BRDF - Revision history

Patapom at 17:16, 4 July 2013

2013-07-04T17:16:30Z

Patapom at 11:54, 3 February 2013

2013-02-03T11:54:49Z

Patapom: /* Warping */

2013-01-05T16:08:45Z

‎Warping

Patapom: /* Warping */

2013-01-05T14:38:14Z

‎Warping

Patapom: /* What can we bring to the table? */

2013-01-05T13:59:08Z

‎What can we bring to the table?

Patapom: /* Diffuse Part */

2013-01-04T14:23:22Z

‎Diffuse Part

Patapom: /* Warping */

2013-01-02T19:33:43Z

‎Warping

Patapom: /* Warping */

2013-01-02T19:12:33Z

‎Warping

Patapom: /* Warping */

2013-01-02T18:08:20Z

‎Warping

Patapom at 16:42, 2 January 2013

2013-01-02T16:42:13Z

@@ Line 81: / Line 81: @@
 Well, it ''can'' work for a few cases.
-For example, in the case of a perfectly diffuse reflector (Lambert model) then the BRDF is a simple constant <math>f_r(x,\omega_o) = \frac{\rho(x)}{\pi}</math> where <math>\rho(x)</math> is called the diffuse reflectance (or diffuse albedo) of the surface. The division by <math>\pi</math> is here to account for our "per steradian" need.
+For example, in the case of a perfectly diffuse reflector (Lambert model) then the BRDF is a simple constant <math>f_r(x,\omega_o) = \frac{\rho(x)}{\pi}</math> where <math>\rho(x)</math> is called the diffuse reflectance (or diffuse albedo) of the surface. The division by <math>\pi</math> is here to account for our "per steradian" need and to keep the BRDF from reflecting more light than came in: integration of a unit reflectance <math>\rho = 1</math> over the hemisphere yields <math>\pi</math>.
 This is okay as long as we don't want to model materials that behave in a more complex manner.

@@ Line 280: / Line 280: @@
 The expressions for the [http://en.wikipedia.org/wiki/Fresnel_equations#Definitions_and_power_equations Fresnel reflectance] are quite complicated and deal with complex numbers to account for light polarity, but thanks to a simplification by [http://www.cs.virginia.edu/~jdl/bib/appearance/analytic%20models/schlick94b.pdf Schlick] (in the same paper where he described his BRDF model!), it can be written:
-:<math>F(\theta_d) = F_0 + (1 - F_0) (1 - \cos \theta_d)^5</math>
+:<math>F(\theta_d) = F_0 + (1 - F_0) (1 - \cos \theta_d)^5~~~~~~~~~\mbox{(5)}</math>
@@ Line 360: / Line 360: @@
 Anyway, it's not compulsory to be exact as long as the expressions for Fresnel/specular peaks and overall reflectances don't yield infinity terms at certain view/light angles and as long as the integral of the BRDF for any possible view direction doesn't exceed 1.
 ===So what?===
@@ Line 475: / Line 473: @@
 \frac{\cos\theta_d}{\cos\phi_d \sin\theta_d} &= \tan\theta_h \\
 \tan(\frac{\pi}{2}-\theta_d) &= \cos\phi_d \tan\theta_h \\
 \end{align}</math>
 We notice that when <math>\phi_d = \frac{\pi}{2}</math> then vectors intersect the surface if <math>\theta_d = \frac{\pi}{2}</math> or <math>\theta_h = \frac{\pi}{2}</math>
@@ Line 499: / Line 499: @@
 This is where the iso-lines of <math>\theta_i</math> and <math>\theta_o</math> are coming to the rescue!
@@ Line 512: / Line 558: @@
 * (right figure) The second set of longitudinal isolines (labeled from 7° to 172°) is constructed by keeping <math>\phi_i = 0</math> and making <math>0 < \theta \le 180°</math> for a <math>\Phi_o = \mbox{iso}</math>.
 ===Layered materials model===

