EE Light & Image Formation

Transcription

1 EE Light & Electric Electronic Engineering Bogazici University January 29, 2018 EE Light &

2 EE Light &

3 The image of a three-dimensional object depends on: 1. Shape 2. Reflectance properties 3. Distribution of light sources EE Light &

4 Reflectance Model Outline Reflectance model - A mathematical formulation of how light energy incident on an object is transferred from the object to the camera sensor. EE Light &

5 Radiance & Irradiance EE Light &

6 Figure: SolidEE angle Light & Outline Solid angle subtended by a patch Proportional to its area A and the cos of the angle of inclination θ Inversely proportional to the distance R as Ω = Acosθ R 2.

7 Radiance & Irradiance Radiance: The amount of light emitted from a surface as measured by power per unit unit area per unit solid angle. (Wm 2 sr 1 watts per square meter per steradian) Irradiance: The amount of light falling on a surface as measured by power per unit area (Wm 2 Watts per square meter) EE Light &

8 Relation btw Radiance & Irradiance EE Light &

9 Brightness Apparent area: Surface area multiplied by the cosine of the angle between the surface normal and the specified direction. Brightness - Amount of energy an imaging system receives per unit apparent area EE Light &

10 Optics Outline Rays passing through the center of the lens are not deflected. The solid angle of the cone of rays leading to the patch on the object = Solid angle of corresponding image patch. Solid angle of the object patch as seen from the lens center: δocosθ (z/cosα) 2 Solid angle subtended by the image patch: δicosα (f/cosα) 2 Equality δo δi = ( z f )2cosα cosθ EE Light &

11 Object Radiance Outline Amount of light emitted by the surface that makes its way through the lens is equal to Ω = (d 2 )2 πcosα (z/cosα) 2 Power of the light originating from the object patch and passing through the lens is δp = LδOΩcosθ = LδO (d 2 )2 πcosαcosθ (z/cosα) 2 EE Light &

12 Image Irradiance Outline The resulting image irradiance of the image patch: E = δp δi = L δo δi Ωcosθ = LδO(d 2 )2 πcosα (z/cosα) 2 = Lπ 4 (d f )2 cos 4 α EE Light &

13 Next Question Scene Radiance The amount of light falling on a surface E The fraction of light that is reflected as well as viewing angle L EE Light &

14 BRDF A measure on the observed brightness from a certain direction θ e when the light is incident in a different direction θ i : f(θ i,φ i,θ e,φ e ) = δl(θ e,φ e ) δe(θ i,φ i ) EE Light &

15 Bidirectional reflectance distribution function Figure: Left: Spherical coordinate system; Right: Bidirectional reflectance distribution function. EE Light &

16 BRDF Models Lambertian Equally bright from all directions Specular The surface luminance is highest when the observer is located at the perfect reflection direction, and falls off sharply. Phong A mixture of Lambertian and Specular EE Light &

17 Light Reflection from a Surface EE Light &

18 Lambertian Surfaces Apparent brightness of the surface to an observer - The same regardless of the observer s angle of view L = Eρcosθ ρ is the albedo EE Light &

19 Specular Reflection The surface luminance is highest when the observer is situated at the perfect reflection direction, and falls off sharply. Law of reflection The angle of incidence equals the angle of reflection. The incident, normal, and reflected directions coplanar. EE Light &

20 Phong Model Lambertian reflection is typically accompanied by specular reflection. L = E(acosθ +bcos n α) a Diffuse albedo b Specular albedo EE Light &

21 Phong Reflection Outline EE Light &

22 Surface Normals Outline EE Light &

23 Surface Normals (cont) Surface normal at each point on the surface Expressed as the directional slope in the X 1 and X 2 directions. f X1 f X2 1 = p q 1 EE Light &

24 Normalized Surface Normals p, q Gradient or gradient space representation for local surface orientation Normalized surface normal 1 1+p 2 +q 2 p q 1 EE Light &

25 Gradient Space The p q plane is called the gradient space and every point corresponds to a particular surface orientation. The point at the origin represents the orientation of all planes normal to the viewing direction A contour map Depict the reflectance map. EE Light &

26 Reflectance Map A viewer-centered representation of reflectance. Expresses the reflectance of a material directly in terms of viewer-centered representation of local surface orientation. Gives us the dependence of the scene radiance on the surface orientation. EE Light &

27 Reflectance Map Outline Figure: Reflectance map. EE Light &

28 Lambertian Reflectance Map Consider Lambertian surface where E = Lρcosθ 1+p s p +q s q cosθ = Lρ 1+p 2 s +qs 2 1+p 2 +q = R(p,q) 2 Grouping L and ρ as a constant, local surface orientations that produce equivalent intensities under the Lambertian reflectance map are quadratic conic section contours in gradient space. (1+p s p +q s q) 2 = c 2 (1+p 2 s +q 2 s)(1+p 2 +q 2 ) EE Light &

29 Reflectance map - Isobrightness Contours EE Light &

30 Reflectance map - Isobrightness Contours (cont.) EE Light &

31 Reflectance map with p s = q s = 0 EE Light &

32 Reflectance map with p s = 0.7 and q s = 0.3 Figure: Reflectance map with p s = 0.7 and q s = 0.3. EE Light &

33 Reflectance map with p s = 2 and q s = 1 EE Light &

34 Figure: Left: Surface normal map; Right: Reflectance EE Light &

35 Surface Description Outline Figure: Reflectance map. EE Light &

36 Surface Description (cont.) EE Light &

37 Surface Normal and Depth Depth Map can be determined from from Normal Map by integrating over gradients p,q across the image. Not all normal maps have a unique depth map. Integrable: A normal map that produces a unique depth map independent of image plane direction used to sum over gradients Integrability is enforced when the following condition holds: p = q X 2 X 1 EE Light &

38 Green s Theorem Outline Note that by using Green s Theorem, this integral is equivalent to a closed contour integral as follows: ( p q ) dx 1 dx2 = pdx 1 + qdx 2 X 2 X 1 EE Light &

39 Violation of Integrability EE Light &