2 Theoretical Background
2.1 Mechanics
In this section we briefly review the essential principles of mechanics needed for
a course in applied robotic computation.
Translational Motion - Consider a body (e.g. a mobile robot platform) that moves


from an initial location x 0 to a new location x1 over a time t. This body has an
average velocity given by,
 
 x1  x0 
v
t

where v is a vector quantity having both a magnitude and a direction. Since it
deals with only the initial and final conditions it does not tell us anything about the


average speed of the body along the path from x 0 to x1 .

x0  x1 , y1

x0  x0 , y0 
We can write the instantanteous velocity as a function of time as,


dx (t )
v (t ) 
dt
where velocity is the time rate of change of the body position. We can express
instantaneous speed as,


dx (t )
s (t )  v (t ) 
dt

The acceleration a (t ) is the time rate of change of the velocity and is given by,


d v (t )
a (t ) 
dt
The position, velocity and acceleration of a body are related by the basic
kinematic equations as shown below:
1
x(t )  x0  v0  t  a  t 2
2
v(t )  v0  a0  t
a(t )  const . acceleration
Dead Reckoning - Computing the position of a moving object based on
measurements of its initial position, velocity and acceleration is called dead
reckoning. Consider the following simple, one-dimensional example.
Example: A body accelerates in the positive x direction with a constant
acceleration of 2 m/s2 for 1 second, then after 5 additional seconds it accelerates
in the negative x direction with a constant acceleration of -1 m/s2 for 2 seconds.
Determine the position and velocity of the body given an initial position of 10 m
and an initial velocity of 0 m/s.
Step 1: Compute the position and velocity at the end of 1 second.
1
x(1)  10  (0)(1)  (2)(1) 2  11 m
2
v(1)  0  (2)(1)  2 m / s
Step 2: Compute the position at the end of 6 seconds (i.e. now consider the 5
seconds of zero acceleration).
1
x(6)  11  (2)(5)  (0)(5) 2  21 m
2
Step 3: Comupute the position and velocity at the end of 8 seconds (i.e. now
consider the last 2 seconds of constant negative acceleration.)
1
x(8)  21  (2)( 2)  (1)( 2) 2  23 m
2
v(8)  2  (1)( 2)  0 m / s
The previous example was for one-dimensional motion. We can perform the
same computations for two- or three-dimensional motion by considering the
motion in each dimension separately. For higher dimensions the quantities x(t),
v(t) and a(t) rather than scalar values. (see Exercise 2.2).
Rotational Motion - Rotational motion is expressed in angular units such as
degrees or radians. The angular velocity as a function of time is given by,
 (t ) 
d (t )
dt
where  (t ) is expressed in units of radians per second. Angular acceleration is
expressed as
 (t ) 
d (t )
dt
Just as with translational motion we can use the angular position, velocity and
acceleration to describe the rotational motion of a rigid body.
1
2
 (t )   0   0  t   0 t 2
 (t )   0   0  t
 (t )  angular acceleration
For our mobile robots will need to relate rotational motion and angular velocity of
the wheels with translational motion and velocity of the robot platform. While we
could compute these relationships based on the geometry of the wheels we will
obtain a more accurate estimate of these parameters through experimentation.
(see Exercise 2).
Torque - Consider the situation in which a lever arm (e.g. a length of rod) is
attached to a rotating shaft. The amount of force that can be generated at the
end of the arm by a motor driving the shaft is a function of the length of the arm.
This force at a distance is called torque and is expressed in units of force times
length (e.g. oz-in, or gm-cm).
force
rotation
Torque = Force x Distance
distance
weight
The servos used to drive the wheels on the SPaRC-I are rated at a torque of
42 oz in. This means that the servo can lift a weight of 42 ounces at the end of a
lever arm 1 inch long.
Example: Compute the maximum (robot + payload) weight that can be pulled up
an 30o incline by the SPaRC-I, given that the two drive wheels produce a torque
of 42 oz-in each and have a diameter of 3 inches.
Step 1: Determine the maximum force exerted at the surface of a wheel, and
total pulling force of the SPaRC-I.
f 
42 oz in
 28 oz
1.5 in
f y  2  28  56 oz
This is the weight that can be lifted vertically by the two drive wheels together.
Step 2: In our example the force fy is the limit of the vertical component of the
force being applied at a 30o angle. Ignoring friction (for now) we can compute the
maximum weight that can be pulled up this inclined plane by,
wmax 
fy
sin 