← Older revision		Revision as of 16:08, 5 January 2013
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−	'''Note:''' From this, we can decide to discard the second half of the table –everything ~~above 90°~~– because it seems to correspond to the portion of tilted hemisphere that is ''always'' above the surface. By the way, it's quite unclear why we lack data (i.e. the black pixels) in the slices of this area. Assuming the isotropic materials were scanned by choosing <math>\phi_h = 0°</math>, then perhaps the authors wanted the table to also account for the opposite and mirror image if <math>\phi_h = 180°</math> and so the table is reversible and reciprocity principle is upheld?	+	'''Note:''' From this, we can decide to discard the second half of the table –everything where <math>\phi_d > \frac{\pi}{2}</math>– because it seems to correspond to the portion of tilted hemisphere that is ''always'' above the surface. By the way, it's quite unclear why we lack data (i.e. the black pixels) in the slices of this area. Assuming the isotropic materials were scanned by choosing <math>\phi_h = 0°</math>, then perhaps the authors wanted the table to also account for the opposite and mirror image if <math>\phi_h = 180°</math> and so the table is reversible and reciprocity principle is upheld?

	Anyway, we can assume the "real data" corresponding to rays actually grazing the surface lie in the [0,90] section.		Anyway, we can assume the "real data" corresponding to rays actually grazing the surface lie in the [0,90] section.

@@ Line 468: / Line 468: @@
 y = \cos\theta_d,
 \end{cases}</math>
 [[File:SlicePhiD0.jpg|thumb|right|Material slice at <math>\phi_d = 0</math>]]
@@ Line 479: / Line 478: @@
 We notice that when <math>\phi_d = \frac{\pi}{2}</math> then vectors intersect the surface if <math>\theta_d = \frac{\pi}{2}</math> or <math>\theta_h = \frac{\pi}{2}</math>
 And when <math>\phi_d = \pi</math> then vectors go below the surface if <math>\frac{\pi}{2} - \theta_d < \theta_h</math>, which is the behavior we observe with slices at <math>\phi_d = 0</math> or <math>\phi_d = \pi</math>.
@@ Line 512: / Line 512: @@
 * (right figure) The second set of longitudinal isolines (labeled from 7° to 172°) is constructed by keeping <math>\phi_i = 0</math> and making <math>0 < \theta \le 180°</math> for a <math>\Phi_o = \mbox{iso}</math>.
 ===Layered materials model===

@@ Line 436: / Line 436: @@
-From these we can deduce that, except for very specular materials, there seems to be a symmetry about the half angle vector so I take the responsibility to assume that half of the table &ndash;the <math>\phi_d \in [0,90[</math>&ndash; is redundant enough to be discarded.
+From these we can deduce that, except for very specular materials, there seems to be a symmetry about the half angle vector so I take the responsibility to assume that half of the table is redundant enough to be discarded.
 ====Warping====
@@ Line 452: / Line 448: @@
 In the figure above, we see that for <math>\phi_d = \frac{\pi}{2}</math>, whatever the combination of <math>\theta_h</math> or <math>\theta_d</math>, we absolutely cannot get below the surface (i.e. the grey area), that's why the characteristic slice has valid pixels everywhere.
-On the other hand, when <math>\phi_d > \frac{\pi}{2}</math>, we see that some combinations of <math>\theta_h > 0</math> and <math>\theta_d</math> give positions that can get below the surface (highlighted in red):
+On the other hand, when <math>\phi_d < \frac{\pi}{2}</math>, we can notice that some combinations of <math>\theta_h > 0</math> and <math>\theta_d</math> give positions that get below the surface (highlighted in red):
 [[File:TiltedSpherePhi120.jpg|600px]]
-We can see the problem in 2D:
+'''Note:''' From this, we can decide to discard the second half of the table &ndash;everything above 90°&ndash; because it seems to correspond to the portion of tilted hemisphere that is ''always'' above the surface. By the way, it's quite unclear why we lack data (i.e. the black pixels) in the slices of this area. Assuming the isotropic materials were scanned by choosing <math>\phi_h = 0°</math>, then perhaps the authors wanted the table to also account for the opposite and mirror image if <math>\phi_h = 180°</math> and so the table is reversible and reciprocity principle is upheld?
 [[File:TiltedSphere2D.jpg|600px]]
@@ Line 511: / Line 512: @@
 * (right figure) The second set of longitudinal isolines (labeled from 7° to 172°) is constructed by keeping <math>\phi_i = 0</math> and making <math>0 < \theta \le 180°</math> for a <math>\Phi_o = \mbox{iso}</math>.
 ===Layered materials model===

