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\Huge

\begin{center}
Computer Vision (II)
\end{center}

\LARGE
\noindent
Course Number: 526 U1200 \\
Credits: 3 \\
Time: Tuesday 6, 7, 8 (2:10PM-5:00PM) \\
Classroom: New CSIE Classroom 309 \\
Classification: Elective for junior, senior, and graduate students \\
Prerequisite: Computer Vision (I) or Digital Image Processing \\
Instructor: Chiou-Shann Fuh \\
Office: New Computer Science and Information Engineering 327 \\
Phone: 23625336 ext. 327, 23630231 ext. 3232 ext. 327 \\
Office Hours: Thursday 3PM-5PM \\
Objective: To learn computer and robot vision through extensive  \\
  \hspace*{1in} course projects. 

\vspace{1cm}
\noindent
Textbook: R. M. Haralick and L. G. Shapiro, {\em Computer and Robot Vision}, \\
 \hspace*{1in} Vol. II, Addison Wesley, Reading, MA, 1993. \\
Reference: B. K. P. Horn, {\em Robot Vision}, MIT Press, Cambridge, MA, 1986. \\
Reference: L. G. Shapiro and G. C. Stockman, {\em Computer Vision,} \\
\hspace*{1in} Prentice-Hall, Upper Saddle River, NJ, 2001. \\
Reference: R. Jain, R. Kasturi, and B. G. Schunck, {\em Machine Vision,} \\
\hspace*{1in} McGraw-Hill, New York, 1995. \\
Projects: assigned weekly on first few weeks (20\%) and a term project (30\%)\\
Examinations: one midterm (20\%) and one final (30\%)

\vspace{1cm}
\noindent
Content: \\
This is the second semester of a fast pace course which covers 
robot and computer vision.
The first semester covers low-level vision and mostly no reference to three 
dimensions. 
This semester covers higher-level techniques: \\
12. Illumination \\
13. Perspective Projective Geometry \\
14. Analytic Photogrammetry \\
15. Motion and Surface Structure from Time Varying Image Sequences \\
16. Image Matching \\
17. The Consistent-Labeling Problem \\
18. Object Models and Matching \\
19. Knowledge-Based Vision \\
20. Accuracy \\
21. Glossary of Computer Vision Terms 

\vspace{1cm}
Bibliography
\begin{itemize}
\item D. H. Ballard and C. M. Brown, {\em Computer Vision}, Prentice-Hall,
      Englewood Cliffs, NJ, 1982.
\item G. A. Baxes, {\em Digital Image Processing,} Wiley, New York, 1994.
\item K. Castleman, {\em Digital Image Processing,} Prentice-Hall,
      Englewood Cliffs, NJ, 1996.      
\item E. R. Davies, {\em Machine Vision: Theory, Algorithms, Practicalities,}
      2nd Ed., Academic Press, San Diego, CA, 1997.
\item R. O. Duda, P. E. Hart and D. G. Stork, {\em Pattern Classification,}
      2nd Ed., Wiley, New York, 2001.      
\item R. C. Gonzalez and R. E. Woods, {\em Digital 
      Image Processing,} Addison Wesley, Reading, MA, 1992.
\item  R. C. Gonzalez and R. E. Woods, {\em Digital Image 
      Processing,} 2nd Ed., Prentice-Hall, Upper Saddle River, NJ, 2002.  
\item E. Gose, R. Johnsonbaugh, and S. Jost, {\em Pattern Recognition 
      and Image Analysis,} Prentice-Hall, Englewood Cliffs, NJ, 1996.
\item A. K. Jain, {\em Fundamentals of Digital Image Processing}, Prentice-Hall,
      Englewood Cliffs, NJ, 1989.
\item J. S. Lim, {\em Two-Dimensional Signal and Image Processing}, Prentice-Hall,
      Englewood Cliffs, NJ, 1990.
\item C. G. Looney, {\em Pattern Recognition Using Neural Networks}, 
      Oxford University Press, Oxford, 1997.
\item D. Marr, {\em Vision}, W. H. Freeman, San Francisco, 1982.
\item V. S. Nalwa, {\em A Guided Tour of Computer Vision}, Addison Wesley, 
      Reading, MA, 1993. 
\item M. Petrou and P. Bosdogianni, {\em Image Processing: The Fundamentals,}
      Wiley, New York, 1999.      
\item W. K. Pratt, {\em Digital Image Processing}, 2nd ed., Wiley-Interscience,
      New York, 1991.
\item R. J. Schalkoff, {\em Digital Image Processing and Computer Vision: An
      Introduction to Theory and Implementations}, Wiley, New York, 1989.
\item R. J. Schalkoff, {\em Pattern Recognition: Statistical, Structural, and 
      Neural Approaches}, Wiley, New York, 1992.
\item M. Sonka, V. Hlavac, and R. Boyle, {\em Image Processing, Analysis, 
      and Machine Vision,} 2nd Ed., Brooks/Cole, Pacific Grove, CA, 1999.      
\item E. Trucco and A. Verri, {\em Introductory Techniques for 3-D Computer
      Vision,} Prentice-Hall, Upper Saddle River, NJ, 1998.      
\item S. E. Umbaugh, {\em Computer Vision and Image Processing: A Practical
      Approach Using CVIPtools,} Prentice-Hall, Englewood Cliffs, NJ, 1998.      
\item B. A. Wandell, {\em Foundations of Vision,} Sinauer, Sunderland,
      MA, 1995.
\end{itemize}