56
 112 oz
sin 30
or around 7 lbs. Remember that the amount of work is defined as force applied
over a distance. Since the maximum force is limited we are using the inclined
plane to trade force for distance to perform a given amount of work.
ftot
fy
fx
30o
Exercises
2.1: A body accelerates in the positive x direction with a constant acceleration of
2 m/s2 for 3 seconds, then after 10 additional seconds it accelerates in the
negative x direction with a constant acceleration of -2 m/s2 for 2 seconds.
Determine the position and velocity of the body given an initial position of 30 m
and an initial velocity of -2 m/s.
2.2: Using dead reckoning determine the final position and final velocity of a
robot undergoing the accelerations shown below.
You may solve this problem analytically or graphically. Show all work.
2.3: Given a torque of 42 oz in for each of the two wheels and a wheel radius of
1.5 inches, determine the maximum incline that the SPaRC-I could pull a total
weight of 20 lbs.
2.4: Given a maximum angular rate of the SPaRC-I drive motors of 270 deg/sec
and a wheel radius of 1.5 inches compute the maximum translational speed of
the robot.
Linear Springs - The force exerted by an ideal linear spring under compression or
tension is proportional to the force causing the distortion. This is Hooke's Law
and it applies to springs in their linear range of operation. This is a circular
definition since the linear range of operation for a spring is defined as the range
of distortion over which Hooke's Law applies. In any case we can express this
relationship as,
Fs  K S x
where Fs is the force being applied to the spring, K s is a constant for the
particular spring being considered and x is the total change in the position of the
free end of the spring being distorted.
x
Fs
Springs can also be distorted by twisting. In this case we refer to the torque
(rather than the force) being applied to the spring and to the spring's restoring
torque. In MKS units torque is measured in newton.meters.
Ts  K ' s 
The expression above give the relationship between the torque on the spring T s ,
the angular stiffness of the spring K's and  , the angle through which the spring
is twisted (measured in radians). As with compression and tension, this
relationship is valid only in the linear range of operation for the spring.
Gears - The most common uses for gears are to transfer or redirect mechanical
motion and to change the rotational speed and torque between a motor and a
drive train or other load. Electric and other types of motors achieve an optimum
performance at rotational speeds that are usually much higher than is needed for
most applications. A gearbox or gear train can be used to reduce the rotational
speed and increase the torque of a motor.
Ignoring frictional losses between the gears we can assume that the power into a
gear train is equal to the power out of the gear train. For rotational motion, the
power is the torque T times the angular velocity , so we can express the power
of a drive motor as
P1  T1 1
The power from the output of a gear train to which this motor is connected can be
expressed as,
P2  T2 2
But since P1=P2 we have,
or
T1 1  T2 2
T1  2

