CS280 "Computer Vision"

Homework 1, Feb-2002.

Siddharth Jain , Cheng Chang , Almudena Konrad
* ( morpheus@eecs.berkeley.edu,
cchang@eecs.berkeley.edu, almudena@cs.berkeley.edu
)
*

** Written items**

- Expression for the radiance transferred from one surface patch of small area
(dA) to another of small area in the presence of this medium

*(Equation 1)*

or

*(Equation 2)*

which is the expression for the radiance transferred from one surface patch to another separated by a distance of x. - Distribution of light.
- If is a very small
positive number, the distribution of light falls off slowly.
- If is a medium sized
positive number, the distribution of light falls off more rapidly than
in the previous case.
- If is a large positive
number, the distribution of light falls off very rapidly.
Consider a cube with source at the center of a face. Let d be the distance from the source to the corners of the opposite face. Then for a small fall off, let's say that 90% of the light is able to reach the corners at the opposite face. Applying Equation 2, we obtain:

where,

and s represents the side of the cube.For a medium sized fall off let's say 50% of the light is able to reach the corners, applying Equation 2, we obtain:

And for a large fall off let's say only 10% of the light is able to reach the corners, we obtain:

Figure 1.1 illustrates the exponential decay of .

- If is a very small
positive number, the distribution of light falls off slowly.

where d is the grating constant and it represents the track spacing in the CD.

Examples of biological surfaces that show similar properties are insect wings (such as butterflies wings), hummingbird throat feathers, and organisms having scales on their skin like snakes and fish - the scales act as the diffraction grating.

A small glass of sufficiently deeply colored red wine looks black because the high concentration of colored pigment compensates for the small distance traveled by the light. We have an exponential attenuation of .

A big glass of lightly colored red wine appears black because even though the absorbance is low due to low concentration of colored pigment, the distance is high because there is more wine in the glass. We have an exponential attenuation of .

but , therefore the attenuation in both cases is relatively the same so the wine appears black in both glasses.

During daytime the sunlight is scattered by the air molecules producing a blur between the observer and the mountains. We see the mountains because of the reflected and scattered light - so they appear as blurred patches of low intensity.

We see the mountains because of the main beam of light so they are much sharper and more visible. Note that light suffers some attenuation in the air due to moisture.

Also note that at sunset an object overhead could appear blurred just as the mountains appeared blurred in the daytime. We estimate the distance for which air can be modeled as vacuum to be a few Kilometers, (<10) .

** Practical exercises **

- The quantization errors is a direct consequence of the
discretization of the area source giving a finite number of grid
points. This means that if we are at a position P
_{1}and suppose n points are visible and m are invisible, then there is some distance d_{min}such that if I move a distance less than d_{min}, m and n do not change - which is the approximation we make and which gives the quantization errors. -
Figure 1 (
parameters), illustrates an area source of
approximately 12x12 grid, Figure 2 (
parameters), illustrates an area source of
approximately 25x25 grid, and Figure 3 (
parameters), illustrates an area source of
approximately 36x36 grid. From these figures, it is apparent that as the number of point
sources goes up, the quantization error goes down (i.e., the image
become more realistic and accurate).
- If we change the geometry keeping the number of points in the
source the same, then we can increase d
_{min}. Figure 4 ( parameters), Figure 5 ( parameters), and Figure 6 ( parameters), show this effect. In these figures the occluder is tilted at an angle of 45 degrees.Thus, we find that Figure 1, gives an image which is more accurate than the corresponding image for Figure 4, because d

_{min}has increased.

- http://fusioned.gat.com/Teachers/Curriculum/Curriculum-HTML/T03S-CD-diffract.html
- http://web.media.mit.edu/~wad/color/constancy.html
- D'Zmura, M. & Iverson, G., A formal approach to color constancy:
the recovery of surface and light source spectral properties using
bilinear models.
*Mahwah, NJ: Lawrence Erlbaum Associates, 99-132*, 1998.