From Vertices to Fragments Chapter 7 Part I

From Vertices to Fragments Chapter 7

Part I Objectives Introduce basic implementation strategies n Clipping n Scan conversion n CS 480/680 Chapter 7 -- From Vertices to Fragments 2

Overview At end of the geometric pipeline, vertices have been assembled into primitives n Must clip out primitives that are outside the view frustum n n n Algorithms based on representing primitives by lists of vertices Must find which pixels can be affected by each primitive Fragment generation n Rasterization or scan conversion n CS 480/680 Chapter 7 -- From Vertices to Fragments 3

Required Tasks Clipping n Rasterization or scan conversion n Transformations n Some tasks deferred until fragment processing n Hidden surface removal n Antialiasing n CS 480/680 Chapter 7 -- From Vertices to Fragments 4

Rasterization Meta Algorithms Consider two approaches to rendering a scene with opaque objects n For every pixel, determine which object that projects on the pixel is closest to the viewer and compute the shade of this pixel n n n Ray tracing paradigm For every object, determine which pixels it covers and shade these pixels Pipeline approach n Must keep track of depths n CS 480/680 Chapter 7 -- From Vertices to Fragments 5

Clipping n n 2 D against clipping window 3 D against clipping volume Easy for line segments polygons Hard for curves and text n Convert to lines and polygons first CS 480/680 Chapter 7 -- From Vertices to Fragments 6

Clipping 2 D Line Segments n Brute force approach: compute intersections with all sides of clipping window n Inefficient: one division per intersection CS 480/680 Chapter 7 -- From Vertices to Fragments 7

Cohen-Sutherland Algorithm Idea: eliminate as many cases as possible without computing intersections n Start with four lines that determine the sides of the clipping window n y = ymax x = xmin x = xmax y = ymin CS 480/680 Chapter 7 -- From Vertices to Fragments 8

The Cases n Case 1: both endpoints of line segment inside all four lines n Draw (accept) line segment as is y = ymax x = xmin x = xmax y = ymin n Case 2: both endpoints outside all lines and on same side of a line n Discard (reject) the line segment CS 480/680 Chapter 7 -- From Vertices to Fragments 9

The Cases n Case 3: One endpoint inside, one outside n n Must do at least one intersection Case 4: Both outside May have part inside n Must do at least one intersection n y = ymax x = xmin CS 480/680 x = xmax Chapter 7 -- From Vertices to Fragments 10

Defining Outcodes n For each endpoint, define an outcode b 0 b 1 b 2 b 3 b 0 = 1 if y > ymax, 0 otherwise b 1 = 1 if y < ymin, 0 otherwise b 2 = 1 if x > xmax, 0 otherwise b 3 = 1 if x < xmin, 0 otherwise Outcodes divide space into 9 regions n Computation of outcode requires at most 4 subtractions n CS 480/680 Chapter 7 -- From Vertices to Fragments 11

Using Outcodes Consider the 5 cases below n AB: outcode(A) = outcode(B) = 0 n n Accept line segment CS 480/680 Chapter 7 -- From Vertices to Fragments 12

Using Outcodes n CD: outcode (C) = 0, outcode(D) 0 Compute intersection n Location of 1 in outcode(D) determines which edge to intersect with n Note if there were a segment from A to a point in a region with 2 ones in outcode, we might have to do two intersections n CS 480/680 Chapter 7 -- From Vertices to Fragments 13

Using Outcodes n EF: outcode(E) logically ANDed with outcode(F) (bitwise) 0 Both outcodes have a 1 bit in the same place n Line segment is outside of corresponding side of clipping window n reject n CS 480/680 Chapter 7 -- From Vertices to Fragments 14

Using Outcodes GH and IJ: same outcodes, neither zero but logical AND yields zero n Shorten line segment by intersecting with one of sides of window n Compute outcode of intersection (new endpoint of shortened line segment) n Reexecute algorithm n CS 480/680 Chapter 7 -- From Vertices to Fragments 15

