Course Note Credit: Some of slides are extracted from the course notes of prof. Mathieu Desburn (USC) and prof. Han-Wei Shen (Ohio State University).

CSC 830 Computer GraphicsLecture 5

Rasterization

Rasterization• Okay, you have three points

(vertices) at screen space. How can you fill the triangle?

v0

v2

E2

E1

E0

v0

v1

E0

E1

E2

v1

v2

E1

1

12

2

Simple Solution

int draw_tri(render,t) GzRender *render; GzTriangle t;{ sort_vert(t); /* Vertices are properly sorted, so edge1 can be composed by ver[0],ver[1] and edge2 by ver[1],[2], and so on */

init_edge(render,E0,t[0],t[1]); init_edge(render,E1,t[1],t[2]); init_edge(render,E2,t[0],t[2]); step_edges(render); return GZ_SUCCESS;}

My simple approach

void init_edge(GzRender *render,int i,GzVertex ver1,GzVertex ver2){ Interpolater *ip; float dy; ip = &render->interp[i]; ip->fromx = ver1.p[x]; ip->fromy = ver1.p[y]; dy = ver2.p[y]-ver1.p[y]; if(dy== 0) ip->dxdy = zmax; else ip->dxdy = (ver2.p[x]-ver1.p[x])/dy; ip->tox = ver2.p[x]; ip->toy = ver2.p[y]; ip->fromz = ver1.p[z]; if(ver2.p[y]-ver1.p[y]== 0) ip->dz = zmax; else ip->dz =(ver2.p[z]-ver1.p[z])/dy; vec_copy(ip->from_norm,ver1.n); ip->dnorm[x]=(ver2.n[x] - ver1.n[x])/dy; ip->dnorm[y]=(ver2.n[y] - ver1.n[y])/dy; ip->dnorm[z]=(ver2.n[z] - ver1.n[z])/dy;}

Interpolater

x1, y1, z1

x2, y2, z2

x3, y3, z3

(xn,yn,zn)

xn = x1 + (yn-y1)*(x2-x1)/(y2-y1)

zn = z1 + (yn-y1)*(z2-z1)/(y2-y1)

v0

v1

v2

v2v2

v0v0

v1

v1

E2

E1

E0

E0

E0

E1

E1

E2

E2

v0

v0

v1

v1

v2v2Step Edges

v0v1

v2

Better & still simple approach

• Find left edge & right edge from v0• You can not tell when y==v0.y• But you can tell by comparing x

values when y == v0.1+1• You need to treat horizontal edges

carefully (divide by zero).

How can you handle this?

Or this?

Back-face Culling• If a surface’s normal is pointing to the

same direction as our eye direction, then this is a back face

• The test is quite simple: if N * V > 0 then we reject the surface

Painters AlgorithmSort objects in depth order

Draw all from Back-to-Front (far-to-near)

Is it so simple?

at z = 22, at z = 18, at z = 10,

1 2 3

X

Y

3D CyclesHow do we deal with cycles?

Deal with intersections

How do we sort objects that overlap in Z?

Z

Z-buffer

Z-buffer is a 2D array that stores a depth value for each pixel.

Invented by Catmull in '79, it allows us to paint objects to the

screen without sorting, without performing intersection

calculations where objects interpenetrate.

InitScreen:

    for i := 0 to N do

        for j := 1 to N do

       Screen[i][j] := BACKGROUND_COLOR;  Zbuffer[i][j] := ;

DrawZpixel (x, y, z, color)

      if (z <= Zbuffer[x][y]) then

             Screen[x][y] := color; Zbuffer[x][y] := z;

Z-buffer: Scanline

I.  for each polygon do        for each pixel (x,y) in the polygon’s projection do             z := -(D+A*x+B*y)/C;             DrawZpixel(x, y, z, polygon’s color);

II.  for each scan-line y do        for each “in range” polygon projection do             for each pair (x1, x2) of X-intersections do                   for x := x1 to x2 do                         z := -(D+A*x+B*y)/C;                         DrawZpixel(x, y, z, polygon’s color);

If we know zx,y at (x,y) than:    zx+1,y = zx,y - A/C

Z-buffer - Example

Z-buffer

Screen

Rectangle: P1(10,5,10), P2(10,25,10), P3(25,25,10),

P4(25,5,10)

Triangle: P5(15,15,15), P6(25,25,5), P7(30,10,5)

Frame Buffer: Background 0, Rectangle 1, Triangle 2

Z-buffer: 32x32x4 bit planes

Non Trivial Example ?

