From Vertices to Fragments II

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From Vertices to Fragments II

• Software College, Shandong University
• Instructor Zhou Yuanfeng
• E-mail yuanfeng.zhou_at_gmail.com

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Review

• Cyrus-Beck clipping algorithm
• Liang-Barsky clipping algorithm
• Sutherland-Hodgman polygon clipping
• DDA line algorithm
• Bresenhams line algorithm

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Objectives

• Rasterization Polygon scan conversion algorithm
• Hidden surface removal
• Aliasing

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Polygon Scan Conversion

• Scan Conversion Fill
• How to tell inside from
• outside
• Convex easy
• Nonsimple difficult
• Odd even test
• Count edge crossings
• Winding number

odd-even fill
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Winding Number

• Count clockwise encirclements of point
• Alternate definition of inside inside if winding
  number ? 0

winding number 1
winding number 2
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OpenGL Concave Polygon

• OpenGL can only fill convex polygon correctly

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OpenGL Concave Polygon

• // create tessellator
• GLUtesselator tess gluNewTess()
• // describe non-convex polygon
• gluTessBeginPolygon(tess, user_data)
• // first contour
• gluTessBeginContour(tess)
• gluTessVertex(tess, coords0,
  vertex_data)
• ...
• gluTessEndContour(tess)
• ...
• gluTessEndPolygon(tess)
• // delete tessellator after processing
• gluDeleteTess(tess)

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Constrained Delaunay Triangulation
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Filling in the Frame Buffer

• Fill at end of pipeline
• Convex Polygons only
• Nonconvex polygons assumed to have been
  tessellated
• Shades (colors) have been computed for vertices
  (Gouraud shading)
• Combine with z-buffer algorithm
• March across scan lines interpolating shades
• Incremental work small

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Using Interpolation

• C1 C2 C3 specified by glColor or by vertex
  shading
• C4 determined by interpolating between C1 and C2
• C5 determined by interpolating between C2 and C3
• interpolate between C4 and C5 along span

C1
C2
C4
scan line
C5
span
C3
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Flood Fill

• Fill can be done recursively if we know a seed
  point located inside (WHITE)
• Scan convert edges into buffer in edge/inside
  color (BLACK)

flood_fill(int x, int y) if(read_pixel(x,y)
WHITE) write_pixel(x,y,BLACK)
flood_fill(x-1, y) flood_fill(x1, y)
flood_fill(x, y1) flood_fill(x,
y-1) //4 directions
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Flood Fill
//8 directions
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Flood Fill with Scan Line
Stack
2
2
2
1
2
2
2
3
2
2
2
2
3
1
0
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Scan Line Fill

• Can also fill by maintaining a data structure of
  all intersections of polygons with scan lines
• Sort by scan line
• Fill each span

vertex order generated by vertex list
desired order
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Singularities
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Data Structure
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Scan line filling

• For each scan line
• Find the intersections of the scan line with all
  edges of the polygon
• Sort the intersections by increasing
  x-coordinate
• Fill in all pixels between pairs of
  intersections.
• Problem
• Calculating intersections is slow.
• Solution
• Incremental computation / coherence

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Coherence of Region
Trapezoid inside and outside
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Coherence of Scan Line

• e is an integer e
• Intersections number is even
• are inside

inside
inside
inside
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Coherence of Edge

• Intersection points
• of d is le and ld
• The number of le ld
• and are on the same
  edge

y63
e
d
y24
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Coherence of Edge

• Observation Not all edges intersect each
  scanline.
• Many edges intersected by scanline i will also be
  intersected by scanline i1
• Formula for scanline s is y s, for an edge is
  y mx b
• Their intersection is s mxs b --gt xs
  (s-b)/m
• For scanline s 1, xs1 (s1 - b)/m xs
  1/m
• Incremental calculation xs1 xs 1/m

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Data Structure
Edge index table ET Active Edge List AEL Node
ymax The max coord y x The bottom x coord
of edge in ET, the intersection x of edge and
scan line in AEL. ?x 1/m next the
next node
Where is the end of one edge while scanning
Initial x
Current scan line and edge intersection
When yy1, xx1/m
Next edge
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Data Structure ET
6
5
2
1
y
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Data Structure AEL
AEL is the edges list which intersect with
current scan line, the next intersection can be
computed by
2
x
1/m
ymax
Active Edge List
Edge list is ordered according x increasing
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Algorithm