Efficiency n In many applications, the clipping window is small relative to the size of the entire data base n n Most line segments are outside one or more side of the window and can be eliminated based on their outcodes Inefficiency when code has to be reexecuted for line segments that must be shortened in more than one step CS 480/680 Chapter 7 -- From Vertices to Fragments 16

Cohen Sutherland in 3 D Use 6 -bit outcodes n When needed, clip line segment against planes n CS 480/680 Chapter 7 -- From Vertices to Fragments 17

Liang-Barsky Clipping n Consider the parametric form of a line segment p(a) = (1 -a)p + ap 1 a 0 1 2 p 1 n We can distinguish between the cases by looking at the ordering of the values of a where the line determined by the line segment crosses the lines that determine CS 480/680 Chapter 7 -- From Vertices to Fragments 18

Liang-Barsky Clipping n In (a): a 4 > a 3 > a 2 > a 1 n n Intersect right, top, left, bottom: shorten In (b): a 4 > a 2 > a 3 > a 1 n Intersect right, left, top, bottom: reject CS 480/680 Chapter 7 -- From Vertices to Fragments 19

Advantages Can accept/reject as easily as with Cohen. Sutherland n Using values of a, we do not have to use algorithm recursively as with C-S n Extends to 3 D n CS 480/680 Chapter 7 -- From Vertices to Fragments 20

Clipping and Normalization General clipping in 3 D requires intersection of line segments against arbitrary plane n Example: oblique view n CS 480/680 Chapter 7 -- From Vertices to Fragments 21

Plane-Line Intersections CS 480/680 Chapter 7 -- From Vertices to Fragments 22

Normalized Form top view before normalization n n after normalization Normalization is part of viewing (pre clipping) but after normalization, we clip against sides of right parallelepiped Typical intersection calculation now requires only a floating point subtraction, e. g. is x > xmax ? CS 480/680 Chapter 7 -- From Vertices to Fragments 23

CS 480/680 Chapter 7 -- From Vertices to Fragments 24

Part II Objectives Introduce clipping algorithms for polygons n Survey hidden-surface algorithms n CS 480/680 Chapter 7 -- From Vertices to Fragments 25

Polygon Clipping n Not as simple as line segment clipping Clipping a line segment yields at most one line segment n Clipping a polygon can yield multiple polygons n n However, clipping a convex polygon can yield at most one other polygon CS 480/680 Chapter 7 -- From Vertices to Fragments 26

Tessellation and Convexity One strategy is to replace nonconvex (concave) polygons with a set of triangular polygons (a tessellation) n Also makes fill easier n Tessellation code in GLU library n CS 480/680 Chapter 7 -- From Vertices to Fragments 27

Clipping as a Black Box n Can consider line segment clipping as a process that takes in two vertices and produces either no vertices or the vertices of a clipped line segment CS 480/680 Chapter 7 -- From Vertices to Fragments 28

Pipeline Clipping of Line Segments n Clipping against each side of window is independent of other sides n Can use four independent clippers in a pipeline CS 480/680 Chapter 7 -- From Vertices to Fragments 29

Pipeline Clipping of Polygons Three dimensions: add front and back clippers n Strategy used in SGI Geometry Engine n Small increase in latency n CS 480/680 Chapter 7 -- From Vertices to Fragments 30

Bounding Boxes n Rather than doing clipping on a complex polygon, we can use an axis-aligned bounding box or extent n Smallest rectangle aligned with axes that encloses the polygon n Simple to compute: max and min of x and y CS 480/680 Chapter 7 -- From Vertices to Fragments 31

Bounding boxes n Can usually determine accept/reject based only on bounding box reject accept requires detailed clipping CS 480/680 Chapter 7 -- From Vertices to Fragments 32

Clipping and Visibility Clipping has much in common with hiddensurface removal n In both cases, we are trying to remove objects that are not visible to the camera n Often we can use visibility or occlusion testing early in the process to eliminate as many polygons as possible before going through the entire pipeline n CS 480/680 Chapter 7 -- From Vertices to Fragments 33

Hidden Surface Removal n Object-space approach: use pair-wise testing between polygons (objects) partially obscuring n can draw independently Worst case complexity O(n 2) for n polygons CS 480/680 Chapter 7 -- From Vertices to Fragments 34

