Lecture 16: Projection and Cameras. October 17, 2017
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1 Lecture 6: Projection and Cameras October 7 207
2 3D Viewing as Virtual Camera To take a picture with a camera or to render an image with computer graphics we need to:. Position the camera/viewpoint in 3D space 2. Orient the camera/viewpoint in 3D space 3. Focus camera ray trace thin lens 4. Crop image to the aperture/window 5. Project scene onto the image plane 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 2
3 Perspective 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 3
4 Orthographic Projection If not for the fog you could see forever and nothing ever would look smaller. 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 4
5 Orthographic / Perspective Think About Rays 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 5
6 Is Perspective Always Better? No! Technical programs including for example Maple often favor orthographic projection. 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 6
7 Math: Orthographic Projection Simply drop a dimension. Think of a bug hitting a windshield. No more z axis! no more bug Photo by Brian Jeff Booth site (creative common License 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 7
8 Perspective Projection Light rays pass through the focal point. a.k.a. Eye principal reference point or PRP. The image plane is an infinite plane in front of (or behind the focal point. Images are formed by rays of light passing through the image plane Common convention: Image points are (uv World points are (xyz 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 8
9 Why Pinhole Camera? Because you can build a camera that exactly fits this description: Create a fully-enclosed black box So that no light enters Put a piece of film inside it facing front Punch a pin-hole in the front face of the box What doesn t this camera have? What is this camera s depth-of-field? Why don t we build cameras this way? 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 9
10 History The Camera Obscura - see Wikipedia Pre-dates photographic cameras. Theory: Mo-Ti (China BC Practice: Abu Ali Al-Hasan Ibn al-haitham (~000 AD Western Painting: Johannes Vermeer (~660 AD 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 0
11 Pinhole Projection Flip the Bear in the Box 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper Slide
12 Human Eye - 4 year old view Drawing by Bryce Beveridge in /7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper Slide 2
13 Room Obscura 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 3
14 Perspective Projection Where we place the origin matters How we handle z values matters Form : Origin at focal point z values constant Form 2: Origin at image center z values are zero Form 3: Origin at focal point z proportional to depth 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 4
15 Perspective Projection Form The key to perspective projection is that all light rays meet at the PRP (E focal point. Notice that we are looking down the Z axis with the origin at the focal point and the image plane at z = d. v P(xyz d P v P y z P z 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 5
16 By similar triangles: horizontal vertical P u d = P x P z P v d = P y P z P u = P x d P z P v = P y d P z P u = P x d P P v = P d y z P z 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 6
17 Perspective Projection Matrix Problem: division of one variable by another is a non-linear operation. Solution: homogeneous coordinates! /d 0 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 7
18 Perspective Matrix (II u v d = x d z y d z d = x y z z d = d 0 x y z Point in (uv coordinates Point in Non-normalized Homogeneous coordinates Projection Matrix times a Point Normalized 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 8
19 What happens to Z? What happens to the Z dimension? u v d = x d z y d z d = x y z z d = d 0 The Z dimension projects to d. Why? Because (u v d is a 3D point on the image plane located at z = d! x y z 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 9
20 Perspective Projection Form 2 P v P y d O P z v d = P d v = P d + P. d + P. 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 20
21 x d d + z! " % y d d + z! " % 0 ' ( * + = x y 0 z + d d ' ( * + = x y 0 z d + ' ( * + = d ' ( * + x y z ' ( * + Leading to the following 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 2 Now look at what happens to depth. Contrast this with previous version.
22 Let distance d go to infinity. Formulation Formulation 2 X Y 0 Recall formulation 2 when considering how projection changes with increased focal length. 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 22
23 Moving to Formulation 3 Review origin at PRP. Review 2 origin at image center. Review and 2 No useful information on the z-axis We now have a new goal Project into a cannonical view volumne A rectangular value with bounds: U: - to V: - to D: 0 to 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 23
24 Remember When We Started What happens if you multiply a point in homogeneous coordinates by a scalar? Nothing! 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 24 x y z w! " % = sx sy sz sw! " % = s x y z w! " %
25 Scalar Multiplication Continued Multiply a homogeneous matrix by a scalar? Again nothing changes.! "! " ax + by + cz + dw ex + fy + gz + hw ix + jy + kz + lw mx + ny + oz + pw ax + by + cz + dw ex + fy + gz + hw ix + jy + kz + lw mx + ny + oz + pw! = % "! = % " a b c d e f g h i j k l m n o p! % " x y z w ( ( s( ix + jy + kz + lw ( s ax + by + cz + dw s ex + fy + gz + hw s mx + ny + oz + pw %! = s % " a b c d e f g h i j k l m n o p! % " x y z w % 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 25
26 Form Textbook Derivation d 0 Lecture Form Equivalent d d d 0 0 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 26
27 Now introduce clipping planes Text introduces n the near clipping plane. Also introduces f the far clipping plane. Sets what first called d to n. The handling of z now carries information? " n n n + f fn % ' ' ' ' " nx ny zn + zf fn z % " ' ' ' = ' ' n n n + f fn %" ' ' ' ' x y z % ' ' ' ' 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 27
28 Visualize View Volume (View 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 28
29 Visualize View Volume (View 2 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 29
30 Consider Some Key Points n f 0 0 f 0 0 n 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 30
31 The Algebra The corner of the near clipping plane maps to on image plane. The corner on the far clipping plane comes toward the optical axis and becomes n/f on the image plane. 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 3
32 Visualize Frustum Change (0n Becomes a rectangle (top down view with space toward the back being compressed to fit larger area into same width. (00 0/7/7 CSU CS 40 Fall 207 Ross Beveridge Bruce Draper 32
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