776 Computer Vision Jan-Michael Frahm Spring 2012

Scheduling January 20 th is the first Friday class 2:00 pm – 3:15 pm in SN 115 3:30pm to 4:45 pm in SN 115

Camera

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

Pinhole camera Add a barrier to block off most of the rays o This reduces blurring o The opening is known as the aperture Slide by Steve Seitz

Pinhole Camera (Camera obscura) Interior of camera obscura (Sunday Magazine, 1838) Camera obscura (France, 1830)

Pinhole camera model Pinhole model: o Captures pencil of rays – all rays through a single point o The point is called Center of Projection (focal point) o The image is formed on the Image Plane Slide by Steve Seitz

Figures © Stephen E. Palmer, 2002 Dimensionality reduction: from 3D to 2D 3D world2D image What is preserved? Straight lines, incidence (except for parallel) What have we lost? Angles, lengths Slide by A. Efros

Projection properties Many-to-one: any points along same visual ray map to same point in image Points → points o But projection of points on focal plane is undefined Lines → lines (collinearity is preserved) o But lines through focal point (visual rays) project to a point Planes → planes (or half-planes) o But planes through focal point project to lines slide: S. Lazebnik

Vanishing points Each direction in space has its own vanishing point o All lines going in that direction converge at that point o Exception: directions parallel to the image plane slide: S. Lazebnik

Vanishing points Each direction in space has its own vanishing point o All lines going in that direction converge at that point o Exception: directions parallel to the image plane How do we construct the vanishing point of a line? o What about the vanishing line of a plane? image plane camera center line on ground plane vanishing point slide: S. Lazebnik

One-point perspective Masaccio, Trinity, Santa Maria Novella, Florence, One of the first consistent uses of perspective in Western art slide: S. Lazebnik

Perspective distortion Problem for architectural photography: converging verticals Source: F. Durand

Perspective distortion Problem for architectural photography: converging verticals Solution: view camera (lens shifted w.r.t. film) Source: F. Durand 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

Perspective distortion Problem for architectural photography: converging verticals Result: Source: F. Durand

Perspective distortion image: Wikipedia

Perspective distortion What does a sphere project to? Image source: F. Durand

Perspective distortion What does a sphere project to? slide: S. Lazebnik

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

Perspective distortion: People slide: S. Lazebnik

Modeling projection Projection equations o Compute intersection with image plane of ray from P = (x,y,z) to O o Derived using similar triangles Source: J. Ponce, S. Seitz We get the projection by throwing out the last coordinate: x y z f

Homogeneous coordinates Is this a linear transformation? no—division by z is nonlinear Slide by Steve Seitz

Affine coordinates e i : affine basis vectors o: coordinate origin e2e2 e3e3 e1e1 o M Vector relative to o: a1a1 a3a3 a2a2 Point in affine coordinates: Vector: relative to some origin Point: absolute coordinates

Homogeneous coordinates e2e2 e3e3 e1e1 o M Unified notation: include origin in affine basis Affine basis matrix Homogeneous Coordinates of M a1a1 a3a3 a2a2

Properties of affine transformation Parallelism is preserved ratios of length, area, and volume are preserved Transformations can be concatenated: Transformation T affine combines linear mapping and coordinate shift in homogeneous coordinates  Linear mapping with A 3x3 matrix  coordinate shift with t 3 translation vector

Special transformation: Rotation Rigid transformation: Angles and lengths preserved R is orthonormal matrix defined by three angles around three coordinate axes ezez eyey exex  Rotation with angle  around e z

Special transformation: Rotation Rotation around the coordinate axes can be concatenated: ezez eyey exex    Inverse of rotation matrix is transpose:

Projective geometry in 2D Projective space is space of rays emerging from O o view point O forms projection center for all rays o rays v emerge from viewpoint into scene o ray g is called projective point, defined as scaled v: g= v x y w O

Projective and homogeneous points Given: Plane  in R 2 embedded in R 3 at coordinates w=1 o viewing ray g intersects plane at v (homogeneous coordinates) o all points on ray g project onto the same homogeneous point v o projection of g onto  is defined by scaling v=g/ = g/w w=1 O  (R 2 ) w x y

Finite and infinite points All rays g that are not parallel to  intersect at an affine point v on . w=1 O  The ray g(w=0) does not intersect . Hence v  is not an affine point but a direction. Directions have the coordinates (x,y,0) T Projective space combines affine space with infinite points (directions). (R 2 ) Remember exception of no vanishing point for directions parallel to the image plane

Perspective projection Perspective projection models pinhole camera:  scene geometry is affine R 3 space with coordinates M=(X,Y,Z,1) T  camera focal point in O=(0,0,0,1) T, camera viewing direction along Z  image plane (x,y) in  (R 2 ) aligned with plane (X,Y) at Z= Z 0  scene point M projects onto point M p on plane surface X Y Z0Z0 O  Image plane Camera center Z (Optical axis) x y (R 2 )

Projective Transformation Projective Transformation maps M onto M p X Y O Projective Transformation linearizes projection

Perspective Projection Dimension reduction from R 3 into R 2 by projection onto  (R 2 ) X Y O  (R 2 )

Perspective Projection Dimension reduction from R 3 into R 2 by projection onto  (R 2 ) X Y O  (R 2 )

