Computer Graphics Inf4/MSc 28/10/08Lecture 91 Computer Graphics Lecture 9 Visible Surface Determination Taku Komura

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Computer Graphics Inf4/MSc

Computer Graphics

Lecture 9

Visible Surface Determination

Taku Komura

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Hidden surface removal

• Drawing polygonal faces on screen consumes CPU cycles– We cannot see every surface in scene– We don’t want to waste time rendering

primitives which don’t contribute to the final image.

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Visibility (hidden surface removal)

• A correct rendering requires correct visibility calculations

• Correct visibility – when multiple opaque polygons cover the same

screen space, only the closest one is visible (remove the other hidden surfaces)

– wrong visibility correct visibility

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Visibility of primitives• A scene primitive can be invisible for 3 reasons:

– Primitive lies outside field of view– Primitive is back-facing– Primitive is occluded by one or more objects nearer the

viewer

• How do we remove these efficiently?• How do we identify these efficiently?

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The visibility problem.Two problems remain: (Clipping we have covered)• Removal of faces facing away from the viewer.• Removal of faces obscured by closer objects.

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Visible surface algorithms.Definitions:•Object space techniques: applied before vertices are mapped to pixels

•Back face culling, Painter’s algorithm, BSP Trees•Image space techniques: applied after vertices have been rasterized

•Z-buffering

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Back face culling.• We saw in modelling, that the vertices

of polyhedra are oriented in an anticlockwise manner when viewed from outside – surface normal N points out.

• Project a polygon.

– Test z component of surface normal. If negative – cull, since normal points away from viewer.

– Or if N.V > 0 we are viewing the back face so polygon is obscured.

• Only works for convex objects without holes, ie. closed orientable manifolds.

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Back face culling

• Back face culling can be applied anywhere in the pipeline: geometry stage or rasterization stage

• If we clip our scene to the view frustrum, then remove all back-facing polygons – are we done?

• NO! Most views involve overlapping polygons.

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Painters algorithm (object space).

• Draw surfaces in back to front order – nearer polygons “paint” over farther ones.

• Supports transparency.• Key issue is order

determination.• Doesn’t always work –

see image at right.

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One Solution

• Binary Space Partitioning Tree (or BSP Tree)

• A data structure that is used to organize objects within a space.

• Polygons are subdivided and queued

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BSP (Binary Space Partitioning) Tree.•One of class of “list-priority” algorithms – returns ordered list of polygon fragments for specified view point (static pre-processing stage).

•Choose polygon arbitrarily

•Divide scene into front (relative to normal) and back half-spaces.

•Split any polygon lying on both sides.

•Choose a polygon from each side – split scene again.

•Recursively divide each side until each node contains only 1 polygon.

3

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View of scene from above

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BSP Tree.






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BSP Tree.






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front

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BSP Tree.






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BSP Tree.






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back

Alternate formulation starting at 5

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Displaying a BSP tree.

• Once we have the regions – need priority list

• BSP tree can be traversed to yield a correct priority list for an arbitrary viewpoint.

• Start at root polygon.– If viewer is in front half-space, draw polygons behind root first,

then the root polygon, then polygons in front.

– If polygon is on edge – either can be used.

– Recursively descend the tree.

• If eye is in rear half-space for a polygon – then can back face cull.

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BSP Tree.

• A lot of computation required at start.– Try to split polygons along good dividing plane– Intersecting polygon splitting may be costly

• Cheap to check visibility once tree is set up.• Can be used to generate correct visibility for

arbitrary views. Efficient when object location don’t change

very often in the scene.

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Z-buffering : image space approach

Basic Z-buffer idea:• rasterize every input polygon• For every pixel in the polygon interior, calculate

its corresponding z value (by interpolation)• Track depth values of closest polygon (smallest z)

so far• Paint the pixel with the color of the polygon

whose z value is the closest to the eye.

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Triangle Rasterization

• Consider a 2D triangle with vertices p0 , p1 , p2. • Let x be any point in the plane. We can always

find a, b, c such that

• We will have if and only if x is inside the triangle.

• We call the barycentric coordinates of x.

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Rasterization

• We assume that p are normalized device coordinates (NDC); that is, the canvas corresponds to the region [-1, 1] × [-1, 1].

-This is what you get after applying all transformation matrices.

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Bounding box of the triangle• First, identify a rectangular region on the

canvas that contains all of the pixels in the triangle (excluding those that lie outside the canvas).

• Calculate a tight bounding box for a triangle: simply calculate pixel coordinates for each vertex, and find the minimum/maximum for each axis

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Scanning inside the triangle

• Once we've identified the bounding box, we loop over each pixel in the box.

• For each pixel, we first compute the corresponding NDC coordinates (x, y).

• Next we convert these into barycentric coordinates for the triangle being drawn.

• Only if the barycentric coordinates are within the range of [0,1], we plot it (and compute the depth)

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Computer Graphics Inf4/MSc Computing the baricentric

coordinates of the interior pixels

• Depth can be computed by • Can do the same thing for color, normals

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Implementation.

• Initialise frame buffer to background colour.

• Initialise depth buffer to z = max. value for far clipping plane

• For each triangle – Calculate value for z for each pixel inside– Update both frame and depth buffer

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Why is z-buffering so popular ?Advantage• Simple to implement in hardware.

– Memory for z-buffer is now not expensive• Diversity of primitives – not just polygons.• Unlimited scene complexity• Don’t need to calculate object-object intersections.Disadvantage• Extra memory and bandwidth• Waste time drawing hidden objectsZ-precision errors• May have to use point sampling

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Z-buffer performance

• Brute-force image-space algorithm scores best for complex scenes – not very accurate but is easy to implement and is very general.

• Storage overhead: O(1)• Time to resolve visibility to screen precision: O(n)

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Exercise• Write a program to load 3D polygons• Use OpenGL’s 2D functions to draw

triangles glOrtho2D(); glColor3f(); glRecti(i,j,i+1,j+1); -Try to use barycentric coordinates to

rasterize the triangle

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References for this lecture• Foley et al. Chapter 15, all of it.

• Introductory text, Chapter 13, all of it

• Baricentric coordinates www.cs.caltech.edu/courses/cs171/barycentric.pdf

• Or equivalents in other texts, look out for:– (as well as the topics covered today)– Depth sort – Newell, Newell & Sancha– Scan-line algorithms

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Computer Graphics Inf4/MSc 28/10/08Lecture 91 Computer Graphics Lecture 9 Visible Surface Determination Taku Komura