\newpage
\Huge
\begin{center}
Chapter 12 Illumination
\end{center}

\LARGE
12.1 Introduction \\
two key questions in understanding 3D image formation
\begin{enumerate}
\item what determines where some point on object will appear on image? \\
      answer: geometric perspective projection model
\item what determines how bright the image of some surface on object will be? \\
      answer: radiometry, general illumination models, diffuse and specular
\end{enumerate}
photographs illustrating how shading dramatically affect perception of shape \\
shading: variation in brightness of a surface, ---caption--- \\
=====Nalwa, {\em A Guided Tour of Computer Vision}, Fig. 5.1===== 

\vspace{1cm}
\noindent
refraction of light bouncing off a surface patch: basic reflection phenomenon\\
=====Figure 12.1===== \\
image intensity $I$: proportional to scene radiance \\
scene radiance depends on
\begin{enumerate}
\item the amount of light that falls on a surface
\item the fraction of the incident light that is reflected
\item the geometry of light reflection, i.e. \\
      viewing direction and illumination directions
\end{enumerate}
image intensity:
\[
I = g J^i f_r C S + b
\]
$J^i$: incident radiance \\
$f_r$: bidirectional reflectance function \\
$C$: lens collection \\
$S$: sensor responsivity \\
$g$: sensor gain \\
$b$: sensor offset \\
=====Kodak CCD sensor=====

\vspace{1cm}
\noindent
The appearance of a surface depends greatly on the lighting conditions. \\
We have to understand how images are formed if we are to recover information
\hspace*{1cm} about the surface from one or more images. \\
three views of the surface of Mars taken by Viking Lander I \\
=====Horn, {\em Robot Vision}, Figure 10.1===== \\
NASA Spirit rover 2004 \\
=====spirit.jpg=== \\
=====Garfield 17:18=====


\vspace{1cm}
12.2 Radiometry \\
Radiometry is the measurement of the flow and transfer of radiant energy in 
terms of both the power emitted from or incident upon an area and the power
radiated within a small solid angle about a given direction. \\
irradiance: the amount of light falling on a surface \\
irradiance: power per unit area of radiant energy falling on a surface \\
irradiance: measured in units of watts per square meter \\
radiance: the amount of light emitted from a surface \\
radiance: power per unit foreshortened area emitted into a unit solid angle \\
radiance: measured in units of watts per square meter per steradian \\
radiant intensity of a point illumination source: power per steradian \\
radiant intensity: measured in units of watts per steradian \\
radiant intensity: may be a function of polar and azimuth angles 

\vspace{1cm}
\noindent
geometry of incident and reflected elementary beams \\
=====Figure 12.2===== \\
$z$-axis: along the normal to the surface element $dA$ at 0 \\
polar angle: measured from the $z$-axis (pointing north) \\
azimuth angle: measured from $x$-axis (pointing east) 