Z-Buffer Advantages

Simple and easy to implement

Amenable to scan-line algorithms

Can easily resolve visibility cycles

Z-Buffer DisadvantagesAliasing occurs! Since not all depth questions can be resolved

Anti-aliasing solutions non-trivial

Shadows are not easy

Higher order illumination is hard in general

Rasterizing PolygonsGiven a set of vertices and edges, find the pixels that fill the polygon.

Scan Line AlgorithmsTake advantage of coherence in “insided-ness”

Inside/outside can only change at edge eventsCurrent edges can only change at vertex events

Scan Line AlgorithmsCreate a list of vertex events (bucket sorted by y)

Scan Line AlgorithmsCreate a list of the edges intersecting the first scanline

Sort this list by the edge’s x value on the first scanline

Call this the active edge list

Scanline Rasterization Special Handling

• Intersection is an edge end point, say: (p0, p1, p2) ??

• (p0,p1,p1,p2), so we can still fill pairwise• In fact, if we compute the intersection of the scanline

with edge e1 and e2 separately, we will get the intersection point p1 twice. Keep both of the p1.

Scanline Rasterization Special Handling

• But what about this case: still (p0,p1,p1,p2)

Rule• Rule:

– If the intersection is the ymin of the edge’s endpoint, count it. Otherwise, don’t.

• Don’t count p1 for e2

Data Structure• Edge table:

– all edges sorted by their ymin coordinates.– keep a separate bucket for each scanline– within each bucket, edges are sorted by

increasing x of the ymin endpoint

Edge Table

Active Edge Table (AET)

• A list of edges active for current scanline, sorted in increasing x

y = 9

y = 8

Penetrating Polygons

S

I

T

a

2

BG

b

c

3

d

e

1

False edges and new polygons!

Compare z value & intersection when

AET is calculated

Polygon Scan-conversion Algorithm

Construct the Edge Table (ET); Active Edge Table (AET) = null;for y = Ymin to Ymax

Merge-sort ET[y] into AET by x valueFill between pairs of x in AETfor each edge in AET

if edge.ymax = yremove edge from AET

elseedge.x = edge.x + dx/dy

sort AET by x value

end scan_fill

For each scanline:1. Maintain active edge list (using vertex events)

2. Increment edge’s x-intercepts, sort by x-intercepts

3. Output spans between left and right edges

Scan Line Algorithms

delete insert replace

Convex PolygonsConvex polygons only have 1 span

Insertion and deletion events happen only once

crow(vertex vList[], int n) int imin = 0;

for(int i = 0; i < n; i++) if(vList[i].y < vList[imin].y) imin = i; scanY(vList,n,imin);

Crow’s AlgorithmFind the vertex with the smallest y value to start

scanY(vertex vList[], int n, int i) int li, ri; // left & right upper endpoint indices int ly, ry; // left & right upper endpoint y values vertex l, dl; // current left edge and delta vertex r, dr; // current right edge and delta int rem; // number of remaining vertices int y; // current scanline

li = ri = i; ly = ry = y = ceil(vList[i].y);

for( rem = n; rem > 0) // find appropriate left edge // find appropriate right edge // while l & r span y (the current scanline) // draw the span

Crow’s AlgorithmScan upward maintaining the active edge list

(1)(3)

(2)

for( ; y < ly && y < ry; y++) // scan and interpolate edges scanX(&l, &r, y); increment(&l,&dl); increment(&r,&dr);

increment(vertex *edge, vertex *delta) edge->x += delta->x;

Crow’s AlgorithmDraw the spans

(2)

Increment the x value

scanX(vertex *l, vertex *r, int y) int x, lx, rx; vertex s, ds;

lx = ceil(l->x); rx = ceil(r->x); if(lx < rx) differenceX(l, r, &s, &ds, lx); for(x = lx, x < rx; x++) setPixel(x,y); increment(&s,&ds);

Crow’s AlgorithmDraw the spans

differenceX(vertex *v1, vertex *v2, vertex *e, vertex *de, int x) difference(v1, v2, e, de, (v2->x – v1->x), x – v1->x);

difference(vertex *v1, vertex *v2, vertex *e, vertex *de, float d, float f) de->x = (v2->x – v1->x) / d; e->x = v1->x + f * de->x;

differenceY(vertex *v1, vertex *v2, vertex *e, vertex *de, int y) difference(v1, v2, e, de, (v2->y – v1->y), y – v1->y);

Crow’s AlgorithmCalculate delta and starting values

df

d

f

while( ly < = y && rem > 0) rem--; i = li – 1; if(i < 0) i = n-1; // go clockwise ly = ceil( v[i].y ); if( ly > y ) // replace left edge differenceY( &vList[li], &vList[i], &l, &dl, y); li = i;