T2  1
N2= # teeth gear 2
D2
T1
T2
D1
N1= # teeth gear 1
The gear ratio for a pair of gears is the ratio of the number of teeth on each gear.
For more than two gears the gear ratio is the product of the successive gear
ratios for each pair of gears.
chord pitch
pitch circles
backlash
Backlash - The backlash is the amount by which the width of a tooth space
exceeds the thickness of the engaging tooth on the pitch circles. Backlash is an
important characteristic of gears because it accumulates through a gear train
resulting in an uncertainty in the position of the output shaft relative to the
position of the input shaft.
Example: Compute the gear ratio for the gear train below comprised of three
gears with N1=25, N2a=75, N2b=30, and N3=80.
N2b
N3
N1
N2a
Gear 1 is meshed with gear 2 with three times as many teeth so the first gear
turns three times for each revolution of the middle gear. The middle gear in this
example is a compound gear in which two simple gears are combined. The
relative rotation rate between the gear 2 and the gear 3 is 30/80. Therefore the
combined gear ratio of this gear train is,
 25  30  1
N tot     
 75  80  8
Which means that a the rotation of the output shaft (gear 3) is 1/8 the rotation
rate of the input shaft (gear 1) and that the torque developed at the output would
be eight times the input torque.
Friction - The force of friction always tends to reduce the relative motion between
objects in contact. This includes contact between solids, liquids or gasses in any
combination. Friction generates sound and heat and thus saps energy from a
system reducing the amount of useful work that can be performed for a given
amount of energy input.
Static Friction acts at the beginning of motion and is proportional to the area of
contact between two surfaces. Static friction is non-zero only when there is no
relative motion between the surfaces.
Coulomb Friction is the component of kinetic friction (or the friction of motion
between two surfaces) that is proportional to the force pressing the two surfaces
together.
Viscous Friction is the component of kinetic friction that is proportional to the
velocity between two surfaces.
Although we usually think of friction as a nuisance we make use of it in many
important ways. Static friction holds a nut on a bolt and enables wheels to pull a
vehicle along a surface. Coulomb friction is used in brakes to stop a vehicle and
viscous friction is used in damper and shock absorbers to reduce vibrations and
smooth erratic motion.
2.2 Electricity and Electronics
In this section we briefly overview the essential principles of electricity and
electronics needed for a course in applied robotic computation.
Direct Current (DC) Basics - The fundamentals of the flow, restriction and
accumulation of electrons in various materials has been well understood for
centuries. However, the development and refinement of electronic devices and
technologies continues. Since we will be using some of these devices in our
robotics laboratory we will review their properties.
Voltage - The voltage of an electrical power source, V (measured in volts) is the
potential for pushing electrons through a conductor. The voltage of a source can
be thought of as an electrical pressure.
Resistance - The resistance to the flow of electrons in a material R (measured in
ohms) is the inverse of conductance (measured in mhos). Generally, metals are
good conductors while wood, plastic and rubber are insulators and other
materials such as carbon and metal oxides are poor conductors (also called
resistors). Conductors have low resistance, resistors can have a moderate to
high resistance and insulators have an infinite resistance.
Current - The current, I (meaured in amps or amperes) is a measure of the flow
of electrons through a material.
Ohm's Law - The relationship between the resistance R in a resistor, the voltage,
V of an electrical source and the current, I in a circuit composed of connecting
the conductor to the source is given by Ohm's Law.
+
V  IR
V
I
R
-
As indicated by this simple expression we can increase the current through the
resistor by increasing the voltage or by reducing the resistance.
Power - The battery or other electrical power source expends energy (MKS units
of electrical energy are Joules) to push electrons through a resistor. The power
P dissipated by a source V sustaining a current I in a resistor R (measured in
watts or Joules/sec) is given by,
V2
P  IV  I 2 R 
R
An important underlying issue of this expression is the consideration of the
maximum power that can be tolerated by an electrical component. The power
generated in an electrical component by the flow of electrons is dissipated in the
form of heat. Commercially produced resistors are rated for a maximum power
dissipation. Typical resistor power ratings are 1/4 watt, 1/2 watt and 1 watt.
Higher wattage resistors are produced for special purposes.
Example: Consider a 100 ohm resistor connected to a 12 volt source in a simple
circuit. Compute the power generated in the resistor.
V2
R
(12) 2
P
 1.44 watts
100
P
Series Circuits - A series circuit is one in which the components are connected
end-to-end as shown below.
Va = V
R1
+
V
Vb = V.R2/(R1+R2)
R2
Vc = 0
The electrons flowing in this circuit must pass through both resistors so the
equivalent resistance of this circuit is the sum of the two resistors R1 and R2.
Rtot  R1  R2
(series)
We can compute the voltage level at any point in this circuit. The voltage V a at
any point from the positive end of the source to the resistor R 1 is equal to the
source voltage. There is a voltage drop through this resistor proportional to the
fraction of the total resistance controbuted by R1, so the voltage Vb is given by,
Vb 
R2
V
R1  R2
We can test the validity of this expression by varying the values of the
resistances R1 and R2. For example, as R1 goes to zero the voltage Vb goes to V
and as R2 goes to zero the voltage Vb goes to zero. This matches our
understanding of voltage drops across resistors in a series circuit. We can
compute the power dissipated in each resistor using the current through the
circuit and the voltage drop (voltage difference) between the ends of each
resistor.
Example: Assuming that R1=100 ohms, R2=200 ohms and V=6 volts of the
series circuit above, compute the voltage Vb and the power dissipated in each
resistor, (P1 for R1 and P2 for R2).
 200 
 2
Vb  (6)
  (6)   4 volts
 100  200 
 3
P1 
P2 
Va  Vb 2
R1
Vb  Vc 2
R1
22

