Announcements. Introduction to Cameras. The Key to Axis Angle Rotation. Axis-Angle Form (review) Axis Angle (4 steps) Mechanics of Axis Angle

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1 Ross Beerige Bruce Draper Introuction to Cameras September th 25 Announcements PA ue eek from Tuesa Q: hat i I mean b robust I/O? Hanle arious numbers of erte/face features Check for count matches Goo error messages Other Questions? 2 Ross Beerige Bruce Draper Ais-Angle Form (reie θ W R = θw = θ r r 2 r 3 r R = 2 r 22 r 23 r 3 r 32 r 33 Pros Cons an going from ais angle to 44 matri form. Ross Beerige Bruce Draper The Ke to Ais Angle Rotation Rotating b θ aroun the Z ais is no problem: cos( θ sin( θ sin( θ cos( θ So to rotate aroun ais W first rotate the coorinate sstems so that W = Z 3 4 Ross Beerige Bruce Draper Mechanics of Ais Angle Ho o e rotate the ata to make the angle of rotation Z? Multiplication is projection onto the ros of M If M is orthonormal it is a rotation matri Magnitue of eer ro is Dot prouct of eer pair of ros is If the thir ro is the ais of rotation Z becomes the ais of rotation Ross Beerige Bruce Draper Ais Angle (4 steps Step : normalie the ais of rotation Write the normalie ais as = ( Step 2: pick an ais M not parallel to W Heuristic: pick the smallest term in set it to an renormalie to create m Step 3: create U = W M Step 4: pick an ais perpenicular to u V = W U (or U W 5 6

2 Ross Beerige Bruce Draper Ais Angle R ω Matri No put those together in a rotation matri: u u u R ω = Ais of Rotation Ross Beerige Bruce Draper Ais Angle (Putting the Pieces Together To rotate b θ aroun ω: P = ( R ω R Zθ R ω P = ( R T ω R Zθ R ω P = MP 7 8 Ross Beerige Bruce Draper 3D Vieing as Virtual Camera To take a picture ith a camera or to rener an image ith computer graphics e nee to: Perspectie Ross Beerige Bruce Draper Position the camera/iepoint in 3D space Orient the camera/iepoint in 3D space Focus camera -- e on t o this step Crop image to the aperture/ino 9 Ross Beerige Bruce Draper Orthographic Projection If not for the fog ou coul see foreer an nothing eer oul look smaller. Ross Beerige Bruce Draper Orthographic / Perspectie Think About Ras 2

3 Ross Beerige Bruce Draper Is Perspectie Alas Better? Ross Beerige Bruce Draper CSU 24 Math: Orthographic Projection Simpl rop a imension. No Technical programs incluing for eample Maple often faor orthographic projection. Think of a bug hitting a inshiel. No more ais (no more bug Photo b Brian Jeff Booth site.jeffbooth.net (creatie common License 3 4 Ross Beerige Bruce Draper Perspectie Projection Light ras pass through the focal point. a.k.a. principal reference point PRP. The image plane is an infinite plane in front of (or behin the focal point. Images are forme b ras of light passing through the image plane Common conention: Image points are (u Worl points are ( Ross Beerige Bruce Draper Wh Pinhole Camera? Because ou can buil a camera that eactl fits this escription: Create a full-enclose black bo So that no light enters Put a piece of film insie it facing front Punch a pin-hole in the front face of the bo What oesn t this camera hae? What is this camera s epth-of-fiel? Wh on t e buil cameras this a? 5 6 Ross Beerige Bruce Draper Histor The Camera Obscura - see Wikipeia Pre-ates photographic cameras. Theor: Mo-Ti (China BC Practice: Abu Ali Al-Hasan Ibn al-haitham (~ AD Western Painting: Johannes Vermeer (~66 AD Ross Beerige Bruce Draper Perspectie Projection: 3 Formulations Where e place the origin matters Ho e hanle alues matters Form : Origin at focal point alues constant Form 2: Origin at image center alues are ero Form 3: Origin at focal point proportional to epth 7 8

4 Perspectie Projection Form The ke to perspectie projection is that all light ras meet at the PRP (focal point. Notice that e are looking on the Z ais ith the origin at the focal point an the image plane at =. P( Ross Beerige Bruce Draper P P P b similar triangles: P u P u = P u = P horiontal = P P P P P Ross Beerige Bruce Draper P P = er+cal = P = P P P P P P 9 2 Ross Beerige Bruce Draper Perspectie Projection Matri Problem: iision of one ariable b another is a non-linear operation. Solution: homogeneous coorinates / Ross Beerige Bruce Draper Perspectie Matri (II = = = u Normalie Projection Matri times a Point Point in Point in (u Non-normalie coorinates Homogeneous coorinates 2 22 Ross Beerige Bruce Draper What happens to Z? What happens to the Z imension? u = = = The Z imension projects to Wh? Because (u is a 3D point on the image ( = plane Perspectie Projection Form 2 P = Ross Beerige Bruce Draper O P P P = P + P + P 23 24

5 Leaing to the folloing * + = + ( ( + Ross Beerige Bruce Draper + * = + ( + No look at hat happens to epth. Contrast this ith preious ersion. * * = + ( + ( * + Ross Beerige Bruce Draper Let istance go to infinit. Formulation Formulation 2 X Y Recall formulation 2 hen consiering ho projection changes ith increase focal length Ross Beerige Bruce Draper Moing to Formulation 3 Reie origin at PRP. Reie 2 origin at image center. Reie an 2 No useful information on the -ais We no hae a ne goal Project into a cannonical ie olumne A rectangular alue ith bouns: U: - to V: - to D: to Ross Beerige Bruce Draper Remember When We Starte What happens if ou multipl a point in homogeneous coorinates b a scalar? Nothing = s s s s = s Ross Beerige Bruce Draper Scalar Multiplication Continue What happens if ou multipl a homogeneous matri b a scalar? Nothing a + b + c + a b c e + f + g +h e f g h i + j + k +l = i j k l m + n + o + p m n o p a + b + c + s a+b+c+ ( a b c e + f + g +h s( e+ f+ g+h i + j + k +l = e f g h = s s( i+ j+ k+l i j k l m + n + o + p s( m + n + o + p m n o p Ross Beerige Bruce Draper Form Tetbook Deriation equialent to Lecture Form p. 52 (er top D 29 3

6 Ross Beerige Bruce Draper No introuce clipping planes Tet introuces n the near clipping plane. Also introuces f the far clipping plane. Sets hat first calle to n. The hanling of no carries information? Ross Beerige Bruce Draper Visualie Vie Volume (Vie n n n + f fn n n n n + f fn = n n + f fn 3 32 Ross Beerige Bruce Draper Visualie Vie Volume (Vie 2 33