\vspace{1cm}
\noindent
The solid angle subtended by a surface patch is defined by the cone whose
vertex is at the point of radiation and whose axis is the line segment going
from the point of radiation to the center of the surface patch. \\
size of solid angle: area intercepted by the cone on a unit radius sphere \\
\hspace*{1cm} centered at the point of radiation \\
solid angle: measured in steradians \\
total solid angle about a point in space: $4 \pi$ steradians \\
determination of the solid angle $\Omega$ subtended by a small surface patch \\
=====Figure 12.3===== 
\[
\Omega = \frac{\cos \theta A}{d^2}
\]
$A$: surface area \\
$d$: distance from surface area to point of radiation ($d^2 >>A$) \\
$\theta$: angle the surface normal makes w.r.t. the cone axis 

\vspace{1cm}
\noindent
surface irradiance (${\rm w/m}^2$):
\[
\frac{I_\circ A \cos \theta_\circ /d^2}{A} 
   = \frac{I_\circ \cos \theta_\circ}{d^2}
\]
$A$: area of surface patch \\
$I_\circ$ (w/sr): constant radiant intensity of point illumination source \\
law of inverse squares: irradiance varies inversely as square of distance  \\
\hspace*{1cm} from illuminated surface to source \\
caption of Figure 12.4 \\
=====Figure 12.4===== 

\vspace{1cm}
\noindent
infinitesimal slice on annulus on sphere of radius $r$, polar angle $\theta$, 
azimuth $\phi$ \\
=====Horn, {\em Robot Vision}, Figure 10.8===== \\
slice subtends solid angle $d \Omega$, since $\cos 0^\circ = 1, d = r, 
A= (r \sin \theta d \phi) * (r d \theta)$
\[
d \Omega = \sin \theta d \theta d \phi
\]

\vspace{1cm}
\noindent
appearance of object depends greatly on attitude in space relative to viewer \\
not only does outline vary, but brightness pattern within silhouette changes \\
=====Horn, {\em Robot Vision}, Figure 10.2=====

\vspace{1cm}
12.2.1 Bidirectional Reflectance Function \\
The bidirectional reflectance distribution function $f_r$ is the fraction of
incident light emitted in one direction when the surface is illuminated from
another direction. \\
light source and sensor positions in spherical coordinates \\
=====Figure 12.2===== \\
$\theta$: polar angle between surface normal and lens center \\
$\phi$: azimuth angle of the sensor \\
$e$: emitting from \\
$i$: incident to \\
$J^i$: irradiance of the incident light at the illuminated surface \\
$J^r$: radiance of the reflected light \\
$f_r$: ratio of the scene radiance to the scene irradiance \\
differential reflectance model:
\[
d J^r(\theta_e, \phi_e, \theta_i, \phi_i)= d J^i(\theta_i, \phi_i) 
       f_r(\theta_i, \phi_i, \theta_e, \phi_e)
\]
The differential emitted radiance (w/m$^2$-sr) in the direction
$(\theta_e, \phi_e)$ due to the incident differential irradiance in the 
direction $(\theta_i, \phi_i)$ is equal to the incident differential
irradiance $d J^i(\theta_i, \phi_i)$ (w/m$^2$) times the bidirectional
reflectance distribution function $f_r(\theta_i, \phi_i, \theta_e, \phi_e)$
(1/sr).

\vspace{1cm}
\noindent
For many surfaces the dependence of $f_r$ on the azimuth angles $\phi_i$ and
$\phi_e$ is only a dependence on their difference.
\[
f_r(\theta_i, \phi_i, \theta_e, \phi_e)=f_r(\theta_i, \theta_e; \phi_e-\phi_i)
\]
except surfaces with oriented microstructure \\
\hspace*{1cm} e.g. mineral called tiger's eye, iridescent feathers of some birds