• Construct the Edge Table (ET)
• Active Edge List (AEL) null
• for y Ymin to Ymax
• Merge-sort ETy into AEL by x value
• Fill between pairs of x in AEL
• for each edge in AEL
• if edge.ymax y
• remove edge from AEL
• else
• edge.x edge.x dx/dy
• sort AEL by x value
• end scan_fill

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Hidden Surface Removal

• Object-space approach use pairwise testing
  between polygons (objects space)
• Worst case complexity O(n2) for n polygons

partially obscuring
can draw independently
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Image Space Approach

• Look at each projector (nm for an n x m frame
  buffer) and find closest of k polygons
• Complexity O(nmk)
• Ray tracing
• z-buffer
• Fast but with low paint quality

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Painters Algorithm

• Render polygons a back to front order so that
  polygons behind others are simply painted over

Fill B then A
B behind A as seen by viewer
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Depth Sort

• Requires ordering of polygons first
• O(n log n) calculation for ordering
• Not every polygon is either in front or behind
  all other polygons
• Order polygons and deal with
• easy cases first, harder later

Polygons sorted by distance from COP
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Easy Cases

• (1) A lies behind all other polygons
• Can render
• (2) Polygons overlap in z but not in either x or
  y
• Can render independently

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Hard Cases
cyclic overlap
(3) Overlap in all directions but can one is
fully on one side of the other
(4)
penetration
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Back-Face Removal (Culling)
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 OpenGL we can simply
enable culling but may not work correctly if we
have nonconvex objects
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z-Buffer Algorithm

• 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

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z-Buffer Algorithm
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z-Buffer Algorithm
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Example
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Example
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Example
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Efficiency

• If we work scan line by scan line as we move
  across a scan line, the depth changes satisfy
  a?xb?yc?z0

Along scan line ?y 0 ?z - ?x
In screen space ?x 1
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Scan-Line Algorithm

• 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
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z-Buffer Scan-Line
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Implementation

• Need a data structure to store
• Flag for each polygon (inside/outside)
• Incremental structure for scan lines that stores
  which edges are encountered
• Parameters for planes for intersections

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Visibility Testing

• In many real-time applications, such as games, we
  want to eliminate as many objects as possible
  within the application
• Reduce burden on pipeline
• Reduce traffic on bus
• Partition space with Binary Spatial Partition
  (BSP) Tree

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Simple Example
consider 6 parallel polygons
top view
The plane of A separates B and C from D, E and F
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BSP Tree

• Can continue recursively
• Plane of C separates B from A
• Plane of D separates E and F
• Can put this information in a BSP tree
• Use for visibility and occlusion testing

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BSP Tree
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BSP Algorithm

• Choose a polygon P from the list.
• Make a node N in the BSP tree, and add P to
  the list of polygons at that node.
• For each other polygon in the list
• If that polygon is wholly in front of the
  plane containing P, move that polygon to the list
  of nodes in front of P.
• If that polygon is wholly behind the
  plane containing P, move that polygon to the list
  of nodes behind P.
• If that polygon is intersected by the
  plane containing P, split it into two polygons
  and move them to the respective lists of polygons
  behind and in front of P.
• If that polygon lies in the plane
  containing P, add it to the list of polygons at
  node N.
• Apply this algorithm to the list of polygons
  in front of P.
• Apply this algorithm to the list of polygons
  behind P.

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BSP display

• Type Tree
• Tree front
• Face face
• Tree back
• End
• Algorithm DrawBSP(Tree T point w)
• //w ???
• If T is null then return endif
• If w is in front of T.face then
• DrawBSP(T.back,w)
• Draw(T.face,w)
• DrawBSP(T.front,w)
• Else // w is behind or on T.face
• DrawBSP(T.front,w)
• Draw(T.face,w)
• DrawBSP(T. back,w)
• Endif
• end

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Aliasing

• Ideal rasterized line should be 1 pixel wide
• Choosing best y for each x (or visa versa)
  produces aliased raster lines

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Antialiasing by Area Averaging

• Color multiple pixels for each x depending on
  coverage by ideal line

antialiased
original
magnified
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Polygon Aliasing

• Aliasing problems can be serious for polygons
• Jaggedness of edges
• Small polygons neglected
• Need compositing so color
• of one polygon does not
• totally determine color of
• pixel

All three polygons should contribute to color
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Time-domain aliasing

• Adding ray tracing line from one pixel