 0.04 watts
100

42
16

0.08 watts
200 200
Alternatively, we could compute the power using P  I 2 R and V  IR to find the
current I in the resistors.
V
6
I 
 0.02 amps
R (100  200)
P1  (0.02) 2 R1  (0.0004)(100)  0.04 watts
P2  (0.0004)( 200)  0.08 watts
Parallel Circuits - The components in a parallel circuit are connected side-byside as illustrated by the two resistors in the circuit below.
+
V
-
I1
R1
R2
I2
The electrons flowing in this circuit can pass through either R 1 or R2 so the total
resistance to electron flow is less than it would be for either one of the resistors
alone. The total resistance of a pair of resistors, R1, R2 connected in parallel is
given by,
RR
1
(parallel)
Rtot 
 1 2
1
1
R1  R2

R1 R2
The power dissipated by each resistor can be computed using P  V 2 / R where
the voltage drop is the same for each resistor and is equal to the total source
voltage. The current in each resistor is inversely proportional to its resistance
and can be determined by Ohm's Law.
Example: Compute the current I1, I2 in each resistor in a simple parallel circuit as
shown above with R1=100 ohms, R2=500 ohms and V=10 volts.
I1 
V
10

 0.10 amps
R1 100
I2 
V
10

 0.02 amps
R2 500
Capacitors - A capacitor is composed of a pair of conducting plates separated by
an insulator. When a voltage is applied to the leads of a capacitor electrons
accumulate on one of the plates and a depletion of electrons forms on the other.
Since the plates are not in contact the electrons cannot continue to flow. The
accumulation of electrons is called a charge.
conductor
+
+
+
+
+
++
+ ++ + +
- - - - -- - -
insulator
(dielectric)
When a capacitor is charged the power source can be removed and it will hold its
charge until the leads of the capacitor are connected to some conductor. The
capacity to hold charge (called capacitance, symbol C) is measured in units of
Farads. We can use a resistor to slow the rate of charge and discharge as shown
in the simple RC-circuit shown below.
When the switch is closed, the capacitor will begin to charge. The voltage drop
across the capacitor will asymptotically approach the source voltage. The
amount of charge (measured in volts) on the capacitor as a function of time is
given by,
t



VC (t )  V  1  e RC 


where V is the source voltage, R is in ohms, C is in Farads and t is the time in
seconds. The product R.C is called the RC-time constant and it represents the
+
Voltage
time required for the voltage across the capacitor to increase by ~62.3% of the
remaining voltage.
C
V
R
RC
2RC
3RC
4RC
5RC
Time
Voltage
Once charged, a capacitor can be discharged by connecting it leads to a
conductor. In the circuit below a capacitor, C with an initial charge producing a
voltage VC is connected to a resistor R. When the switch is closed the capacitor
will be discharged at a rate determined by the RC-time constant.
+
VC
C
-
R
RC
2RC
3RC
4RC
5RC
Time
V (t )  VC  e

t
RC
Example: Compute the voltage drop across the capacitor and the resistor in the
circuit below 3 seconds after the switch is closed. C=10F, R=100K, V=10 v
+
C
V
R
The instant the switch is closed, the voltage drop across the capacitor is zero,
which means that the voltage drop across the resistor is V=10v. As the capacitor
charges the voltage drop across the capacitor increases asymptotically while the
voltage drop across the resistor decreases by a proportional amount.
The RC-time constant is equal to (100x103)(100x10-6)=10 seconds, so we can
compute the voltage drop across the capacitor after 3 seconds by,
t
3

 



RC 
10 


VC  V  1  e
  (10) 1  e   (10)(1  0.741)  2.59 volts




When two capacitors are connected in parallel the equivalent capacitance is the
sum of the capacitance of each. When they are connected in series the
equivalence capacitance is reduced.
C parallel  C1  C 2
C series 
1
1
1