Projection in general pose Rotation [R] Projection center C M World coordinates Projection: mpmp

X Y Image center c= (c x, c y ) T Projection center Z (Optical axis) Pixel scale f= (f x,f y ) T x y Pixel coordinates m = (y,x) T Image plane and image sensor A sensor with picture elements (Pixel) is added onto the image plane Image sensor Image-sensor mapping: Pixel coordinates are related to image coordinates by affine transformation K with five parameters:  Image center c=(c x,c y ) T defines optical axis  Pixel size and pixel aspect ratio defines scale f=(f x,f y ) T  image skew s to model angle between pixel rows and columns

Projection matrix P Camera projection matrix P combines:  inverse affine transformation T cam -1 from general pose to origin  Perspective projection P 0 to image plane at Z 0 =1  affine mapping K from image to sensor coordinates World to camera coord. trans. matrix (4x4) Perspective projection matrix (3x4) Camera to pixel coord. trans. matrix (3x3) = 2D point (3x1) 3D point (4x1)

Orthographic Projection Special case of perspective projection o Distance from center of projection to image plane is infinite o Also called “parallel projection” o What’s the projection matrix? Image World Slide by Steve Seitz

Home-made pinhole camera Why so blurry? Slide by A. Efros

Shrinking the aperture Why not make the aperture as small as possible? o Less light gets through o Diffraction effects… Slide by Steve Seitz

Shrinking the aperture

Adding a lens A lens focuses light onto the film o Thin lens model: Rays passing through the center are not deviated (pinhole projection model still holds) Slide by Steve Seitz

Adding a lens A lens focuses light onto the film o 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 focal point f

Adding a lens A lens focuses light onto the film o There is a specific distance at which objects are “in focus” other points project to a “circle of confusion” in the image “circle of confusion” Slide by Steve Seitz

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’)? f D D’ Slide by Frédo Durand object image plane lens

Thin lens formula f D D’ Similar triangles everywhere! Slide by Frédo Durand object image plane lens

Thin lens formula f D D’ Similar triangles everywhere! y’ y y’/y = D’/D Slide by Frédo Durand object image plane lens

Thin lens formula f D D’ Similar triangles everywhere! y’ y y’/y = D’/D y’/y = (D’-f)/f Slide by Frédo Durand object image plane lens

Thin lens formula f D D’ 1 D 11 f += Any point satisfying the thin lens equation is in focus. Slide by Frédo Durand object image plane lens

Depth of Field Slide by A. Efros

How can we control the depth of field? Changing the aperture size affects depth of field o A smaller aperture increases the range in which the object is approximately in focus o But small aperture reduces amount of light – need to increase exposure Slide by A. Efros

Varying the aperture Large aperture = small DOFSmall aperture = large DOF Slide by A. Efros

Field of View Slide by A. Efros What does FOV depend on?

f Field of View Smaller FOV = larger Focal Length Slide by A. Efros f FOV depends on focal length and size of the camera retina

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

Same effect for faces standard wide-angletelephoto Source: F. Durand

The dolly zoom Continuously adjusting the focal length while the camera moves away from (or towards) the subject slide: S. Lazebnik

The Dolly Zoom

Real Lenses Zoom lens image: Simal

slide: S. Lazebnik Lens Flaws: Chromatic Aberration Lens has different refractive indices for different wavelengths: causes color fringing Near Lens Center Near Lens Outer Edge

Lens flaws: Spherical aberration Spherical lenses don’t focus light perfectly Rays farther from the optical axis focus closer slide: S. Lazebnik

Lens flaws: Vignetting slide: S. Lazebnik

No distortionPin cushionBarrel Radial Distortion o Caused by imperfect lenses o Deviations are most noticeable near the edge of the lens slide: S. Lazebnik

Digital camera A digital camera replaces film with a sensor array o Each cell in the array is light-sensitive diode that converts photons to electrons o Two common types Charge Coupled Device (CCD) Complementary metal oxide semiconductor (CMOS) o Slide by Steve Seitz

Color sensing in camera: Color filter array Source: Steve Seitz Estimate missing components from neighboring values (demosaicing) Why more green? Bayer grid Human Luminance Sensitivity Function

Problem with demosaicing: color moire Slide by F. Durand

The cause of color moire detector Fine black and white detail in image misinterpreted as color information Slide by F. Durand

Color sensing in camera: Prism Requires three chips and precise alignment More expensive CCD(B) CCD(G) CCD(R) slide: S. Lazebnik

Color sensing in camera: Foveon X3 Source: M. Pollefeys CMOS sensor Takes advantage of the fact that red, blue and green light penetrate silicon to different depths better image quality

Digital camera artifacts Noise low light is where you most notice noisenoise light sensitivity (ISO) / noise tradeoff stuck pixels In-camera processing oversharpening can produce haloshalos Compression JPEG artifacts, blocking Blooming charge overflowing into neighboring pixelsoverflowing Smearing o columnwise overexposue Color artifacts purple fringing from microlenses, purple fringing white balance Slide by Steve Seitz

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) Niepce, “La Table Servie,” 1822 CCD chip Alhacen’s notes

Early color photography Sergey Prokudin-Gorskii ( ) Photographs of the Russian empire ( ) Lantern projector

First digitally scanned photograph 1957, 176x176 pixels