Introduction to Computer Vision Radiometry

Transcription

1 Radiometry Image: two-dimensional array of 'brightness' values. Geometry: where in an image a point will project. Radiometry: what the brightness of the point will be. Brightness: informal notion used to describe both scene and image brightness. Image brightness: related to energy flux incident on the image plane: IRRADIANCE Scene brightness: brightness related to energy flux emitted (radiated) from a surface. RADIANCE

2 Light Electromagnetic energy Wave model Light sources typically radiate over a frequency spectrum Φ watts radiated into 4π radians Φ watts r Φ = dφ sphere dω R = Radiant Intensity = (of source) dφ dω Watts/unit solid angle (steradian)

3 Irradiance Light falling on a surface from all directions. How much? dφ da Irradiance: power per unit area falling on a surface. dφ Irradiance E = dα watts/m

4 Inverse Square Law Relationship between radiance (radiant intensity) and irradiance Φ watts da dω = r da dω r dφ E = dα R: Radiant Intensity E: Irradiance Φ: Watts ω : Steradians dφ r dφ R = = = dω dα E = R r r E

5 Surface Radiance Surface acts as light source Radiates over a hemisphere Surface Normal dω R: Radiant Intensity E: Irradiance L: Scene radiance da f Radiance: power per unit foreshortened area emitted into a solid angle L = d Φ da dω f (watts/m - steradian)

6 Pseudo-Radiance Consider two definitions: Radiance: power per unit foreshortened area emitted into a solid angle Pseudo-radiance power per unit area emitted into a solid angle Why should we work with radiance rather than pseudoradiance? Only reason: Radiance is more closely related to our intuitive notion of brightness.

7 Lambertian Surfaces A particular point P on a Lambertian (perfectly matte) surface appears to have the same brightness no matter what angle it is viewed from. Piece of paper Matte paint Doesn t depend upon incident light angle. What does this say about how they emit light?

8 Lambertian Surfaces Equal Amounts of Light Observed From Each Vantage Point Theta Area of black box = 1 Area of orange box = 1/cos(Theta) Foreshortening rule.

9 Lambertian Surfaces Relative magnitude of light scattered in each direction. Proportional to cos (Theta).

10 Lambertian Surfaces Equal Amounts of Light Observed From Each Vantage Point Theta Area of black box = 1 Area of orange box = 1/cos(Theta) Foreshortening rule. Radiance = 1/cos(Theta)*cos(Theta) = 1

11 Geometry Goal: Relate the radiance of a surface to the irradiance in the image plane of a simple optical system. Φ α: Solid angle of patch da s : Area on surface da i : Area in image das Lens Diameter d i α e α dai

12 Light at the Surface dφ E = flux incident on the surface (irradiance) = dα i = incident angle e = emittance angle g = phase angle ρ = surface reflectance dω Φ watts Incident Ray r da s ρ i e g N (surface normal) Emitted ray We need to determine dφ and da

13 Reflections from a Surface I da da = da s cos i {foreshortening effect in direction of light source} dφ dφ = flux intercepted by surface over area da da subtends solid angle dω = da s cos i / r dφ = R dω = R da s cos i / r E = dφ / da s Surface Irradiance: E = R cos i / r

14 Reflections from a Surface II Now treat small surface area as an emitter.because it is bouncing light into the world How much light gets reflected? Incident Ray da s i e g N (surface normal) Emitted Ray E is the surface irradiance L is the surface radiance = luminance They are related through the surface reflectance function: L s E = ρ(i,e,g,λ) May also be a function of the wavelength of the light

15 das Power Concentrated in Lens Lens Diameter d θ α Image Plane α dai L = s d Φ da dω s Z -f Luminance of patch (known from previous step) What is the power of the surface patch as a source in the direction of the lens? d Φ = L s da dω s

16 Through a Lens Darkly In general: L is a function of the angles i and e. s Lens can be quite large Hence, must integrate over the lens solid angle to get dφ dφ = da s Ω L s dω

17 Simplifying Assumption Lens diameter is small relative to distance from patch dφ = da s Ω L s dω L s is a constant and can be removed from the integral dφ = da sl s Ω dω Surface area of patch in direction of lens Solid angle subtended by lens in direction of patch = da scos e = Area of lens as seen from patch (Distance from lens to patch) = π (d/) cos α (z / cos α)

18 Putting it Together dφ = da s L s Ω dω = da cos e L s s π (d/) cos α (z / cos α) Power concentrated in lens: π d dφ = L s da cos e cos 3 s α 4 Z Assuming a lossless lens, this is also the power radiated by the lens as a source.

19 das Lens Diameter d Through a Lens Darkly θ α Image Plane α dai Image irradiance at da = Z -f i dφ da i =E i E = i Ls da s π 4 3 da i d Z cos e cos α ratio of areas

20 das Lens Diameter d Patch ratio θ α Image Plane α dai Z -f The two solid angles are equal da scos e da i cos α da (Z / cos α) = s cos α (-f / cos α) = da i cos e Z -f

21 The Fundamental Result Source Radiance to Image Sensor Irradiance: da s da i = cos α cos e Z -f E = i Ls da s π 4 3 da i d Z cos e cos α E = i Ls cos α cos e Z -f π 4 d Z 3 cos e cos α E = i L s π 4 d -f 4 cos α

22 Radiometry Final Result E = i L s π 4 d -f 4 cos α Image irradiance is a function of: Scene radiance L Focal length of lens f Diameter of lens d s f/d is often called the 'effective focal length' of the lens Off-axis angle α

23 4 Cos α Light Falloff Lens Center Top view shaded by height y π/ π/ π/ x