Image formation ECE 847 Digital Image Processing Stan

Image formation ECE 847: Digital Image Processing Stan Birchfield Clemson University

Cameras • First photograph due to Niepce • Basic abstraction is the pinhole camera – lenses required to ensure image is not too dark – various other abstractions can be applied F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Image formation overview Image formation involves • geometry – path traveled by light • radiometry – optical energy flow • photometry – effectiveness of light to produce “brightness” sensation in human visual system • colorimetry – physical specifications of light stimuli that produce given color sensation • sensors – converting photons to digital form

Pinhole camera D. Forsyth, http: //luthuli. cs. uiuc. edu/~daf/bookpages/slides. html

Parallel lines meet: vanishing point • each set of parallel lines (=direction) meets at a different point – The vanishing point for this direction • Sets of parallel lines on the same plane lead to collinear vanishing points. – The line is called the horizon for that plane

Perspective projection Properties of projection: • Points go to points • Lines go to lines • Planes go to whole image • Polygons go to polygons • Degenerate cases k P – line through focal point to point – plane through focal point to line C f O j p q i Q F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Perspective projection (cont. ) F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Weak perspective projection • perspective effects, but not over the scale of individual objects • collect points into a group at about the same depth, then divide each point by the depth of its group D. Forsyth, http: //luthuli. cs. uiuc. edu/~daf/bookpages/slides. html

Weak perspective (cont. ) F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Orthographic projection Let Z 0=1: F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Pushbroom cameras

Pinhole size Pinhole too big many directions are averaged, blurring the image Pinhole too smalldiffraction effects blur the image Generally, pinhole cameras are dark, because a very small set of rays from a particular point hits the screen. D. Forsyth, http: //luthuli. cs. uiuc. edu/~daf/bookpages/slides. html

The reason for lenses D. Forsyth, http: //luthuli. cs. uiuc. edu/~daf/bookpages/slides. html

The thin lens focal points D. Forsyth, http: //luthuli. cs. uiuc. edu/~daf/bookpages/slides. html

Focusing http: //www. theimagingsource. com

Thick lens • thick lens has 6 cardinal points: – two focal points (F 1 and F 2) – two principal points (H 1 and H 2) – two nodal points (N 1 and N 2) • complex lens is formed by combining individual concave and convex lenses http: //physics. tamuk. edu/~suson/html/4323/thick. html D. Forsyth, http: //luthuli. cs. uiuc. edu/~daf/bookpages/slides. html

Complex lens All but the simplest cameras contain lenses which are actually composed of several lens elements http: //www. cambridgeincolour. com/tutorials/camera-lenses. htm

Choosing a lens • How to select focal length: – x=f. X/Z – f=x. Z/X • Lens format should be >= CCD format to avoid optical flaws at the rim of the lens http: //www. theimagingsource. com/en/resources/whitepapers/download/choosinglenswp. en. pdf

Lenses – Practical issues • standardized lens mount has two varieties: – C mount – CS mount • CS mount lenses cannot be used with C mount cameras http: //www. theimagingsource. com/en/resources/whitepapers/download/choosinglenswp. en. pdf

Spherical aberration perfect lens actual lens On a real lens, even parallel rays are not focused perfectly http: //en. wikipedia. org/wiki/Spherical_aberration

Chromatic aberration On a real lens, different wavelengths are not focused the same http: //en. wikipedia. org/wiki/Chromatic_aberration

Radial distortion straight lines are curved: uncorrected

Radial distortion (cont. ) Two types: barrel distortion pincushion distortion (more common) barrel pincushion http: //en. wikipedia. org/wiki/Image_distortion http: //foto. hut. fi/opetus/260/luennot/11/atkinson_6 -11_radial_distortion_zoom_lenses. jpg

Vignetting vignetting – reduction of brightness at periphery of image D. Forsyth, http: //luthuli. cs. uiuc. edu/~daf/bookpages/slides. html

