3D Computer Vision. Photometric stereo. Prof. Didier Stricker
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1 3D Computer Vision Photometric stereo Prof. Didier Stricker Kaiserlautern University DFKI Deutsches Forschungszentrum für Künstliche Intelligenz 1
2 Physical parameters of image forma2on Op2cal Sensor s lens type focal length, field of view, aperture Geometric Type of projec2on Camera calibra2on Two/Mul2- views geometry Photometric (radiometry) Type, direc2on, intensity of light reaching sensor Surfaces reflectance proper2es Inference from shading
3 Image forma2on What determines the brightness of an image pixel? Light source proper.es Surface shape and orienta.on Sensor characteris.cs Exposure Op.cs Surface reflectance proper.es Slide by L. Fei- Fei
4 Radiance and irradiance Radiance (L) energy exiting a source or surface Irradiance (E) incoming energy E L E E L L E L Which (E or L) does a camera sensor array directly measure?
5 Radiometry Radiometry is the part of image forma.on concerned with the rela.on among the amounts of q q q light energy emi@ed from light sources, reflected from surfaces, and registered by sensors. 5
6 Foreshortening A big source, viewed at a glancing angle, must produce the same effect as a small source viewed frontally. This phenomenon is known as foreshortening. 6
7 Solid Angle Solid angle is defined by the projected area of a surface patch onto a unit sphere of a point. (Solid angle is subtended by a point and a surface patch.) * da projected onto surface of the sphere of radius r: da cos(theta) * Ra.o of surface areas of spheres of radii 1 and r: 4 pi / (4 pi r^2) = 1/r^2 7
8 Solid Angle Arc length r dφ dφ r 8
9 Solid Angle Solid angle is defined by the projected area of a surface patch onto a unit sphere of a point. da = rdθ r sin θ dφ = r 2 sin θ dθ dφ da 2π TotalArea = π φ θ r 2 sin θ dθ dφ = 4π r 2 =0 =0 da dw = 2 = sin θ dθ dφ r 9
10 Solid Angle Similarly, solid angle due to a line segment is θ dl dφ r 10
11 Radiance The distribu.on of light in space is a func.on of posi.on and direc.on. The appropriate unit for measuring the distribu.on of light in space is radiance, which is defined as the power (the amount of energy per unit.me) traveling at some point in a specified direc.on, per unit area perpendicular to the direc7on of travel, per unit solid angle. In short, radiance is the amount of light radiated from a point (into a unit solid angle, from a unit area). Radiance = Power / (solid angle x foreshortened area) W/sr/m2 W is Wa@, sr is steradian, m2 is meter- squared 11
12 Radiance Radiance from ds to dr Radiance = Power / (solid angle x foreshortened area) 12
13 Radiance Example: Infinitesimal source and surface patches Radiance = Power / (solid angle x foreshortened area) Power or radiant flux emi@ed of the source Radiance leaving x1 in the direc.on of x2 Illuminated surface dψ r 2 dψ L(x1,x1 x 2 ) = = dw2 cosθ1da1 da2 cosθ 2 cosθ1da1 dw2 = Source da2 cosθ 2 r2 13
14 Radiance Radiance = Power / (solid angle x foreshortened area) Power at x1 leaving to x2 Illuminated surface dψ = L(x1,x1 x 2 )dw2 cosθ1da1 = Source L(x1,x1 x 2 )da2 cosθ 2 cosθ1da1 r2 dw2 = da2 cosθ 2 r2 14
15 Radiance Illuminated surface Source Let the radiance arriving at x2 from the direc.on of x1 is dψ r 2 dψ L(x 2,x1 x 2 ) = = dw1 cosθ 2 da2 da1 cosθ1 cosθ 2 da2 dw1 = da1 cosθ1 r2 15
16 Radiance The medium is vacuum, that is, it does not absorb energy. Therefore, the power reaching point x2 is equal to the power leaving for x2 from x1. Radiance is constant along a straight line. Illuminated surface L(x1, x1 x2 ) = L(x2, x1 x2 ) Source 16
17 Radiance If the medium is vacuum, power is preserved. 17
