Overview. The pinhole projection model. Cameras with lenses. Qualitative properties. Field of view Lens aberrations. Sensors.
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1 Cameras
2 Overview The pinhole projection model Qualitative properties Perspective projection matrix Cameras with lenses Depth of focus Field of view Lens aberrations Digital it cameras Sensors Color Artifacts
3 Let s design a camera Idea 1: put a piece of film in front of an object Do we get a reasonable image? Slide by Steve Seitz
4 Pinhole camera Add a barrier to block off most of the rays This reduces blurring The opening is known as the aperture Slide by Steve Seitz
5 Pinhole camera model Pinhole model: Captures pencil of rays all rays through a single point The point is called Center of Projection (focal point) The image is formed on the Image Plane Slide by Steve Seitz
6 Dimensionality reduction: from 3D to 2D 3D world 2D image Point of observation What is preserved? Straight lines, incidence What have we lost? Angles, lengths Slide by A. Efros Figures Stephen E. Palmer, 2002
7 Projection properties Many-to-one: any points along same visual ray map to same point in image Points points But projection of points on focal plane is undefined Lines lines (collinearity is preserved) ed) But lines through focal point (visual rays) project to a point Planes planes (or half-planes) But planes through focal point project to lines
8 Vanishing points Each direction in space has its own vanishing point All lines going in that direction converge at that point Exception: directions parallel to the image plane
9 Vanishing points Each direction in space has its own vanishing point All lines going in that direction converge at that point Exception: directions parallel to the image plane How do we construct the vanishing point of a line? What about the vanishing line of a plane? image plane vanishing point camera center line on ground plane
10 One-point perspective Masaccio, Trinity, Santa Maria Novella, Florence, One of the first consistent uses of perspective in Western art
11 Perspective distortion Problem for architectural photography: converging verticals Source: F. Durand
12 Perspective distortion Problem for architectural photography: converging verticals Tilting the camera upwards results in converging verticals Keeping the camera level, with an ordinary lens, captures only the bottom portion of the building Shifting the lens upwards results in a picture of the entire subject Solution: view camera (lens shifted w.r.t. film) Source: F. Durand
13 Perspective distortion Problem for architectural photography: converging verticals Result: Source: F. Durand
14 Perspective distortion What does a sphere project to? Image source: F. Durand
15 Perspective distortion What does a sphere project to?
16 Perspective distortion The exterior columns appear bigger The distortion is not due to lens flaws Problem pointed out by Da Vinci Slide by F. Durand
17 Perspective distortion: People
18 Modeling projection f y z x The coordinate system The optical center (O) is at the origin The image plane is parallel to xy-plane (perpendicular to z axis) Source: J. Ponce, S. Seitz
19 Modeling projection y f z P j ti ti x Projection equations Compute intersection with image plane of ray from P = (x,y,z) to O Derived using similar triangles ),, ( ),, ( f z y f z x f z y x We get the projection by throwing out the last coordinate: Source: J. Ponce, S. Seitz We get the projection by throwing out the last coordinate: ), ( ),, ( z y f z x f z y x
20 Homogeneous coordinates x y ( x, y, z) ( f, f ) z z Is this a linear transformation? no division by z is nonlinear Trick: add one more coordinate: homogeneous image coordinates homogeneous scene coordinates Converting from homogeneous coordinates Slide by Steve Seitz
21 Perspective Projection Matrix Projection is a matrix multiplication using homogeneous coordinates
22 Perspective Projection Matrix Projection is a matrix multiplication using homogeneous coordinates x y x ) ( y f x f divide by the third = f z y z y f / 1 0 1/ ), ( z y f z f coordinate 1
23 Perspective Projection Matrix Projection is a matrix multiplication using homogeneous coordinates x x y x y = y ( f, f ) z z z 0 0 1/ f 0 z / f 1 coordinate In practice: lots of coordinate transformations divide by the third 2D point (3x1) = Camera to pixel coord. trans. matrix (3x3) Perspective projection matrix (3x4) World to camera coord. trans. matrix (4x4) 3D point (4x1)
24 Orthographic Projection Special case of perspective projection Distance from center of projection to image plane is infinite Image World Also called parallel projection What s the projection matrix? Slide by Steve Seitz
25 Building a real camera
26 Camera Obscura Basic principle known to Mozi ( BCE), Aristotle ( BCE) Drawing aid for artists: described by Leonardo da Vinci ( ) Gemma Frisius, 1558 Source: A. Efros
27 Abelardo Morell From Grand Images Through a Tiny Opening, Photo District News, February 2005 Camera Obscura Image of Manhattan View Looking South in Large Room,
28 Home-made pinhole camera Why so blurry? Slide by A. Efros
29 Shrinking the aperture Why not make the aperture as small as possible? Less light gets through Diffraction effects Slide by Steve Seitz