\vspace{1cm}
\noindent
An ideal Lambertian surface is one that appears equally bright from all viewing directions and reflects all incident light, absorbing none. \\
Lambertian surface: perfectly diffusing surface, with matte appearance \\
reflectivity $r$: unitless fraction called reflectance factor \\
white blotting paper: $r=0.8$ \\
white writing paper: $r=0.68$ \\
white ceilings or yellow paper: $r=0.6$ \\
dark brown paper: $r=0.13$ \\
dark velvet: $r=0.004$ \\
bidirectional reflectance distribution function for Lambertian surface
\[
f_r(\theta_i, \phi_i, \theta_e, \phi_e) = \frac{r}{\pi}
\]
shading as a cue to surface shape, ---caption--- \\
=====Nalwa, {\em A Guided Tour of Computer Vision}, Fig. 5.2===== \\
boundary shape as a factor in 3D interpretation by humans, ---caption--- \\
=====Nalwa, {\em A Guided Tour of Computer Vision}, Fig. 5.3===== \\
Lambertian sphere illuminated in direction -[1 0.5 -1], ---caption--- \\
=====Nalwa, {\em A Guided Tour of Computer Vision}, Fig. 5.6===== \\
reflectance map: plot of brightness as function of surface orientation \\
reflectance map of a Lambertian surface (direction [1 0.5 -1]) \\
=====Nalwa, {\em A Guided Tour of Computer Vision}, Fig. 5.7===== \\
reflectance map of a Lambertian surface (direction -[0 0 -1]) \\
=====Nalwa, {\em A Guided Tour of Computer Vision}, Fig. 5.8===== \\
differential relationship for emitted radiance for Lambertian surface 
\[
d J^r(\theta_e, \phi_e) = \frac{r d J^i}{\pi} \ \ \ {\rm w/m}^2 {\rm -sr}
\]
Lambertian surface: consistent brightness no matter what viewing direction \\
power radiated  into a fixed solid angle: same in any direction \\
=====Example 12.1===== \\
=====joke=====

\vspace{1cm}
12.2.2 Photometry \\
photometry: study of radiant light energy resulting in physical sensation \\
brightness: attribute of sensation by which observer aware of differences of \\
\hspace*{1cm} observed radiant energy \\
radiometry $\leftrightarrow$ radiant energy, photometry $\leftrightarrow$
luminous energy \\
radiometry $\leftrightarrow$ power, photometry $\leftrightarrow$
luminous flux \\
lumen: unit of luminous flux \\
luminous intensity: luminous flux leaving point source per unit solid angle \\
luminous intensity: has units of lumens per steradian \\
candela: one lumen per steradian \\
illuminance: luminous flux per unit area incident upon a surface \\
illuminance: in units of lumens per square meter \\
one lux: one lumen per square meter \\
foot-candle: one lumen per square foot \\
one foot = 0.3048 meter 
\[
1 {\rm foot-candle} = \frac{1}{(0.3048)^2} {\rm lux} = 10.76 {\rm lux}
\]
luminance: luminous flux per unit solid angle per unit of projected area \\
luminance: in units of lumens per square meter per steradian

\vspace{1cm}
12.2.3 Torrance-Sparrow Model \\
$J_s^r$: specular reflection from mirrorlike surface facets \\
$J_d^r$: diffuse reflection from Lambertian surface facets \\
$J^r$: reflected light from roughened surface \\
$J_s^r$: dependent on the view point, whereas $J_d^r$ is not \\
consider surfaces: $f_r(\theta_i, \phi_i, \theta_e, \phi_e) =
f_r(\theta_i, \theta_e; \phi_e - \phi_i)$, let $\phi = \phi_e-\phi_i$ \\
light source and sensor positions specified by pair of angles ---caption---\\
=====Figure 12.5===== \\
$\vec{N}$: unit surface normal \\
$\vec{L}$: unit positional vector of the light source \\
$\vec{V}$: unit positional vector of the sensor
\[
\cos \theta_i = \vec{N} \cdot \vec{L}
\]
\[
\cos \theta_e = \vec{N} \cdot \vec{V}
\]

\vspace{1cm}
\noindent
Torrance-Sparrow model:
\[
J^r(\theta_i; \theta_e,\phi;\lambda)=s J_s^r(\theta_i;\theta_e,\phi;\lambda)+
   (1-s)J_d^r(\theta_i; \lambda)
\]
s ($0 \leq s \leq 1$): proportion of specular reflection depending on surface\\
$\lambda$: wavelength of light

\vspace{1cm}
12.2.4 Lens Collection \\
lens collection: portion of reflected light coming through lens to film\\
$f$: distance between the image plane and the lens \\
$r_1$: distance between the object and the lens \\
$r_2$: distance between the lens and the image of the object \\
$a$: diameter of the lens \\
$\alpha$: angle between the ray from the object patch to the lens center \\
=====Figure 12.6=====