18 Point Source Many light sources are physically small compared with the environment in which they stand. Such a light source is approximated as an extremely small sphere, in fact, a point. Such a light source is known as a point source. 18
19 Radiance Intensity If the source is a point source, we use radiance intensity. Radiance intensity = Power / (solid angle) Illuminated surface dψ r 2 dψ I= = dw da2 cosθ 2 dw = da2 cosθ 2 r2 Source 19
20 Light at Surfaces When light strikes a surface, it may be absorbed, transmi@ed, or sca@ered; usually, combina.on of these effects occur. It is common to assume that all effects are local and can be explained with a local interac2on model. In this model: q q q The radiance leaving a point on a surface is due only to radiance arriving at this point. Surfaces do not generate light internally and treat sources separately. Light leaving a surface at a given wavelength is due to light arriving at that wavelength. 20
21 Light at Surfaces In the local interac.on model, fluorescence, [absorb light at one wavelength and then radiate light at a different wavelength], and emission [e.g., warm surfaces emits light in the visible range] are neglected. 21
22 Irradiance (E) Irradiance is the total incident power per unit area. Irradiance = Power / Area 22
23 Irradiance What is the irradiance due to source from angle? 23
24 Irradiance What is the irradiance due to source from angle? da Irradiance = dψ Li ( x,θ i, φi )dw cos θ i da = = Li ( x,θ i, φi )dw cos θ i da da radiance foreshortening factor Solid angle 24
25 Irradiance What is the total irradiance? Integrate over the whole hemisphere. E= 2π 0 π 2 0 L cosθ sin θ dθ dφ solid angle Since dω = sin θ dθ dφ 25
26 Irradiance Exercise: Calculate the irradiance at O due to a plate source at O. 26
27 Irradiance due to a Point Source For a point source, Radiance intensity = Power / (solid angle) dψ r 2 dψ I= = dw dai cosθ i dw = da cosθ dψ = I i 2 i r dai cosθ i r2 cosθ i dψ Irradiance = =I 2 dai r 27
28 The Relationship Between Image Intensity and Object We assume that there is no power loss Diameter of Radiance in the lens. lens The power to the lens is dψ = Lobject da0 cos α dw0 Radiance of object 28
29 Area of the lens The Relationship Between with diameter d Image Intensity and Object The solid angle for the en.re lens is Diameter of Radiance lens dw0 πd ( = 2 / 4 ) cosθ r2 The power emi@ed to the lens is dψ = Lobject da0 cos α dw0 π d 2 cosθ = Lobject da0 cos α 4r 2 29
30 The Relationship Between Image Intensity and Object Diameter of The solid angle at O can be wri@en in Radiance lens two ways. da0 cos α dap cos θ = 2 r2 OA ' Note that OA ' = f / cosθ Therefore 3 da0 cos α dap cos θ = 2 r f2 30
31 The Relationship Between Image Intensity and Object Diameter of Combine Radiance lens 3 da0 cos α dap cos θ = 2 r f2 dψ = Lobject da0 cos α π d 2 cosθ 4r 2 to get 2 d π dψ = Lobject cos 4 θ dap 4 f 31
32 The Relationship Between Image Intensity and Object Diameter of Therefore the irradiance on the image Radiance lens plane is 2!d $ dψ! π $ Irradiance = = # & Lobject # & cos 4 θ dap " 4 % "f% The irradiance is converted to pixel intensi.es, which is directly propor.onal to the radiance of the object. 32
33 Image irradiance on the image plane The image irradiance (E) is proportional to the object radiance (L) Lens diameter Angle off op.cal axis ' π! d $2 * E = ) # & cos 4 θ, L )( 4 " f %,+ Focus distance What the image reports to us via pixel values What we really want to know
34 Fundamental radiometric rela2on ' π! d $2 * E = ) # & cos 4 θ, L )( 4 " f %,+ S. B. Kang and R. Weiss, Can we calibrate a camera using an image of a flat, textureless Lamber.an surface? ECCV 2000.