30 Shrinking the aperture
31 Adding a lens A lens focuses light onto the film Thin lens model: Rays passing through the center are not deviated (pinhole projection model still holds) Slide by Steve Seitz
32 Adding a lens focal point f A lens focuses light onto the film Thin lens model: Rays passing through the center are not deviated (pinhole projection model still holds) All parallel rays converge to one point on a plane located at the focal length f Slide by Steve Seitz
33 Adding a lens circle of confusion A lens focuses light onto the film There is a specific distance at which objects are in focus other points project to a circle of confusion in the image Slide by Steve Seitz
34 Thin lens formula What is the relation between the focal length (f), the distance of the object from the optical center (D), and the distance at which the object will be in focus (D )? D f D image plane lens object Slide by Frédo Durand
35 Thin lens formula Similar triangles everywhere! D f D image plane lens object Slide by Frédo Durand
36 Thin lens formula Similar triangles everywhere! y /y = D /D D D f y y image plane lens object Slide by Frédo Durand
37 Thin lens formula Similar triangles everywhere! D D f y y /y = D /D y /y y/y = (D -f)/f y image plane lens object Slide by Frédo Durand
38 Thin lens formula = 1 D D f Any point satisfying the thin lens equation is in focus. D f D image plane lens object Slide by Frédo Durand
39 Depth of Field Slide by A. Efros
40 How can we control the depth of field? Changing the aperture size affects depth of field A smaller aperture increases the range in which the object is approximately in focus But small aperture reduces amount of light need to increase exposure Slide by A. Efros
41 Varying the aperture Large aperture = small DOF Small aperture = large DOF Slide by A. Efros
42 Field of View Slide by A. Efros
43 Field of View What does FOV depend on? Slide by A. Efros
44 Field of View f f FOV depends on focal length and size of the camera retina Smaller FOV = larger Focal Length Slide by A. Efros
45 Field of View / Focal Length Large FOV, small f Camera close to car Small FOV, large f Camera far from the car Sources: A. Efros, F. Durand
46 Same effect for faces wide-angle standard telephoto Source: F. Durand
47 Approximating an affine camera Source: Hartley & Zisserman
48 The dolly zoom Continuously adjusting the focal length while the camera moves away from (or towards) the subject
49 The dolly zoom Continuously adjusting the focal length while the camera moves away from (or towards) the subject The Vertigo shot Examples of dolly zoom from movies (YouTube)
50 Real lenses
51 Lens Flaws: Chromatic Aberration Lens has different refractive indices for different wavelengths: causes color fringing Near Lens Center Near Lens Outer Edge
52 Lens flaws: Spherical aberration Spherical lenses don t focus light perfectly Rays farther from the optical axis focus closer
53 Lens flaws: Vignetting
54 Radial Distortion Caused by imperfect lenses Deviations are most noticeable near the edge of the lens No distortion Pin cushion Barrel
55 Digital camera A digital camera replaces film with a sensor array Each cell in the array is light-sensitive diode that converts photons to electrons Two common types Charge Coupled Device (CCD) Complementary metal oxide semiconductor (CMOS) Slide by Steve Seitz
56 Color sensing in camera: Color filter array Bayer grid Estimate missing components from neighboring values (demosaicing) Why more green? Human Luminance Sensitivity Function Source: Steve Seitz
57 Problem with demosaicing: color moire Slide by F. Durand
58 The cause of color moire detector Fine black and white detail in image misinterpreted i t as color information Slide by F. Durand
59 Color sensing in camera: Prism Requires three chips and precise alignment More expensive CCD(R) CCD(G) CCD(B)
60 Color sensing in camera: Foveon X3 CMOS sensor Takes advantage of the fact that red, blue and green light penetrate silicon to different depths better image quality Source: M. Pollefeys
61 Digital camera artifacts Noise low light is where you most notice noise light sensitivity (ISO) / noise tradeoff stuck pixels In-camera processing oversharpening can produce halos Compression JPEG artifacts, blocking Blooming charge overflowing into neighboring pixels Color artifacts purple fringing from microlenses, white balance Slide by Steve Seitz
62 Historic milestones Pinhole model: Mozi ( BCE), Aristotle ( BCE) Principles of optics (including lenses): Alhacen ( CE) Camera obscura: Leonardo da Vinci ( ), Johann Zahn ( ) First photo: Joseph Nicephore Niepce (1822) Daguerréotypes (1839) Photographic film (Eastman, 1889) Cinema (Lumière Brothers, 1895) Color Photography (Lumière Brothers, 1908) Television (Baird, Farnsworth, Zworykin, 1920s) First consumer camera with CCD Sony Mavica (1981) First fully digital camera: Kodak DCS100 (1990) Alhacen s notes Niepce, La Table Servie, 1822 CCD chip
63 Early color photography Sergey Prokudin-Gorskii ( ) Photographs of the Russian empire ( ) Assignment 1 (due February 1)! Lantern projector
64 First digitally scanned photograph 1957, 176x176 pixels
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