\vspace{1cm}
\noindent
irradiance incident on differential area $d a_2$ coming from differential
area $d a_1$, having radiance $d J^i$, and passing through a lens having 
aperture area $A = \pi a^2 / 4$: \\
$A \cos \alpha$: foreshortened area of aperture stop seen by $d a_1$ \\
$r_1=s_1/\cos \alpha$: distance from $d a_1$ to the aperture \\
solid angle $\Omega$ subtended by aperture stop as seen from $d a_1$:
\[
\Omega = \frac{A \cos \alpha}{r_1^2}=\frac{A \cos^3 \alpha}{s_1^2}
\]
differential radiant power $d \Phi$ passing through aperture due to $da_1$
\[
d \Phi = d J^i \Omega da_1 \cos \alpha
\]
radiant power passing through aperture from $da_1$
\[
d \Phi = \frac{d J^i A \cos^4 \alpha d a_1}{s_1^2}
\]
irradiance incident to $d a_2$: (radiant power reaching $da_2$ is $d\Phi$)
\[
d J^r = \frac{d \Phi}{d a_2} = \frac{d J^i A \cos^4 \alpha d a_1}{s_1^2 d a_2}
\]
assume $s_1 >> s_2$, then $s_2 = f$, thus lens magnification is $s_1/s_2$ \\
hence $d a_1/d a_2=(s_1/s_2)^2=s_1^2/f^2$, therefore 
\[
d J^r = \frac{d J^i A \cos^4 \alpha}{s_1^2}\frac{s_1^2}{f^2}
      = \frac{d J^i A \cos^4 \alpha}{f^2}
\]
since $A = \pi a^2/4$
\[
d J^r = \frac{\pi d J^i \cos^4 \alpha}{4}(\frac{a}{f})^2
\]
then the lens collection $C$ is given by
\[
C = \frac{dJ^r}{dJ^i}=\frac{\pi}{4}(\frac{a}{f})^2 \cos^4 \alpha
\]

\vspace{1cm}
12.2.5 Image Intensity \\
The image intensity gray level $I$ associated with some small area of the
image plane can then be represented as the integral of all light collected at
the given pixel position coming from the observed surface patch, modified by
sensor gain $g$ and bias $b$.
\[
I(\theta, \phi)=g \int_\lambda C S(\lambda) J^r(\theta, \phi; \lambda)\Omega^2
r^2 d \lambda +b
\]
$\lambda$: light wavelength \\
$S(\lambda)$: sensor responsivity to light at wavelength $\lambda$ \\
$J^r(\theta, \phi; \lambda)$: radiance of observed surface patch 
(watts/m$^2$-sr-m) \\
$\Omega$: solid angle subtended by the viewing cone of camera for the pixel \\
$r$: distance to the observed patch \\
$J^r(\theta, \phi; \lambda)\Omega(\Omega r^2)$: power received for the pixel 
position

\vspace{1cm}
12.3 Photometric Stereo \\
In photometric stereo there is one camera but $K$ light sources having known
intensities $i_1,...,i_K$ and incident vectors $v_1,...,v_K$ to a given 
surface patch. \\
In photometric stereo the camera sees the surface patch $K$ times, one time
when each light source is activated and the remaining ones are deactivated. \\
$n$: surface normal vector of the surface patch having Lambertian reflectance\\
$r$: reflectivity of the Lambertian surface reflectance \\
$f_1,...,f_K$: observed gray levels produced \\
by the model of Lambertian reflectance
\[
f_k = g r i_k v_k \cdot n + b, \ \ \ k = 1,...,K
\]
$g$: sensor gain \\
$b$: sensor offset

\vspace{1cm}
\noindent
if camera has been photometrically calibrated, $g, b$ known \\
let $f_k^*=\frac{f_k - b}{g i_k}$ and
\[
f^*= \left (
\begin{array}{c}
f_1^* \\
\vdots \\
f_K^*
\end{array}
\right ) \ \ \ 
V= \left (
\begin{array}{c}
v_1' \\
\vdots \\
v_K'
\end{array}
\right ) 
\]
in matrix form
\[
f^* = rVn
\]
if surface normal $n$ known, then least-squares solution for reflectivity $r$:
\[
r = \frac{{f^*}' Vn}{(Vn)'(Vn)}
\]
if $K=3$, a solution for unit surface normal $n$:
\[
n = \frac{V^{-1}f^*}{||V^{-1}f^*||}
\]
if $K>3$, a least-squares solution:
\[
n = \frac{(V'V)^{-1} V'f^*}{||(V'V)^{-1} V'f^*||}
\]