Painter’s Algorithm n Render polygons a back to front order so that polygons behind others are simply painted over B behind A as seen by viewer CS 480/680 Chapter 7 -- From Vertices to Fragments Fill B then A 35

Depth Sort n Requires ordering of polygons first O(n log n) calculation for ordering n Not every polygon is either in front or behind all other polygons n Order polygons and deal with easy cases first, harder later n Polygons sorted by distance from COP CS 480/680 Chapter 7 -- From Vertices to Fragments 36

Easy Cases n A lies behind all other polygons n n Can render Polygons overlap in z but not in either x or y n Can render independently CS 480/680 Chapter 7 -- From Vertices to Fragments 37

Hard Cases cyclic overlap Overlap in all directions but one is fully on one side of the other penetration CS 480/680 Chapter 7 -- From Vertices to Fragments 38

Back-Face Removal (Culling) n n n n face is visible iff 90 -90 equivalently cos 0 or v • n 0 plane of face has form ax + by +cz +d =0 but after normalization n = ( 0 0 1 0)T need only test the sign of c In Open. GL we can simply enable culling but may not work correctly if we have nonconvex objects CS 480/680 Chapter 7 -- From Vertices to Fragments 39

Image Space Approach Look at each projector (nm for an n x m frame buffer) and find closest of k polygons n Complexity O(nmk) n Ray tracing n z-buffer n CS 480/680 Chapter 7 -- From Vertices to Fragments 40

z-Buffer Algorithm n n n Use a buffer called the z or depth buffer to store the depth of the closest object at each pixel found so far As we render each polygon, compare the depth of each pixel to depth in z buffer If less, place shade of pixel in color buffer and update z buffer CS 480/680 Chapter 7 -- From Vertices to Fragments 41

Efficiency n If we work scan line by scan line as we move across a scan line, the depth changes satisfy a x+b y+c z=0 Along scan line y = 0 z = In screen space CS 480/680 x x = 1 Chapter 7 -- From Vertices to Fragments 42

Scan-Line Algorithm n Can combine shading and hsr through scan line algorithm scan line i: no need for depth information, can only be in no or one polygon scan line j: need depth information only when in more than one polygon CS 480/680 Chapter 7 -- From Vertices to Fragments 43

Implementation n Need a data structure to store Flag for each polygon (inside/outside) n Incremental structure for scan lines that stores which edges are encountered n Parameters for planes n CS 480/680 Chapter 7 -- From Vertices to Fragments 44

Visibility Testing n In many real-time applications, such as games, we want to eliminate as many objects as possible within the application Reduce burden on pipeline n Reduce traffic on bus n n Partition space with Binary Spatial Partition (BSP) Tree CS 480/680 Chapter 7 -- From Vertices to Fragments 45

Simple Example consider 6 parallel polygons top view The plane of A separates B and C from D, E and F CS 480/680 Chapter 7 -- From Vertices to Fragments 46

BSP Tree n Can continue recursively Plane of C separates B from A n Plane of D separates E and F n n Can put this information in a BSP tree n Use for visibility and occlusion testing CS 480/680 Chapter 7 -- From Vertices to Fragments 47

CS 480/680 Chapter 7 -- From Vertices to Fragments 48

Part III Objectives n Survey Line Drawing Algorithms DDA n Bresenham n CS 480/680 Chapter 7 -- From Vertices to Fragments 49

Rasterization n Rasterization (scan conversion) Determine which pixels that are inside primitive specified by a set of vertices n Produces a set of fragments n Fragments have a location (pixel location) and other attributes such color and texture coordinates that are determined by interpolating values at vertices n n Pixel colors determined later using color, texture, and other vertex properties CS 480/680 Chapter 7 -- From Vertices to Fragments 50

Scan Conversion of Line Segments Start with line segment in window coordinates with integer values for endpoints n Assume implementation has a write_pixel function n y = mx + h CS 480/680 Chapter 7 -- From Vertices to Fragments 51