Crow’s AlgorithmFind the appropriate next left edge

(3)

difference(vertex *v1, vertex *v2, veretx *e, vertex *de, float d, float f) de->x = (v2->x – v1->x) / d; e->x = v1->x + f * de->x; de->r = (v2->r – v1->r) / d; e->r = v1->r + f * de->r; de->g = (v2->g – v1->g) / d; e->g = v1->g + f * de->g; de->b = (v2->b – v1->b) / d; e->b = v1->b + f * de->b;

increment( vertex *v, vertex *dv) v->x += dv->x; v->r += dv->r; v->g += dv->g; v->b += dv->b;

Crow’s AlgorithmInterpolating other values

Scan Line Algorithm• Low memory cost• Uses scan-line coherence

– but not vertical coherence• Has several side advantages:

– filling the polygons– reflections– texture mapping

• Renderman (Toy Story) = scan line

Visibility Clipping Culling

ClippingObjects may lie partially inside and

partially outside the view volumeWe want to “clip” away the parts

outside

Simple approach - checking if (x,y) is inside of screen or not What is wrong with it?

Cohen-Sutherland Clipping

Clip line against convex region Clip against each edge

Line crosses edge replace outside vertex with intersection

Both endpoints outside trivial reject

Both endpoints inside trivial accept

Cohen-Sutherland Clipping

Cohen-Sutherland Clipping

Store inside/outside bitwise for each edge

Trivial acceptoutside(v1) | outside(v2) == 0

Trivial rejectoutside(v1) & outside(v2)

Compute intersection (eg. x = a)(a, y1 + (a-x1) * (y2-y1)/(x2-x1))

• Classifies each vertex of a primitive, by generating an outcode. An outcode identifies the appropriate half space location of each vertex relative to all of the clipping planes. Outcodes are usually stored as bit vectors.

Outcode Algorithm

if (outcode1 == '0000' and outcode2 == ‘0000’) then line segment is insideelse if ((outcode1 AND outcode2) == 0000) then line segment potentially crosses clip region else line is entirely outside of clip region endifendif

If neither trivial accept nor reject:Pick an outside endpoint (with nonzero outcode)

Pick an edge that is crossed (nonzero bit of outcode)

Find line's intersection with that edgeReplace outside endpoint with intersection point

Repeat outcode test until trivial accept or reject

The maybe case?

The Maybe case

The Maybe Case

CullingRemoving completely invisible

objects/polygons

CullingCheck whether all vertices lie outside the

clip plane

Speed up: first check bounding box extents

Backface CullingFor closed objects, back facing

polygons are not visible

nv

0nv

Backfacing iff:

Flood FillHow to fill polygons whose edges are already drawn?

Choose a point inside, and fill outwards

floodFill(int x, int y, color c) if(stop(x,y,c)) return;

setPixel(x,y,c); floodFill(x-1,y,c); floodFill(x+1,y,c); floodFill(x,y-1,c); floodFill(x,y+1,c);

int stop(int x, int y, color c) return colorBuffer[x][y] == c;

Flood FillFill a point and recurse to all of its neighbors

Area Subdivision (Warnock)

(mixed object/image space)

Clipping used to subdivide polygons that are across regions

Area Subdivision (Warnock)

1. Initialize the area to be the image plane

2. Four cases:1. No polygons in area: done2. One polygon in area: draw it3. Pixel sized area: draw closest polygon4. Front polygon covers area: draw it

Otherwise, subdivide and recurse

BSP (Binary Space Partition) Trees

Partition space into 2 half-spaces via a hyper-plane

a

b

cd

e

a

cb

e d

BSPNode* BSPCreate(polygonList pList) if(pList is empty) return NULL;

pick a polygon p from pList; split all polygons in pList by p and insert pieces into pList; polygonList coplanar = all polygons in pList coplanar to p; polygonList positive = all polygons in pList in p’s positive halfspace; polygonList negative = all polygons in pList in p’s negative halfspace;

BSPNode *b = new BSPNode; b->coplanar = coplanar; b->positive = BSPCreate(positive); b->negative = BSPCreate(negative);

return b;

BSP TreesCreating the BSP Tree

BSPRender(vertex eyePoint, BSPNode *b) if(b == NULL) return;

if(eyePoint is in positive half-space defined by b->coplanar) BSPRender(eyePoint,b->negative); draw all polygons in b->coplanar; BSPRender(eyePoint,b->positive); else BSPRender(eyePoint,b->positive); draw all polygons in b->coplanar; BSPRender(eyePoint,b->negative);

BSP TreesRendering the BSP Tree

BSP Trees

Advantages

view-independent tree

anti-aliasing (see later)

transparency

Disadvantages

many small polygons

over-rendering

hard to balance tree

Spatial Data Structures• Data structures for efficiently storing

geometric information. They are useful for– Collision detection (will the spaceships collide?)– Location queries (which is the nearest post office?)– Chemical simulations (which protein will this drug

molecule interact with?)– Rendering (is this aircraft carrier on-screen?), and

more

• Good data structures can give speed up rendering by 10x, 100x, or more

Bounding Volume• Simple notion: wrap things that are hard to

check for ray intersection in things that are easy to check.– Example: wrap a complicated polygonal mesh

in a box. Ray can’t hit the real object unless it hits the box

• Adds some overhead, but generally pays for itself .

