The CameraHow do we see the world?Pinhole cameraPinhole camera modelDimensionality Reduction Machine (3D to 2D)Funny things happen…Parallel lines aren’t…Distances can’t be trusted...…but humans adopt!Building a real cameraCamera ObscuraHome-made pinhole cameraShrinking the apertureSlide 14The reason for lensesSlide 16FocusFocus and DefocusVarying FocusDepth Of FieldDepth of FieldAperture controls Depth of FieldVarying the apertureNice Depth of Field effectField of View (Zoom)Slide 26Slide 27FOV depends of Focal LengthSlide 29Field of View / Focal LengthFun with Focal Length (Jim Sherwood)Large Focal Length compresses depthLens FlawsLens Flaws: Chromatic AberrationChromatic AberrationRadial Distortion (e.g. ‘Barrel’ and ‘pin-cushion’)Radial DistortionSlide 38Modeling ProjectionsModeling projectionSlide 41Homogeneous coordinatesPerspective ProjectionOrthographic ProjectionSpherical ProjectionProgramming Assignment #1The Camera15-463: Computational PhotographyAlexei Efros, CMU, Fall 2005How do we see the world?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 SeitzPinhole cameraAdd a barrier to block off most of the rays•This reduces blurring•The opening known as the aperture•How does this transform the image?Slide by Steve SeitzPinhole camera modelPinhole model:•Captures pencil of rays – all rays through a single point•The point is called Center of Projection (COP)•The image is formed on the Image Plane•Effective focal length f is distance from COP to Image PlaneSlide by Steve SeitzPoint of observationFigures © Stephen E. Palmer, 2002Dimensionality Reduction Machine (3D to 2D)3D world 2D imageWhat have we lost?•Angles•Distances (lengths)Funny things happen…Parallel lines aren’t…Figure by David ForsythDistances can’t be trusted...Figure by David Forsyth…but humans adopt!http://www.michaelbach.de/ot/sze_muelue/index.html Müller-Lyer IllusionWe don’t make measurements in the image planeBuilding a real cameraCamera ObscuraThe first camera•Known to Aristotle•Depth of the room is the effective focal lengthCamera Obscura, Gemma Frisius, 1558Home-made pinhole camera http://www.debevec.org/Pinhole/Why soblurry?Shrinking the apertureWhy not make the aperture as small as possible?•Less light gets through•Diffraction effects…Less light gets throughSlide by Steve SeitzShrinking the apertureThe reason for lensesSlide by Steve SeitzIdeal Lens: Same projection as pinhole but gathers more light!foi111ioLens Formula:• f is the focal length of the lens – determines the lens’s ability to bend (refract) light • f different from the effective focal length f discussed before!PP’fImage Formation using LensesSlide by Shree NayarFocusFocus and DefocusA 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•How can we change focus distance?“circle of confusion”Slide by Steve SeitzVarying FocusRen NgDepth Of FieldDepth of Fieldhttp://www.cambridgeincolour.com/tutorials/depth-of-field.htmAperture controls Depth of FieldChanging 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 exposureVarying the aperture f/2.8Large apeture = small DOFf/22Small apeture = large DOFNice Depth of Field effectField of View (Zoom)Field of View (Zoom)Field of View (Zoom)fFOV depends of Focal LengthSmaller FOV = larger Focal LengthFrom Zisserman & HartleyField of View / Focal LengthLarge FOVCamera close to carSmall FOVCamera far from the carFun with Focal Length (Jim Sherwood)http://www.hash.com/users/jsherwood/tutes/focal/Zoomin.movLarge Focal Length compresses depth© 1995-2005 Michael Reichmann 400 mm 200 mm 100 mm 50 mm 28 mm 17 mmLens FlawsLens Flaws: Chromatic AberrationDispersion: wavelength-dependent refractive index•(enables prism to spread white light beam into rainbow)Modifies ray-bending and lens focal length: f()color fringes near edges of imageCorrections: add ‘doublet’ lens of flint glass, etc.Chromatic Aberration Near Lens CenterNear Lens CenterNear Lens Outer EdgeNear Lens Outer EdgeRadial Distortion (e.g. ‘Barrel’ and ‘pin-cushion’)straight lines curve around the image centerRadial DistortionRadial distortion of the image•Caused by imperfect lenses•Deviations are most noticeable for rays that pass through the edge of the lensNo distortion Pin cushion BarrelRadial DistortionModeling ProjectionsModeling projectionThe coordinate system•We will use the pin-hole model as an approximation•Put the optical center (Center Of Projection) at the origin•Put the image plane (Projection Plane) in front of the COP–Why?•The camera looks down the negative z axis–we need this if we want right-handed-coordinates– Slide by Steve SeitzModeling projectionProjection equations•Compute intersection with PP of ray from (x,y,z) to COP•Derived using similar triangles (on board)•We get the projection by throwing out the last coordinate:Slide by Steve SeitzHomogeneous coordinatesIs this a linear transformation?Trick: add one more coordinate:homogeneous image coordinateshomogeneous scene coordinatesConverting from homogeneous coordinates•no—division by z is nonlinearSlide by Steve SeitzPerspective ProjectionProjection is a matrix multiply using homogeneous coordinates:divide by third coordinateThis is known as perspective projection•The matrix is the projection matrix•Can also formulate as a 4x4divide by fourth coordinateSlide by Steve SeitzOrthographic ProjectionSpecial case of perspective projection•Distance from the COP to the PP is infinite•Also called “parallel projection”•What’s the projection matrix?ImageWorldSlide by Steve SeitzSpherical ProjectionWhat if PP is spherical with center at COP?In spherical coordinates, projection is trivial:dNote: doesn’t depend on focal length d!Programming Assignment #1Out tonight, due Sept. 12, 11:59pmEasy stuff to get you started with MatlabDistance Functions•SSD•Anything else?Bells and Whistles•Use your own photos /
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