C1 C 2
C1
C1
C2
C2
Exercises
2.5: Given R1 through R6 = 100 and V=10v, find the voltage drop across and
current through each resistor in the circuit below.
R4
R1
+
R5
V
R2
R3
R6
2.6: What is the power dissipated by resistor R3 in the circuit above?
2.7: In the circuit below, the switch is flipped up for 1 second and then back
down. Compute the voltage drop across the capacitor 5 seconds after the switch
is flipped down.
+
100F
10 v
100K
-
10K
2.8: Compute the equivalent resistance of the following resistor networks.
R
R
R
R
a.
R
R
R
R
R
b.
R
R
R
R
R
R
R
R
R
R
c.
R
R
R
R
R
d.
2.9: Compute the equivalence capacitance of three capacitors C each with 60F
capacitance connected in series.
Diode - A diode is an electronic component that allows electrons to flow in one
direction but not in the other direction.
1N4004
cathode
anode
anode
cathode
The direction of current is from the anode to the cathode. There are a wide
variety of uses for and types of diodes. Normally diodes are used to redirect or
restrict electron flow. A zener diode is used to drop the voltage of a source to
some lower level. A light-emitting diode (LED) as shown below is used as an
indicator.
+
long lead
is cathode
V
R
-
There is a dual red-green LED on the BasicX-24. These devices are composed
of two separate LEDs of different colors. When applied to a voltage source with
a particular polarity one of the LEDs lights, and when the polarity is reversed the
other lights. If alternatic current is applied both LED's light giving an impression
of another color (e.g. yellow for a red-green dual LED).
V
R
2.3 Optics
The Electromagnetic Spectrum - An electromagnetic (EM) wave is comprised of
an oscillating electric field and a magnetic field. EM waves propagate through a
vaccuum at around 3x108 meters/sec or 186,000 miles/hour.
Electric Field
E
Magnetic Field
B
Different EM waves are characterized by their rates of oscillation which can be
quantified as the frequency of the EM wave measured in Hertz or
cycles/persecond. More commonly we refer to the distance traveled by the wave
during one period of its oscillation also called the wavelength. EM waves can
vary greatly in length. The units of wavelength vary according to the region of the
electromagnetic spectrum we are considering. For example radio waves range
from several hundred meters down to less than 1 meter in length; radiant heat
(infrared energy) is comprised of EM waves measured in millionths of a meter or
microns and can range from a few hundred microns down to around 1 micron;
visible light is measured in Angstroms or nanometers (1x10-9 meters) and
ranges from between 780 and 380 nanometers.
When we refer to visible light we mean light that is visible to humans, however
electro-optical components and some animals can see in the near-IR and ultraviolet (UV) regions of the spectrum. Photoresistors and photovoltaics can be
made that respond to IR, NIR, visible and UV wavelengths.
Sources of Light - Sources of light can be natural or artificial. The distribution of
energies at the various wavelengths is referred to as the spectral power
distribution of the light source.
Spectral Responsivity - The sensitivity of a light sensor as a function of the
wavelength of the light is called the spectral responsivity of the sensor. It is
important to match the spectral responsivity of the light sensor to the spectral
power distribution of the light source.
Blackbody - A blackbody radiator is a theoretical material that reflects emits
100% of its thermal energy as radiant energy.
Planckian Radiators - Planck's Law gives the relationship between the spectral
power distribution of a blackbody radiator and its temperature. The distribution of
EM energy emitted from a blackbody as a function of wavelength for various
temperatures is shown below.
Color Temperature - Color temperature refers to the heat of a light source. As
color temperatures vary, so does the distribution of energy at each wavelength.
This distribution is quantified by Planck's Law.
The following tutorial is from
http://www.adobe.com/support/techguides/color/colortheory/vision.html
The Interaction of Light and Matter - The nature of light and the visible spectrum
one of the three factors that permit us to see colors and light. The second factor
has to do with the interaction of light and matter, for when we see an object as
blue or red or purple, what we're really seeing is a partial reflection of light from
that object. The color we see is what's left of the spectrum after part of it is
absorbed by the object.
First, let's look at the general properties of light interacting with matter. When
light strikes an object it will react in one or more of the following ways depending
on whether the object is transparent, translucent, opaque, smooth, rough, or
glossy:
• It will be wholly or partly transmitted.
• It will be wholly or partly reflected.
• It will be wholly or partly absorbed.
Transmission - Transmission takes place when light passes through an object
without being essentially changed; the object, in this case, is said to be
transparent:
Some alteration does take place, however, according to the refractive index of
the material through which the light is transmitted.
Refractive Index is the ratio of the speed of light in a vacuum to the speed of light
in a given transparent material (e.g., air, glass, water). For example, the RI of air
is 1.0003. If light travels through space at 186,000 miles per second, it travels
through air at 185,944 miles per second—a very slight difference. By