• Can build bounding volume hierarchies

Bounding Volumes• Choose Bounding Volume(s)

– Spheres– Boxes– Parallelepipeds– Oriented boxes– Ellipsoids– Convex hulls

Quad-trees• Quad-tree is the 2-D

generalization of binary tree– node (cell) is a square– recursively split into four

equal sub-squares– stop when leaves get

“simple enough”

Octrees• Octree is the 3-D generalization of quad-tree• node (cell) is a cube, recursively split into eight

equal sub- cubes– stop splitting when the number of objects

intersecting the cell gets “small enough” or the tree depth exceeds a limit

– internal nodes store pointers to children, leaves store list of surfaces

• more expensive to traverse than a grid• adapts to non-homogeneous, clumpy scenes

better

K-D tree• The K-D approach is to

make the problem space a rectangular parallelepiped whose sides are, in general, of unequal length.

• The length of the sides is the maximum spatial extent of the particles in each spatial dimension.

K-D tree

K-D Tree in 3-D• Similarly, the

problem space in three dimensions is a parallelepiped whose sides are the greatest particle separation in each of the three spatial dimensions.

Motivation for Scene Graph

• Three-fold– Performance– Generality– Ease of use

• How to model a scene ?– Java3D, Open Inventor, Open

Performer, VRML, etc.

Scene Graph Example

Scene Graph Example

Scene Graph Example

Scene Graph Example

Scene Description• Set of Primitives• Specify for each primitive

• Transformation• Lighting attributes• Surface attributes

• Material (BRDF)• Texture• Texture transformation

Scene Graphs• Scene Elements

– Interior Nodes• Have children that inherit state• transform, lights, fog, color, …

– Leaf nodes• Terminal• geometry, text

– Attributes• Additional sharable state (textures)

Scene Element Class Hierarchy

Scene Graph• Graph Representation

– What do edges mean?– Inherit state along edges

• group all red object instances together• group logical entities together

– parts of a car

– Capture intent with the structure

Scene Graph

Scene Graph (VRML 2.0)

Example Scene Graph

• Simulation– Animation

• Intersection– Collision detection– Picking

• Image Generation– Culling– Detail elision– Attributes

Scene Graph Traversal

Scene Graph Considerations

• Functional Organization– Semantics

• Bounding Volumes– Culling– Intersection

• Levels of Detail– Detail elision– Intersection

• Attribute Management– Eliminate redundancies

Functional Organization• Semantics:

– Logical parts– Named parts

Functional Organization• Articulated Transformations

– Animation– Difficult to optimize animated objects

Bounding Volume Hierarchies

View Frustum Culling

Level Of Detail (LOD)• Each LOD

nodes have distance ranges

Attribute Management• Minimize transformations

– Each transformation is expensive during rendering, intersection, etc. Need automatic algorithms to collapse/adjust transform hierarchy.

Attribute Management• Minimize attribute changes

– Each state change is expensive during rendering

Question: How do you manage your light

sources?• OpenGL supports only 8 lights. What if

there are 200 lights? The modeler must ‘scope’ the lights in the scene graph?

Sample Scene Graph

Think!• How to handle optimization of

scene graphs with multiple competing goals– Function– Bounding volumes– Levels of Detail– Attributes

Scene Graphs Traversal• Perform operations on graph with

traversal– Like STL iterator– Visit all nodes– Collect inherited state while

traversing edges

• Also works on a sub-graph

Typical Traversal Operations

• Typical operations– Render– Search (pick, find by name)– View-frustum cull– Tessellate– Preprocess (optimize)

Scene Graphs Organization

• Tree structure best– No cycles for simple traversal– Implied depth-first traversal (not essential)– Includes lists, single node, etc as

degenerate trees• If allow multiple references (instancing)

– Directed acyclic graph (DAG)• Difficult to represent cell/portal

structures

PortalsSeparate environment into cellsPreprocess to find potentially visible polygons from any cell

PortalsTreat environment as a graphNodes = cells, Edges = portalsCell to cell visibility must go along edges

Portals

Advantages

pre-process

very efficient

Disadvantages

static

only a partial solution

not appropriate for most scenes