ELSEIF all vertices of P1 lie beyond the plane of P2 THEN render P1 before P2 Two problems: Cyclically overlapping polygons require cutting Piercing polygons require clipping 17/02/2019 Interactive Computer Graphics Contents

Interactive Computer Graphics Contents Scan-Line Algorithm Most efficient before advent of hardware Rasterize one screen scan-line at a time Exploit: Incremental depth calculations for polygons on the scan-line Coherence, since visibility status only changes at edges Example: line i , only one polygon ever active at a time so no depth comparisons needed Line j , enter a (one poly), enter c (two polys active so check depth), leave d (one poly) , leave b (background) Differs from z-buffer because it works on scan-lines rather than polygons 17/02/2019 Interactive Computer Graphics Contents

Interactive Computer Graphics Contents DDA Line Drawing DDA = Digital Differential Analyzer, named after an early differential equation solver. Generating a line is equivalent to solving a simple differential equation Assume slope 0 ≤ m ≤ 1 m = ∆y / ∆x => ∆y = m ∆x Increase ∆x by 1 on each iteration  ∆y = m Algorithm: for(i = x1; i <= x2; i++) { y += m; write_pixel(x, round(y), line_colour); } For different slopes change the roles of x and y 17/02/2019 Interactive Computer Graphics Contents

Bresenham Line Drawing Avoid floating point calculations Again assume slope 0 ≤ m ≤ 1 Next pixel can only be horizontal across or diagonally up Use a decision variable d = a - b, where a = distance to upper pixel, b = distance to lower pixel If d > 0 then across else diagonal Replace with d = ∆x(a - b) for integer ops (does not affect sign of discriminator) Calculating each successive pixel requires only an addition and a sign test 17/02/2019 Interactive Computer Graphics Contents

Bresenham Decision Variable Compute d incrementally: If y incremented on the previous step then a = a + 1 - m, b = m - 1 + b If y NOT incremented previously then a = a - m, b = b + m dk+1 = dk - { 2 ∆y if dk > 0 { 2(∆y - ∆x) otherwise 17/02/2019 Interactive Computer Graphics Contents

Drawing Polygons: Inside-Outside Tests To fill a polygon determine if a point P is inside or outside Crossing or Odd-Even Test: If P is inside then a ray emanating from P going to infinity will cross the polygon an odd number of times Winding Test: Fill the inside of non-simple polygons Winding number = num times that a point is encircled by the edges of the polygon Inside = winding num > 0 Filling with the odd-even test 17/02/2019 Interactive Computer Graphics Contents

Drawing Polygons: Flood Fill Rasterize edges of polygons into frame buffer Assume background is WHITE and drawing colour is BLACK Grow colouring out from a seed point inside the polygon until an edge is encountered Recursive Algorithm: flood_fill(int x, int y) { if(read_pixel(x, y) == WHITE) { write_pixel(x, y, BLACK); flood_fill(x-1, y); flood_fill(x+1, y); flood_fill(x, y-1); flood_fill(x, y+1); } } 17/02/2019 Interactive Computer Graphics Contents

Scan-Line Polygon Drawing I Generate pixels in the same order as they are displayed Group pixels into contiguous spans on a scan line by finding where polygon edges intersect the scan line The natural intersection order is based on processing successive polygon edges But, would like intersections sorted first by scan line and then by increasing x on the scan line With potentially many edge intersections an O(n log n) sort is burdensome Use an incremental y-x algorithm 17/02/2019 Interactive Computer Graphics Contents

Scan-Line Polygon Drawing II F E Scan-Line Polygon Drawing II C B A EF DE Use incremental calculations and exploit coherence Create a global Edge Table (ET): Edges are sorted into scan line buckets using the smallest vertex y value Within a scan line bucket do an insertion sort on increasing x Algorithm: use Active Edge Table (AET) to draw scan line y+1 Edges in AET with ymax = y are deleted Edges in ET with ymin = y+1 are added Update the x-intersection values of active edges using an incremental approach Fill pixels between pairs of active edge x-intersects yminx ymax yminx ymax 7 6  yminx ymax 5 CD 4  yminx ymax 3 FA 2  yminx ymax yminx ymax 1  AB BC 17/02/2019 Interactive Computer Graphics Contents

Interactive Computer Graphics Contents Singularities Polygon drawing breaks down if a scan line exactly intersects a vertex (singularity) Cases: Must count as 2 intersections Must count as 1 intersection Solution: Perturb any vertex that has an integer y value 17/02/2019 Interactive Computer Graphics Contents

Aliasing Original Scene Sampling Pixel Centres Rendered Image Occurs when the sampling inherent in rendering does not contain enough information for an accurate image. Original Scene Sampling Pixel Centres Rendered Image Scanline Luminosity Sampled Signal Luminosity Signal 17/02/2019 Interactive Computer Graphics Contents

Interactive Computer Graphics Contents Effects of Aliasing Common aliasing errors (called artefacts) are: jagged profiles, disappearing or improper fine detail, and disintegrating textures. Jagged Profiles Loss of Detail 17/02/2019 Interactive Computer Graphics Contents

Disintegrating Texture The checkers on a plane should become smaller with distance. But aliasing causes them to become larger and/or irregular. Increasing resolution only moves the artefact closer to the horizon. 17/02/2019 Interactive Computer Graphics Contents

Interactive Computer Graphics Contents Antialiasing Developed to combat aliasing Applies to all types of aliasing – PSC, RT and temporal Prefiltering: Treat a pixel as an area. Calculate overlap of polygon with pixel. Costly Postfiltering (supersampling): Compute pixel colour by averaging multiple samples. Preferred Method Sample the scene at n times the display resolution. For example, suppose the display resolution is 512x512. Sampling at three times the width and three times the height of the display resolution would yield 1536x1536 samples A filter provides the weights used to compute the average 17/02/2019 Interactive Computer Graphics Contents

Interactive Computer Graphics Contents Prefiltering No Antialiasing Prefiltering 17/02/2019 Interactive Computer Graphics Contents

Interactive Computer Graphics Contents Postfiltering Sampling: take jittered, regular or random samples. Averaging: use an filter to assign weights to each sample. Sum all the weighted samples to obtain a pixel. Jittered Regular 17/02/2019 Interactive Computer Graphics Contents

Interactive Computer Graphics Contents Filtering Example 17/02/2019 Interactive Computer Graphics Contents

Antialiasing Examples 17/02/2019 Interactive Computer Graphics Contents

Interactive Computer Graphics Contents 17/02/2019 Interactive Computer Graphics Contents

Interactive Computer Graphics Contents 17/02/2019 Interactive Computer Graphics Contents

Interactive Computer Graphics Contents 17/02/2019 Interactive Computer Graphics Contents