comparison, the RI of water is 1.333 and the RI of glass will vary from 1.5 to
1.96—a considerable slowing of light speed.
The point where two substances of differing RI meet is called the boundary
surface. At this point, a beam of transmitted light (the incident beam) changes
direction according to the difference in refractive index and also the angle at
which it strikes the transparent object. This is called refraction.
Light striking the surface of an object straight on (that is, at normal incidence) will
pass through without refraction (as in the illustration above). But light striking at
any other angle will be refracted as well as partially reflected:
The RI of a substance is further affected by the wavelength of the light striking it.
The RI of a transparent object is higher for shorter wavelengths and lower for
longer ones. This is most apparent in the refraction of a light beam through a
prism. The red end of the visible spectrum does not refract as much as the violet
end. The effect is a visible separation of the wavelengths. The rainbow is another
example, where sunlight is refracted through raindrops in a manner similar to the
refraction of light through a glass prism.
If light is only partly transmitted by the object (the rest being absorbed), the
object is translucent. The degree of absorption is the only essential difference.
Light transmitted through a translucent object reflects and refracts according to
the same principles as light transmitted through a transparent object.
Reflection - As we've seen above, light that strikes a transparent object is
transmitted in part and reflected in part. But when light strikes an opaque object
(that is, an object that does not transmit light), the object's surface plays an
important role in determining whether the light is fully reflected, fully diffused, or
some of both.
A smooth or shiny surface is one made up of particles of equal, or nearly equal,
refractive index. These surfaces reflect light at an intensity and angle equal to the
incident beam:
Scattering, or diffusion, is another aspect of reflection. When a substance
contains particles of a different refractive index, a light beam striking the
substance will be scattered. The amount of light scattered depends on the
difference in the two refractive indices and also on the size of the particles.
Most commonly, light striking an opaque object will be both reflected and
scattered. This happens when an object is neither wholly glossy nor wholly
rough.
Absorption - Finally, some or all of the light may be absorbed depending on the
pigmentation of the object. Pigments are natural colorants that absorb some or
all wavelengths of light. What we see as color, are the wavelengths of light that
are not absorbed.
However, the wavelengths of light that concern us most are the red, green, and
blue wavelengths. These are the basis for the tristimulus response in human
vision, as well as a significant part of color reproduction.
Spectral Reflectance/Transmittance Curve - Just as spectral power distributions
are a property of a light source, the spectral reflectance or transmittance curve is
a property of a colored object. Spectral reflectance refers to the amount of light at
each wavelength reflected from an object as compared to a pure reflection (e.g.,
from a pure white object that reflects 100% at all wavelengths). Spectral
transmittance refers to the amount of light at each wavelength that is transmitted
through a transparent colored object as compared to the amount transmitted
through a clear medium such as air.
Below are some examples of spectral reflectance curves for objects that appear
red, yellow, blue, and purple:
The importance of spectral reflectance or transmittance curves lies in their
contribution toward the definition of color. As we've mentioned, seeing color
depends on the triad of light source, colored object, and the human eye. The
wavelengths reflected or transmitted from or through an object determine the
stimulus to the retina that provokes the optical nerve into sending responses to
our brains that indicate color.
The Physiology of Human Vision - The third part of the color triad is human
vision. After all consideration has been made to the nature of the light and the
spectral reflectance of the object being viewed, how you see color depends on
the combination of three distinct stimuli of the retina. For this reason, human
vision is often referred to as a tristimulus response.
This aspect of seeing color was well described by British physicist James Clerk
Maxwell who wrote in 1872,
We are capable of feeling three different color sensations. Light of different kinds
excites these sensations in different proportions, and it is by the different
combinations of these three primary sensations that all the varieties of visible
color are produced. Maxwell's studies, along with those of Thomas Young and
Hermann von Helmholtz, form the basis for all currently held views on human
color vision.
Vision Basics - The simple mechanics of human vision are as follows:
• The cornea draws light and focuses it on the lens, which adjusts for distance.
As it travels from the cornea to the lens, the light passes through an aperture
called the pupil. This aperture narrows and widens in response to the brightness
or dimness of the surrounding light by the action of the iris (the colored part of the
eye).
• The lens then passes the light through a transparent gel called the vitreous
humor and focuses an inverted image of the object being viewed on the retina at
