Computer Vision Radiometry

Bahadir K. Gunturk2 Radiometry Radiometry is the part of image formation concerned with the relation among the amounts of  light energy emitted from light sources,  reflected from surfaces,  and registered by sensors.

Bahadir K. Gunturk3 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.

Bahadir K. Gunturk4 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.)

Bahadir K. Gunturk5 Solid Angle Arc length r

Bahadir K. Gunturk6 Solid Angle Solid angle is defined by the projected area of a surface patch onto a unit sphere of a point.

Bahadir K. Gunturk7 Solid Angle Similarly, solid angle due to a line segment is r

Bahadir K. Gunturk8 Foreshortening A point on a surface sees the world along a hemisphere of directions centered at the point.

Bahadir K. Gunturk9 Radiance 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/m2 W is Watt, sr is steradian, m2 is meter-squared

Bahadir K. Gunturk10 Radiance Radiance from dS to dR Radiance = Power / (solid angle x foreshortened area)

Bahadir K. Gunturk11 Radiance Example: Infinitesimal source and surface patches Source Illuminated surface Radiance at x1 leaving to x2 Radiance = Power / (solid angle x foreshortened area)

Bahadir K. Gunturk12 Radiance Source Illuminated surface Power at x1 leaving to x2 Radiance = Power / (solid angle x foreshortened area)

Bahadir K. Gunturk13 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. Power at x2 from direction x1 is Let the radiance arriving at x2 from the direction of x1 is Source Illuminated surface

Bahadir K. Gunturk14 Radiance Radiance is constant along a straight line. Source Illuminated surface

Bahadir K. Gunturk15 Radiance If the medium is vacuum, power is preserved.

Bahadir K. Gunturk16 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.

Bahadir K. Gunturk17 Radiance Intensity If the source is a point source, we use radiance intensity. Radiance intensity = Power / (solid angle) Illuminated surface Source

Bahadir K. Gunturk18 Light at Surfaces 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:  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. Gunturk19 Light at Surfaces 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. Gunturk20 Irradiance Irradiance is the total incident power per unit area. Irradiance = Power / Area

Bahadir K. Gunturk21 Irradiance What is the irradiance due to source from angle ?

Bahadir K. Gunturk22 Irradiance What is the irradiance due to source from angle ?

Bahadir K. Gunturk23 Irradiance What is the total irradiance? Integrate over the whole hemisphere. Exercise: Suppose the radiance is constant from all directions. Calculate the irradiance.

Bahadir K. Gunturk24 Irradiance Exercise: Calculate the irradiance at O due to a plate source at O’.

Bahadir K. Gunturk25 Irradiance due to a Point Source For a point source,

Bahadir K. Gunturk26 The Relationship Between Image Intensity and Object Radiance We assume that there is no power loss in the lens. The power emitted to the lens is Diameter of lens Radiance of object

Bahadir K. Gunturk27 The Relationship Between Image Intensity and Object Radiance The solid angle for the entire lens is The power emitted to the lens is Diameter of lens

Bahadir K. Gunturk28 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. Gunturk29 The Relationship Between Image Intensity and Object Radiance Diameter of lens Combine to get

Bahadir K. Gunturk30 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. Gunturk31 Surface Characteristics 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. Gunturk32 Bidirectional Reflectance Distribution Function (BRDF) BRDF is defined as the ratio of the radiance in the outgoing direction to the incident irradiance.

Bahadir K. Gunturk33 Bidirectional Reflectance Distribution Function (BRDF) The radiance leaving a surface due to irradiance in a particular direction is easily obtained from the definition of BRDF:

Bahadir K. Gunturk34 Bidirectional Reflectance Distribution Function (BRDF) The radiance leaving a surface due to irradiance in all incoming directions is where Omega is the incoming hemisphere.

Bahadir K. Gunturk35 Lambertian Surface A Lambertian surface has constant BRDF. constant

Bahadir K. Gunturk36 Lambertian Surface 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. Gunturk37 Lambertian Surface For a Lambertian surface, the outgoing radiance is proportional to the incident radiance. If the light source is a point source, a pixel intensity will only be a function of constant Remember, for a point source

Bahadir K. Gunturk38 Specular Surface 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. Gunturk39 Specular Surface Few surfaces are ideally specular. Specular surfaces commonly reflect light into a lobe of directions around the specular direction.

Bahadir K. Gunturk40 Lambertian + Specular Model 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. Gunturk41 Lambertian + Specular Model Lambertian Lambertian + Specular

Bahadir K. Gunturk42 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.

Part II Shading

Bahadir K. Gunturk44 Point Source For a point source,

Bahadir K. Gunturk45 A Point Source at Infinity The radiosity due to a point source at infinity is

Bahadir K. Gunturk46 Local Shading Models for Point Sources The radiosity due to light generated by a set of point sources is Radiosity due to source s

Bahadir K. Gunturk47 Local Shading Models for Point Sources If all the sources are point sources at infinity, then

Bahadir K. Gunturk48 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 illumination term to each patch’s radiosity. + B0

Bahadir K. Gunturk49 Photometric Stereo If we are given a set of images of the same scene taken under different given lighting sources, can we recover the 3D shape of the scene?

Bahadir K. Gunturk50 Photometric Stereo For a point source and a Lambertian surface, we can write the image intensity as 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 s i.

Bahadir K. Gunturk51 Photometric Stereo Stack the pixel intensities to get a vector The surface normal can be found as Since n is a unit vector

Bahadir K. Gunturk52 Photometric Stereo If we have more than three sources, we can find the least squares estimate using the pseudo inverse: As a result, we can find the surface normal of each point, hence the 3D shape

Bahadir K. Gunturk53 Photometric Stereo When the source directions are not given, they can be estimated from three known surface normals.

Bahadir K. Gunturk54 Photometric Stereo

Bahadir K. Gunturk55 Photometric Stereo Surface normals 3D shape

Bahadir K. Gunturk56 Photometric Stereo (by Xiaochun Cao)

Bahadir K. Gunturk57 Photometric Stereo (by Xiaochun Cao)