@@ Line 333: / Line 333: @@
-A very interesting fact I learned from Naty Hoffman's [http://blog.selfshadow.com/publications/s2012-shading-course/hoffman/s2012_pbs_physics_math_notes.pdf talk] is that metals completely absorb photons if they are not reflected specularly: metals have (almost) no diffuse components. It can be clearly seen in the characteristic slices of several metals:
+A very interesting fact I learned from Naty Hoffman's [http://blog.selfshadow.com/publications/s2012-shading-course/hoffman/s2012_pbs_physics_math_notes.pdf talk] is that, because of free electrons, metals completely absorb photons if they are not reflected specularly: metals have (almost) no diffuse components. It can be clearly seen in the characteristic slices of several metals:
 [[File:MetalsNoDiffuse.jpg|800px]]
@@ Line 341: / Line 341: @@
 [[File:BrassColoredSpecular.jpg]]
 ====Shadowing====

@@ Line 502: / Line 502: @@
 [[File:HalfVectorSpaceIsoLines.jpg]]
 All isolines show <math>\theta_i = \theta_o \in [0,90]</math>.
-The first set of radial latitudinal isolines (marked from 25° to 88°) is constructed by keeping <math>\phi_i = 0</math> and making <math>0 < \phi_o \le 180°</math> for a <math>\theta = \mbox{iso}</math>.
+* (left figure) The first set of radial latitudinal isolines (labeled from 25° to 88°) is constructed by keeping <math>\phi_i = 0</math> and making <math>0 < \phi_o \le 180°</math> for a <math>\theta = \mbox{iso (non linear)}</math>.
-The second set of longitudinal isolines (marked from 7° to 172°) is constructed by keeping <math>\phi_i = 0</math> and making <math>0 < \theta \le 180°</math> for a <math>\Phi_o = \mbox{iso}</math>.
+* (right figure) The second set of longitudinal isolines (labeled from 7° to 172°) is constructed by keeping <math>\phi_i = 0</math> and making <math>0 < \theta \le 180°</math> for a <math>\Phi_o = \mbox{iso}</math>.
 ===Layered materials model===

@@ Line 117: / Line 117: @@
 The BRDF can then be seen as the infinitesimal amount of reflected radiance (<math>W.m^{-2}.sr^{-1}</math>) by the infinitesimal amount of incoming '''ir'''radiance (<math>W.m^{-2}</math>) and thus has the final units of <math>sr^{-1}</math>.
 ==Physically==
@@ Line 137: / Line 138: @@
 I wrote that energy conservation is difficult to enforce but many models represent a single specular highlight near the mirror direction so, instead of testing the integral of the BRDF for all <math>\omega_o</math>, it's only necessary to ensure it returns a correct value in the mirror direction, hence reducing the problem to a single integral evaluation. This usually gives us a single value that we can later use as a normalization factor.
 == BRDF Models ==
@@ Line 196: / Line 198: @@
 We're going to use these ''characteristic slices'' and their important areas a lot in the following section that will treat of analytical models.
 ===Analytical models of BRDF===
@@ Line 253: / Line 256: @@
 You can find more interesting comparisons of the various NDF in the [http://blog.selfshadow.com/publications/s2012-shading-course/hoffman/s2012_pbs_physics_math_notes.pdf talk] by Naty Hoffman.
@@ Line 299: / Line 287: @@
 [[File:FresnelComparison.jpg]]
 ====Diffuse Part====
@@ Line 352: / Line 341: @@
 [[File:BrassColoredSpecular.jpg]]
 ===So what?===