DDA Algorithm n Digital Differential Analyzer n DDA was a mechanical device for numerical solution of differential equations n Line y=mx+ h satisfies differential equation dy/dx = m = y/ x = y 2 -y 1/x 2 -x 1 n Along scan line x = 1 For(x=x 1; x<=x 2, ix++) { y+=m; write_pixel(x, round(y), line_color) } CS 480/680 Chapter 7 -- From Vertices to Fragments 52

Problem n DDA = for each x plot pixel at closest y n Problems for steep lines CS 480/680 Chapter 7 -- From Vertices to Fragments 53

Using Symmetry Use for 1 m 0 n For m > 1, swap role of x and y n n For each y, plot closest x CS 480/680 Chapter 7 -- From Vertices to Fragments 54

Bresenham’s Algorithm n n n DDA requires one floating point addition per step We can eliminate all fp through Bresenham’s algorithm Consider only 1 m 0 n Other cases by symmetry Assume pixel centers are at half integers If we start at a pixel that has been written, there are only two candidates for the next pixel to be written into the frame buffer CS 480/680 Chapter 7 -- From Vertices to Fragments 55

Candidate Pixels 1 m 0 candidates last pixel CS 480/680 Chapter 7 -- From Vertices to Fragments Note that line could have passed through any 56 part of this pixel

Decision Variable d = x(b-a) d is an integer d > 0 use upper pixel d < 0 use lower pixel CS 480/680 Chapter 7 -- From Vertices to Fragments 57

Incremental Form n More efficient if we look at dk, the value of the decision variable at x = k dk+1= dk – 2 y, if dk <0 dk+1= dk – 2( y- x), otherwise For each x, we need do only an integer addition and a test n Single instruction on graphics chips n CS 480/680 Chapter 7 -- From Vertices to Fragments 58

Polygon Scan Conversion = Fill n How to tell inside from outside n Convex easy n Nonsimple difficult n Odd even test n n Count edge crossings n Winding CS 480/680 number Chapter 7 -- From Vertices to Fragments odd-even fill 59

Winding Number n Count clockwise encirclements of point winding number = 1 winding number = 2 n Alternate definition of inside: inside if winding number 0 CS 480/680 Chapter 7 -- From Vertices to Fragments 60

Filling in the Frame Buffer n Fill at end of pipeline Convex Polygons only n Nonconvex polygons assumed to have been tessellated n Shades (colors) have been computed for vertices (Gouraud shading) n Combine with z-buffer algorithm n n March across scan lines interpolating shades n Incremental work small CS 480/680 Chapter 7 -- From Vertices to Fragments 61

Using Interpolation C 1 C 2 C 3 specified by gl. Color or by vertex shading C 4 determined by interpolating between C 1 and C 2 C 5 determined by interpolating between C 2 and C 3 interpolate between C 4 and C 5 along span C 1 C 4 C 2 scan line C 5 span C 3 CS 480/680 Chapter 7 -- From Vertices to Fragments 62

Flood Fill can be done recursively if we know a seed point located inside (WHITE) n Scan convert edges into buffer in edge/inside color (BLACK) n 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); } } CS 480/680 Chapter 7 -- From Vertices to Fragments 63

Scan Line Fill n Can also fill by maintaining a data structure of all intersections of polygons with scan lines n Sort by scan line n Fill each span vertex order generated by vertex list Chapter 7 -- From Vertices to CS 480/680 Fragments desired order 64

Data Structure CS 480/680 Chapter 7 -- From Vertices to Fragments 65

Aliasing n Ideal rasterized line should be 1 pixel wide n Choosing best y for each x (or visa versa) produces aliased raster lines CS 480/680 Chapter 7 -- From Vertices to Fragments 66

Antialiasing by Area Averaging n Color multiple pixels for each x depending on coverage by ideal line antialiased original magnified CS 480/680 Chapter 7 -- From Vertices to Fragments 67

Polygon Aliasing problems can be serious for polygons Jaggedness of edges n Small polygons neglected n Need compositing so color of one polygon does not totally determine color of pixel n All three polygons should contribute to color CS 480/680 Chapter 7 -- From Vertices to Fragments 68

CS 480/680 Chapter 7 -- From Vertices to Fragments 69