35 Surface Characteristics We want to describe the rela.onship between incoming light and reflected light. This is a func.on of both the direc.on in which light arrives at a surface and the direc.on in which it leaves. 35
36 Bidirectional Reflectance Distribution Function (BRDF) BRDF is defined as the ra.o of the radiance in the outgoing direc.on to the incident irradiance. 36
37 Bidirectional Reflectance Distribution Function (BRDF) The radiance leaving a surface due to irradiance in a par7cular direc7on is easily obtained from the defini.on of BRDF: 37
38 Bidirectional Reflectance Distribution Function (BRDF) The radiance leaving a surface due to irradiance in all incoming direc7ons is where Omega is the incoming hemisphere. 38
39 BRDFs can be incredibly complicated
40 Lambertian Surface A Lamber7an surface has constant BRDF. A Lamber.an surface looks equally bright from any view direc.on. The image intensi.es of the surface only changes with the illumina.on direc.ons. constant 40
41 Lambertian Surface For a Lamber.an surface, the outgoing radiance is propor.onal to the incident radiance. constant If the light source is a point source, a pixel intensity will only be a func.on of Remember, for a point source cosθ i dψ Irradiance = =I 2 dai r BRDF is constant, we speak about Albedo Albedo: frac.on of incident irradiance reflected by the surface 41
42 Specular Surface The glossy or mirror like surfaces are called specular surfaces. Radia.on arriving along a par.cular direc.on can only leave along the specular direc.on, obtained from the surface normal. *The term Specular comes from the La.n word speculum, meaning mirror. 42
43 Specular Surface Few surfaces are ideally specular. Specular surfaces commonly reflect light into a lobe of direc.ons around the specular direc.on. 43
44 Lambertian + Specular Model Rela.vely few surfaces are either ideal diffuse or perfectly specular. The BRDF of many surfaces can be approximated as a combina.on of a Lamber.an component and a specular component. 44
45 Lamber2an + Specular Model Lamber.an Lamber.an + Specular 45
46 Radiosity Radiosity, defined as the total power leaving a point. To obtain the radiosity of a surface at a point, we can sum the radiance leaving the surface at that point over the whole hemisphere. 46
47 Part II Shading
48 Point Source For a point source, Radiance intensity = Power / (solid angle) dψ r 2 dψ I= = dw dai cosθ i dw = da cosθ dψ = I i 2 i r dai cosθ i r2 cosθ i dψ Irradiance = =I 2 dai r 48
49 A Point Source at Infinity The radiosity due to a point source at infinity is S( x) N( x) x B: radiosity (total power leaving the surface per unit area) ρ: albedo (frac.on of incident irradiance reflected by the surface) N: unit normal S: source vector (magnitude propor.onal to intensity of the source) 49
50 Local Shading Models for Point Sources The radiosity due to light generated by a set of point sources is Radiosity due to source s 50
51 Local Shading Models for Point Sources If all the sources are point sources at infinity, then 51
52 Ambient Illumination For some environments, the total irradiance a patch obtains from other patches is roughly constant and roughly uniformly distributed across the input hemisphere. In such an environment, it is possible to model the effect of other patches by adding an ambient illumina7on term to each patch s radiosity. + B0 52
53 Photometric Stereo If we are given a set of images of the same scene taken under different given ligh.ng sources, can we recover the 3D shape of the scene? 53
54 Photometric Stereo For a point source and a Lamber.an surface, we can write the image intensity as Suppose we are given the intensi.es under three ligh.ng condi.ons: Camera and object are fixed, so a par.cular pixel intensity is only a func.on of ligh.ng direc.on si. 54
55 Photometric Stereo Stack the pixel intensi.es to get a vector The surface normal can be found as Since n is a unit vector As a result, we can find the surface normal of each point, hence the 3D shape 55
56 More than Three Light Sources Get better results by using more lights T ' I1 $ ' s 1 $ %! " = %! " ρn % " % " %& I N "# %&stn "# Least squares solu.on: ~ I = Sn ~ ST I = ST Sn 1 T T ~ n= S S S I N 1 = (N 3)(3 1) ( ) Solve for ρ, n as before pseudo inverse
57 Photometric Stereo When the source direc.ons are not given, they can be es.mated from three known surface normals. 57
58 Photometric Stereo 58
59 Photometric Stereo Surface normals 3D shape 59
60 Photometric Stereo (by Xiaochun Cao) 60
61 Results Estimate light source directions Compute surface normals Compute albedo values Estimate depth from surface normals Relight the object (with original texture and uniform albedo)
62 Computer vision applica2on Finding the direc.on of light source P. Nillius and J.- O. Eklundh, Automa.c es.ma.on of the projected light source direc.on, CVPR 2001
63 Computer vision applica2on Detec.ng composite photos: Fake photo Real photo M. K. Johnson and H. Farid, Exposing Digital Forgeries by Detec.ng Inconsistencies in Ligh.ng, ACM Mul.media and Security Workshop, 2005.
64 Applica2on: Detec2ng composite photos Fake photo Real photo
65 Thank you!
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