\vspace{1cm}
\noindent
if $g, b$ unknown, camera must be calibrated as follows: \\
geometric setup with known incident angle of light source to surface normal \\
surfaces of known reflectivities illuminated by known intensity light source \\
$i_k$: known intensity of light source for $k$th trial \\
$v_k$: known incident direction of light source for $k$th trial \\
$n$: known unit length surface normal vector \\
$r_k$: known reflectivity of surface illuminated for $k$th trial \\
$y_k$: observed value from the camera \\
let $x_k = i_k r_k v_k \cdot n $
then unknown gain $g$ and offset $b$ satisfy
\[
\left (
\begin{array}{cc}
x_1 & 1 \\
x_2 & 1 \\
\vdots & \vdots \\
x_K & 1 
\end{array}
\right ) \left (
\begin{array}{c}
g \\
b
\end{array} 
\right ) = \left (
\begin{array}{c}
y_1 \\
y_2 \\
\vdots \\
y_K
\end{array} \right )
\]
this leads to the least-squares solution for $(g,b)$
\[
\left (
\begin{array}{c}
g \\
b
\end{array}
\right ) = \left (
\begin{array}{cc}
\sum_{k=1}^K x_k^2 & \sum_{k=1}^K x_k \\
\sum_{k=1}^K x_k   & K
\end{array}
\right )^{-1} \left (
\begin{array}{c}
\sum_{k=1}^K x_k y_k \\
\sum_{k=1}^K y_k
\end{array}
\right )
\]
=====Oldie 33:29=====

\vspace{1cm}
12.4 Shape from Shading \\
nonplanar Lambertian surfaces of constant reflectance factor: appear shaded \\
this shading: secondary clue to shape of the observed surface \\
shape from shading: recovers shape of Lambertian surface from image shading \\
$(a,b,c)$: unit vector of distant point light source direction \\
assume surface viewed by distant camera so perspective projection \\
\hspace*{1cm} approximated by orthographic projection \\
surface point position $(x,y,z)$: projected to image position $(x,y)$ \\
$z=g(x,y)$: surface expression \\
unit vector normal to the surface at $(x,y)$:
\[
\frac{1}{\sqrt{(\frac{\partial g}{\partial x})^2 + (\frac{\partial g}{\partial y
})^2+1}} \left (
\begin{array}{c}
\frac{\partial g}{\partial x} \\
\frac{\partial g}{\partial y} \\
-1
\end{array}
\right )
\]
gray level at $(x,y)$, within multiplicative constant
\[
I(x,y) = \frac{a p(x, y) + b q(x, y) -c}{\sqrt{p^2(x,y)+q^2(x,y)+1}}
\]
where $p = (\partial g/\partial x) $ and $q = (\partial g/\partial y)$ \\
$R(p,q)$: reflectance map 
\[
R(p,q) = \frac{a p + b q -c}{\sqrt{p^2+q^2+1}}
\]
$\lambda$: penalty constant \\
relaxation method: minimizing original error and a smoothness term \\
criterion function to be minimized by choice of $p,q$
\[
\epsilon^2 = \sum_r \sum_c \{I(r,c)-R[p(r,c),q(r,c)]\}^2
+\lambda \{[p(r+1,c)-p(r,c)]^2
\]
\[
                              +[p(r,c+1)-p(r,c)]^2
	  +[q(r+1,c)-q(r,c)]^2+[q(r,c+1)-q(r,c)]^2\}
\]
two orthographic shaded view of the same surface, ---caption--- \\
=====Horn, {\em Robot Vision}, Fig. 10.19===== \\
a block diagram of Dent de Morcles region in southwestern Switzerland \\
=====Horn, {\em Robot Vision}, Fig. 10.18===== \\
shape-from-shading method applied to recovery of nose shape \\
=====Horn, {\em Robot Vision}, Fig. 11.7=====