the back of the eyeball.
• The retina is the light-sensitive part of the eye and its surface is composed of
photoreceptors or nerve endings. These receive the light and pass it along
through the optic nerve as a stimulus to the brain. The photoreceptors are of two
types, rods and cones. The greatest concentration of rods and cones is in an
area of the retina called the fovea. In the very center of the fovea is an area
called the foveola composed entirely of cones. The area of the fovea/foveola is
the most light- and color-sensitive part of the retina.
Cut-Away View of the Human Eye
Anatomy of the Human Vision System
Stimulus - The stimulus received by the brain is what we see as color.
Thespectral power distribution of the light source, times the spectral reflectance
of the colored object, times the spectral sensitivity of the cones in the human eye
equals the stimulus of color that we see.
http://www.adobe.com/support/techguides/color/colortheory/vision.html
The CIE (Commision Internatinale de L'Eclairage) Standard Observer Curve This curve shows that humans are most sensitive to green light and least
sensitive to red and blue. This curve also closely matches the sensitivity of the
monochromatic sensor used in black-and-white film and in black-and-white video
cameras.
Spectral Sensitivity - Similar to the spectral power distributions and spectral
reflectance curves we discussed in the preceeding sections, visual sensitivity to
colored light is also characterized by a graph called a spectral response or
sensitivity curve. We mentioned above that certain cones are sensitive to red,
green, or blue light. However, the sensitivities don't actually peak at these
wavelengths; instead, the curves cover portions of the spectrum, which could be
called reddish, greenish, and bluish. For example, the r sensitivity curve covers
the wavelengths from 475nm to about 700nm and peaks at roughly 590nm which
is yellow light. Below are the sensitivity curves for the r, g, and b cones as well
as the curve for the scotopic vision of the rods:
Spatial Acuity - Another measure of your vision is the spatial resolution or acuity.
This is what is measured by the standard eye chart. Your ability to resolve
(recognize) objects at a distance is typically stated in relative terms. For example
a person with normal sight is said to have 20/20 vision. This means that your
ability to regonize images (at 20 feet) is what is normal for humans. A person
with 20/400 vision is able to recogize objects at 20 feet that are recognizible at
400 feet by a person with "normal vision".
What is really being measured here is the angular resolution, or the ability to
resolve two lines separated by a given angle. As range to the test object
increases the effective angular separation decreases.
The sharpest vision (for normal 20/20 vision) or highest angular resolution is
around 1 line-pair per arcmin or 1/60 of a degree. Human visual acuity drops off
quickly as we move away from the visual axis.
The image transmitted from the eye to the visual cortex of the brain undergoes a
form of compression. This natural image compression take the form of a bandpass filter.
Lateral inhibition and excitation together lead to a bandpass characteristic of the
contrast sensitivity function of the human visual system.
This image compression is lossy. One of the functions of the visual cortex is to
reconstruct the image from this compressed information. Usually this
reconstruction works well but we can set up examples the illustrate the limitations
this processing using some simple optical illusions.
Optical Illusions
Sitting within 2 feet of this image, try to count the black dots at the intersections
of the gray lines. The width of the gray lines is less than your visual acuity in your
peripheral vision but greater than your visual acuity in the region of sharp focus.
Therefore you will experience a "ringing" in the image near an abrupt change in
contrast in your peripheral vision.
The gray lines in the image below are all horizontal and parallel to each other.
The skewed alternating black and white boxes interfere with our ability to
properly reconstruct this image.
Finally, we can test our ability to correctly process moving images. Look at the
black dot in the center of the image below as you move your head toward and
away from the image.
Creating Images
The synthetic camera model for creating images simulates the casting of light
rays onto the image plane through an optical imaging system.
The Pinhole Camera - The simplest camera is a pinhole in an enclosure through
which light passes from the subject (object) to cast an inverted image on the
opposite side of the enclosure. The size of the image relative to the object is a
function of the ratio of the distance from the pinhole to the image and from the
pinhole to the object. Due to the geometry and the fact that light (in a
homogeneous medium) travels in a straigh line the angular subtense of the
image to the pinhole always equals the angular subtense of the object to the
pinhole.
The Synthetic Camera Model - The synthetic camera model requires us to make
a number of decisions concerning image size and format. In addition to deciding
the physical size of the image we are creating, we need to choose a field of view
(FOV). This is the angular subtense of the scene to be represented in the image.
When we use the synthetic camera model for generating images with the
computer we will need to choose the type "lens" system to model.