Normalized Image coordinates 1 O u=X/Z = dimensionless ! P F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Pixel units Pixels are on a grid of a certain dimension f O u=k f X/Z = in pixels ! P [f] = m (in meters) [k] = pixels/m F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Pixel coordinates We put the pixel coordinate origin on topleft f O u=u 0 + k f X/Z P F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Pixel coordinates in 2 D (0. 5, 0. 5) 480 640 (u 0, v 0) i (640. 5, 480. 5) j F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Summary: Intrinsic Calibration skew 5 Degrees of Freedom ! F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Camera Pose In order to apply the camera model, objects in the scene must be expressed in camera coordinates. Camera Coordinates World Coordinates Calibration target looks tilted from camera viewpoint. This can be explained as a difference in coordinate systems. F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Rigid Body Transformations • Need a way to specify the six degrees-of-freedom of a rigid body. • Why are there 6 DOF? A rigid body is a collection of points whose positions relative to each other can’t change Fix one point, three DOF Fix second point, two more DOF (must maintain distance constraint) Third point adds one more DOF, for rotation around line F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Notations • Superscript references coordinate frame • AP is coordinates of P in frame A • BP is coordinates of P in frame B • Example: F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Translation F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Translation • Using homogeneous coordinates, translation can be expressed as a matrix multiplication. • Translation is commutative F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Rotation means describing frame A in The coordinate system of frame B F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Rotation Orthogonal matrix! F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Example: Rotation about z axis What is the rotation matrix? F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Rotation in homogeneous coordinates • Using homogeneous coordinates, rotation can be expressed as a matrix multiplication. • Rotation is not commutative F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Rigid transformations F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Rigid transformations (con’t) • Unified treatment using homogeneous coordinates. F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Projective Camera Matrix 5+6 DOF = 11 ! F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Projective Camera Matrix 5+6 DOF = 11 ! F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Columns & Rows of M m 2 P=0 O F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Effect of Illumination (Subject 8 from the Yale face database due to P. Belhumeur et. al. ) Light source strength and direction has a dramatic impact on distribution of brightness in the image (e. g. shadows, highlights, etc. ) F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Image formation • Light source emits photons • Absorbed, transmitted, scattered • fluorescence source Camera F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Surfaces receives and emits • Incident light from lightfield • Act as a light source • How much light ? F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Irradiance • Irradiance – amount of light falling on a surface patch • symbol=E, units = W/m 2 d. A F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Radiosity • power leaving a point per area • symbol=B, units = W/m 2 d. A F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Light = Directional • Light emitted varies w. direction F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Steradians (Solid Angle) • 3 D analogue of 2 D angle A R F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Steradians (cont’d) F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Polar Coordinates F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Intensity • Intensity – amount of light emitted from a point per steradian • symbol=I, units = W/sr F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Irradiance and Intensity d. A F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Radiance • Radiance – amount of light passing through an area d. A and • symbol=L, units = W x m-2 x sr-1 F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Radiance is important • Response of camera/eye is proportional to radiance • Pixel values • Constant along a ray F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Lightfield = Gibson optic array ! • 5 DOF: Position = 3 DOF, 2 DOF for direction F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Lightfield Sampler F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Lightfield Sample F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Lambertian Emitters • • Lambertian = constant radiance More photons emitted straight up Oblique: see fewer photons, but area looks smaller Same brightness ! Total power is proportional to wedge area “Cosine law” Sun approximates Lambertian: Different angle, same brightness • Moon should be less bright at edges, as gets less light from sun. • Reflects more light at grazing angles than a Lambertian reflector F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Radiance Emitted/Reflected • Radiance – amount of light emitted from a surface patch per steradian per area • foreshortened ! d. A F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Calculating Radiosity If reflected light is not dependent on angle, then can integrate over angle: radiosity is an approximate radiometric unit F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Example: Sun • • Power= 3. 91 1026 W Surface Area: 6. 07 1018 m 2 Power = Radiance. Area. L = 2. 05 107 W/m 2. sr Example from P. Dutre SIGGRAPH tutorial F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Irradiance (again) • Integrate incoming radiance over hemisphere F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Example: Sun F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

BRDF E L Symmetric in incoming and outgoing directions – this is the Helmholtz reciprocity principle F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

BRDF Example F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Lambertian surfaces and albedo • For some surfaces, the DHR is independent of illumination direction too – cotton cloth, carpets, matte paper, matte paints, etc. • For such surfaces, radiance leaving the surface is independent of angle • Called Lambertian surfaces (same Lambert) or ideal diffuse surfaces • Use radiosity as a unit to describe light leaving the surface • DHR is often called diffuse reflectance, or albedo • for a Lambertian surface, BRDF is independent of angle, too. • Useful fact: F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Specular surfaces • Another important class of surfaces is specular, or mirror-like. – radiation arriving along a direction leaves along the specular direction – reflect about normal – some fraction is absorbed, some reflected – on real surfaces, energy usually goes into a lobe of directions – can write a BRDF, but requires the use of funny functions F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Lambertian + specular • Widespread model – all surfaces are Lambertian plus specular component • Advantages – easy to manipulate – very often quite close true • Disadvantages – some surfaces are not • e. g. underside of CD’s, feathers of many birds, blue spots on many marine crustaceans and fish, most rough surfaces, oil films (skin!), wet surfaces – Generally, very little advantage in modelling behaviour of light at a surface in more detail -- it is quite difficult to understand behaviour of L+S surfaces F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Radiometry vs. Photometry http: //www. optics. arizona. edu/Palmer/rpfaq. htm F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Sensors • CCD vs. CMOS • Types of CCDs: linear, interline, fullframe, frame-transfer • Bayer filters • progressive scan vs. interlacing • NTSC vs. PAL vs. SECAM • framegrabbers • blooming F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Bayer color filter http: //en. wikipedia. org/wiki/Bayer_filter

Adobe’s plenoptic lens captures multiple views of the scene from slightly different viewpoints David Salesin and Todor Georgiev F. Dellaert, http: //www. cc. gatech. edu/~dellaert/vision/html/materials. html

Raw image from plenoptic system http: //livesmooth. istreamplanet. com/nvidia 100921/

Reconstructed image

Change the focus

Lenticular display http: //en. wikipedia. org/wiki/Lenticular_printing