Announcements. Focus! Thin Lens Models. New Topic. Intensity Image Formation. Bi-directional: two focal points! Thin Lens Model

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1 Focus! Models Lecture #17 Tuesda, November 1 st, 2011 Announcements Programming Assignment #3 Is due a week rom Tuesda Midterm #2: two weeks rom Tuesda GTA survers: 2 New Topic Toda we will talk about ocus, and introduce a thin lens model Intensit Image Formation To be clear: You will NOT have to implement a thin lens model You ma (probabl will) be asked about them on the midterm Screen Aperture 3 4 Model Ras entering parallel on one side converge at ocal point. Ras diverging rom the ocal point become parallel. Bi-directional: two ocal points! R R 5 6

2 Constraints #1 All ras emanating rom a single point in space must converge on a single point in the image plane (deinition o ocus) Fundamental Equation o es P + Q #2 An ra entering the lens parallel to the optical axis on one side goes through the ocus point on the other side S O s #3 An ra entering the lens rom the ocus point on one side emerges parallel to the axis on the other side R Note: P is not too ar rom optical axis p 7 8 Fundamental Equation (II) The ra PQ (parallel to the optical axis) must be delected to pass through b propert #2 The ra PR must be delected so that it becomes parallel to the optical axis b propert #3 Ater delection, PQ & PR must intersect at p, b propert #1. Now, use similar triangles. Fundamental Eq. (III) Y P S Y = R Q Y O Substitute or and solve: 2 =, or Y = s p = Constraints Questions #1 For ever point on the image plane, there exists a point in space such that all ras emanating rom that point converge on the point in the image plane (deinition o ocus) Proo that Qp & R intersect: n QP deines a plane n Since R is a continuation o Q, QPR is a plane n Since Rp is parallel to QP, QPRp is a plane n Since is parallel to QP, QPRpis a plane n Qp and R are not parallel. QED. What happens to out-o-ocus points? Wh (and how) does a thin lens implement perspective projection? How do ou ocus on objects that are ver ar awa? You can recover the depth o a static scene with a single camera how? 11 12

3 Out o Focus Images What happens when +? + + FL Spherical Blurring Out-o-ocus image planes Implementing a thin lens Given pixel (a,b), what point is in ocus? Intersect two ras: One ra starts at (a, b, +), goes parallel to the optical axis until it hits the lens at (a,b,0), and then heads toward the let ocal point (0,0,) Another ra goes rom (a,b,+) toward the right ocal point (0, 0, -). It intersects the lens at (c, d, 0), and then proceeds in the direction (0,0,1). These two ras intersect at the point in ocus (x,,) Ra Tracing + How do ou do ra tracing with a thin lens? Man ras o light converge on ever pixel, so a pixel value is now the average o the light returned along man ras. How man? How much time do ou have. Pick a set o points on the lens, compute their 3D position once, and use them or ever pixel For ever pixel: Compute the point (X, Y, ) where the ras converge See last slide Note that this point ma not be on a surace in our model Throw a ra rom ever lens point toward (X, Y, ) Recursivel compute the light returned along each ra Previous Slide, Illustrated Focused Point in 3D World Image Plane Model The basic algorithm or simulating a thin lens thereore looks like: For ever pixel { Find point in 3D world that is in ocus Shoot multiple ras rom lens surace to point Ra Trace each ra Average returned colors } Model (II) Or, in more detail: Select set o lens surace points For ever pixel { Find point in 3D world that is in ocus Intersect 2 ras (w/o suraces) a) through right p & lens b) through lens & let p For ever lens surace point Shoot ra rom lens surace to point Average returned colors } 17 18

4 Model IIIa 1) Compute equation o lens plane (trivial) 2) Compute positions o lens points 3) Compute the center o image plane 4) For ever pixel : a) Compute pixel position b) Compute ra rom pixel to lens surace c) Compute ra rom lens surace point through ront FP d) Compute ra rom pixel through back FP to lens e) Compute ra rom lens surace parallel to optic axis ) Intersect ras c & e, ind (in ocus) point P Model IIIb g) For ever lens point L i) Compute ra R rom L toward P ii) Ra Trace R as i rom pinhole camera h) Average values returned or all L s. i) Store averaged value in pixel Wait! What about the aperture? Field o View Screen Aperture: limits the eective diameter o the lens Aperture FOV = tan 1 d 2 d α Field o View (II) The calculation o (X, Y, ) is theoretical, and ma proceed even i the two ras used to calculate it are blocked b the aperture. However, the ras rom some lens points to (X, Y, ) ma be blocked These ras contribute (0,0,0) into the pixel average Points near the edge o the ield o view are thereore darker than those near the middle. Partial Obscuration Partiall obscured objects ma be dimmer than other objects; I out-o-ocus, blurring ma be non-smmetric 23 24

5 Chromatic Aberration The ocal length o a lens is a unction o the wavelength o the light. Thereore, dierent colors o light rom a single point will be out-o-ocus Spherical Distortion Most lenses exhibit spherical (a.k.a. radial) distortion which creates a slightl curved ocal plane Curved Focal Plane 11/01/ /01/10 26 Spherical Distortion (II) Spherical distortion is both geometric & ocal Points are distorted geometricall rom the true perspective position as a result o lattening out the plane. 1) convert points to 2D spherical coordinates, based in the middle o the image 2 3 2) ρ = ρ' ( αρ' + βρ' + χρ' +...) Points also become more blurred as the approach the edge o the image Atmospheric Eects So ar, we have assumed air is invisible For short distances, this is true For long distances, not so much Distant objects turn vaguel blue Hence purple mountain s majest, the blue ridge mountains, etc. It s a unction o humidit In smogg environments, objects turn brown In smoke environments, gra 11/01/ /01/10 28 Atmospheric Eects (II) Particles (e.g. water/smog) in the air relect light toward our camera. The extent depends on the densit o particles (e.g. humidit) and the length o the viewing ra. To implement atmospheric eects: Where D is densit (a ver small number) P-L is length o viewing ra P-L D is << 1 (Simple linear model) 11/01/10 29