\vspace{1cm}
\noindent
uniform brightness if planar surfaces since $p(x,y), q(x,y)$ constant \\
surfaces with curvature: surfaces with $p(x,y), q(x,y)$ not constant \\
brightness has some connection with $p(x,y), q(x,y)$  \\
$p(x,y), q(x,y)$ provide information about surface height $g(x,y)$ \\
first-order Taylor expression for g:
\[
g(x+1,y)=g(x,y)+\frac{\partial g}{\partial x} \Rightarrow
g(x,y) = g(x+1, y) -p(x,y)
\]
\[
g(x,y+1)=g(x,y)+\frac{\partial g}{\partial y} \Rightarrow
g(x,y)=g(x,y+1)-q(x,y)
\]
with boundary conditions on $g(x,y)$, we can solve unknown surface height and\\
\hspace*{1cm} partial derivatives $p(x,y), q(x,y)$

\vspace{1cm}
12.4.1 Shape from Focus \\
possible to recover shape from the shading profile of object edges \\
basic idea: cameras do not have infinite depth of field \\
The degree to which edges may be defocused is related to how far the 3D edge
is away from the depths at which the edges are sharply in focus. 

\vspace{1cm}
12.5 Polarization \\
illumination source characterized by four factors
\begin{enumerate}
\item directionality: relative to surface normal in bidirectional reflectance
\item intensity: energy coming out from source 
\item spectral distribution: function of wavelength $\lambda$
\item polarization: time varying vibration of light energy in certain direction
\end{enumerate}
polarization: time-varying vibration of the light energy in certain direction\\
linearly polarized: changes direction by $180^\circ$ every period \\
circularly polarized: phase angle difference of $90^\circ$, thus 
$\cos wt+i \sin wt$ \\
elliptically polarized: phase angle difference of $90^\circ$ and different
amplitude $a\cos wt+i b \sin wt$ 

\vspace{1cm}
Mathematical Meaning of Polarization \\
polarization of light mathematically described by using wave theory

\vspace{1cm}
Usefulness of Polarization in Machine Vision \\
At Brewster's angle, the parallel polarized light is totally transmitted and
the perpendicularly polarized light is partially transmitted and 
partially reflected. \\
This effect can be used to remove the specular reflections from the window or
metal surfaces by looking through them at Brewster's angle. \\
=====polarizer=====

\vspace{1cm}
12.5.1 Representation of Light Using the Coherency Matrix \\
natural light: completely unpolarized 

\vspace{1cm}
12.5.2 Representation of Light Intensity \\
S-pol: component polarized perpendicularly to the incidence plane \\
P-pol: component polarized parallel to the incidence plane

\vspace{1cm}
12.6 Fresnel Equation \\
Fresnel law of reflectance, ---caption--- \\
=====Figure 12.11=====

\vspace{1cm}
12.7 Reflection of Polarized Light \\
ergodic light: time average of the light equivalent to its ensemble average

\vspace{1cm}
12.8 A New Bidirectional Reflectance Function \\
bright-field: rays reflected back into objective illuminate entire surface \\
dark-field: reflects back into objective only rays from surface irregularities\\
=====Elliott, Integrated Circuit Fabrication Technology, Fig. 5.4=====

\vspace{1cm}
12.9 Image Intensity \\
image intensity can be written in terms of:
\begin{itemize}
\item illumination parameters
\item sensor parameters
\item bidirectional reflectance function
\end{itemize}

\vspace{1cm}
12.10 Related Work \\
reflectance models: have been used in computer graphics and image analysis \\
=====joke=====













\vspace{0.7cm}
Project due Mar. 8: \\
Use correlation to do image matching \\
find $(dx,dy)$ to minimize
\[
\sum_{(x,y) \in R} |PIX(ima,x,y)-PIX(imb,x+dx, y+dy)|
\]
=====trucka.im===== \\
=====truckb.im===== \\
=====toy.trancd.block.ps===== \\
pageview toy.trancd.block.ps \\
Adobe Systems Inc., {\em Postscript Language Tutorial and Cookbook}, \\
\hspace*{1cm} Addison Wesley, Reading, MA, 1985. \\
=====lpr1l toy.trancd.block.ps===== 

\end{document}
