Computer Vision Radiometry Bahadir K Gunturk Radiometry n

Computer Vision Radiometry Bahadir K. Gunturk

Radiometry n Radiometry is the part of image formation concerned with the relation among the amounts of q q q light energy emitted from light sources, reflected from surfaces, and registered by sensors. Bahadir K. Gunturk 2

Foreshortening n n 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. Bahadir K. Gunturk 3

Solid Angle n 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. ) Bahadir K. Gunturk 4

Solid Angle n Arc length r Bahadir K. Gunturk 5

Solid Angle n Solid angle is defined by the projected area of a surface patch onto a unit sphere of a point. Bahadir K. Gunturk 6

Solid Angle n Similarly, solid angle due to a line segment is r Bahadir K. Gunturk 7

Radiance n n n The distribution of light in space is a function of position and direction. The appropriate unit for measuring the distribution of light in space is radiance, which is defined as the power (the amount of energy per unit time) traveling at some point in a specified direction, per unit area perpendicular to the direction 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/m 2 W is Watt, sr is steradian, m 2 is meter-squared Bahadir K. Gunturk 9

Radiance n Radiance from d. S to d. R Radiance = Power / (solid angle x foreshortened area) Bahadir K. Gunturk 10

Radiance n Example: Infinitesimal source and surface patches Radiance = Power / (solid angle x foreshortened area) Radiance at x 1 leaving to x 2 Illuminated surface Source Bahadir K. Gunturk 11

Radiance = Power / (solid angle x foreshortened area) Power at x 1 leaving to x 2 Illuminated surface Source Bahadir K. Gunturk 12

Radiance n The medium is vacuum, that is, it does not absorb energy. Therefore, the power reaching point x 2 is equal to the power leaving for x 2 from x 1. Power at x 2 from direction x 1 is Illuminated surface Let the radiance arriving at x 2 from the direction of x 1 is Source Bahadir K. Gunturk 13

Radiance n Radiance is constant along a straight line. Illuminated surface Source Bahadir K. Gunturk 14

Point Source n n n 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. Bahadir K. Gunturk 16

Radiance Intensity n If the source is a point source, we use radiance intensity. Radiance intensity = Power / (solid angle) Illuminated surface Source Bahadir K. Gunturk 17

Light at Surfaces n n When light strikes a surface, it may be absorbed, transmitted, or scattered; usually, combination of these effects occur. It is common to assume that all effects are local and can be explained with a local interaction 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. Bahadir K. Gunturk 18

Light at Surfaces n In the local interaction 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. Bahadir K. Gunturk 19

Irradiance n Irradiance is the total incident power per unit area. Irradiance = Power / Area Bahadir K. Gunturk 20

Irradiance n What is the irradiance due to source from angle Bahadir K. Gunturk ? 21

Irradiance n What is the irradiance due to source from angle Bahadir K. Gunturk ? 22

Irradiance n What is the total irradiance? Integrate over the whole hemisphere. Exercise: Suppose the radiance is constant from all directions. Calculate the irradiance. Bahadir K. Gunturk 23

Irradiance n Exercise: Calculate the irradiance at O due to a plate source at O’. Bahadir K. Gunturk 24

Irradiance due to a Point Source n For a point source, Bahadir K. Gunturk 25

The Relationship Between Image Intensity and Object Radiance Diameter of lens We assume that there is no power loss in the lens. The power emitted to the lens is Radiance of object Bahadir K. Gunturk 26

The Relationship Between Image Intensity and Object Radiance Diameter of lens The solid angle for the entire lens is The power emitted to the lens is Bahadir K. Gunturk 27

The Relationship Between Image Intensity and Object Radiance Diameter of lens The solid angle at O can be written in two ways. Note that Therefore Bahadir K. Gunturk 28

The Relationship Between Image Intensity and Object Radiance Diameter of lens Combine to get Bahadir K. Gunturk 29

The Relationship Between Image Intensity and Object Radiance Diameter of lens Therefore the irradiance on the image plane is The irradiance is converted to pixel intensities, which is directly proportional to the radiance of the object. Bahadir K. Gunturk 30

Surface Characteristics n n We want to describe the relationship between incoming light and reflected light. This is a function of both the direction in which light arrives at a surface and the direction in which it leaves. Bahadir K. Gunturk 31

Bidirectional Reflectance Distribution Function (BRDF) n BRDF is defined as the ratio of the radiance in the outgoing direction to the incident irradiance. Bahadir K. Gunturk 32

Bidirectional Reflectance Distribution Function (BRDF) n The radiance leaving a surface due to irradiance in a particular direction is easily obtained from the definition of BRDF: Bahadir K. Gunturk 33

Bidirectional Reflectance Distribution Function (BRDF) n The radiance leaving a surface due to irradiance in all incoming directions is where Omega is the incoming hemisphere. Bahadir K. Gunturk 34

Lambertian Surface n A Lambertian surface has constant BRDF. constant Bahadir K. Gunturk 35

Lambertian Surface n n A Lambertian surface looks equally bright from any view direction. The image intensities of the surface only changes with the illumination directions. constant Bahadir K. Gunturk 36

Lambertian Surface n For a Lambertian surface, the outgoing radiance is proportional to the incident radiance. constant n If the light source is a point source, a pixel intensity will only be a function of Remember, for a point source Bahadir K. Gunturk 37

Specular Surface n n The glossy or mirror like surfaces are called specular surfaces. Radiation arriving along a particular direction can only leave along the specular direction, obtained from the surface normal. *The term Specular comes from the Latin word speculum, meaning mirror. Bahadir K. Gunturk 38

Specular Surface n Few surfaces are ideally specular. Specular surfaces commonly reflect light into a lobe of directions around the specular direction. Bahadir K. Gunturk 39

Lambertian + Specular Model n n Relatively few surfaces are either ideal diffuse or perfectly specular. The BRDF of many surfaces can be approximated as a combination of a Lambertian component and a specular component. Bahadir K. Gunturk 40

Lambertian + Specular Model Lambertian Bahadir K. Gunturk Lambertian + Specular 41

Radiosity n n 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. Bahadir K. Gunturk 42

Part II Shading Bahadir K. Gunturk

Point Source n For a point source, Bahadir K. Gunturk 44

A Point Source at Infinity n The radiosity due to a point source at infinity is Bahadir K. Gunturk 45

Local Shading Models for Point Sources n The radiosity due to light generated by a set of point sources is Radiosity due to source s Bahadir K. Gunturk 46

Local Shading Models for Point Sources n If all the sources are point sources at infinity, then Bahadir K. Gunturk 47

Ambient Illumination n n 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 illumination term to each patch’s radiosity. + B 0 Bahadir K. Gunturk 48

Photometric Stereo n If we are given a set of images of the same scene taken under different given lighting sources, can we recover the 3 D shape of the scene? Bahadir K. Gunturk 49

Photometric Stereo n For a point source and a Lambertian surface, we can write the image intensity as n Suppose we are given the intensities under three lighting conditions: Camera and object are fixed, so a particular pixel intensity is only a function of lighting direction si. Bahadir K. Gunturk 50

Photometric Stereo n Stack the pixel intensities to get a vector n The surface normal can be found as n Since n is a unit vector Bahadir K. Gunturk 51

Photometric Stereo n If we have more than three sources, we can find the least squares estimate using the pseudo inverse: n As a result, we can find the surface normal of each point, hence the 3 D shape Bahadir K. Gunturk 52