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					                               Center for Imaging Science
                                SIMG 215 - Laboratory
                Shadowgrams: Geometric Laws of Image Formation

Objective: To explore the geometric and algebraic laws of shadow imaging, and to
design a shadow box imaging system.

(I) Background

       Nearly everyone has made shadow puppets by folding their hands and casting
images on a wall. Shadow imaging is a very ancient form of imaging technology. It was
well known and understood in classical Greece. Plato, for example, used shadow
imaging as a metaphor for some of his most significant philosophical concepts. During
the 18th century, shadow imaging was used to make silhouettes, as illustrated in Figure 1.

            Figure 1: Shadow imaging technique and silhouettes of George
           and Martha Washington (from www.WallaceNuttingLibrary.com).




        In Asia, shadow imaging was used to produce the first moving pictures. This was
done by casting shadows of puppets onto a translucent screen. A modern form of this
technique is illustrated in Figure 2. The screen assembly is often called a shadow box.
The screen is generally an oiled paper or cloth so the shadow cast by the puppet is easily
viewed on the side of the screed facing away from the light. The puppet master works on
the illuminated side of the screen and the audience watches the show on the other side of
the screen.

           Figure 2: Shadow box for puppet shows (www.osv.org)
       Shadow imaging of the kind illustrated in Figure 2 was highly developed as an art
form in Asia by the eleventh century. The technology was introduced into Europe
through Persian scholars about this time, as illustrated by Omar Kahyyam's reference to
"a magic shadow show".

           "For in and out, above, about, below,
           'Tis nothing but a Magic Shadow-show
           Play'd in a Box whose Candle is the Sun,
           Round which we Phantom Figures come and go"
               (Omar Kahyyam, 11th century Persia)

      Shadow imaging is not only a beautiful art form, but the basis for several modern
imaging technologies. X-ray, computed tomography (CT), and magnetic resonance
imaging (MRI) are all based on the geometric theory of shadow imaging.

       The laboratory experiment described below explores the geometry of shadow
imaging in a simulated design problem. A shadow box is to be designed for a puppet
master who provides practical design specifications. Before designing the shadow box,
experiments are done to test the geometric theory of shadow imaging.

(II) The Test System

        The system used in this experiment is shown schematically in Figure 3. The light
source is a white circle on a computer monitor, and the diameter of the circle, d, can be
adjusted.

                        Figure 3: Diagram of the shadow imaging
                             system used in this experiment.
                                             L


                 Light Source              1              2


                                    d            h1                    h2
                                                Puppet
                                                Test Pattern
                    Monitor with a white
                    circle of diameter d                 Translucent Screen

       A bright image is, of course, desirable. In addition, a sharp image is needed. A
sharp image requires the puppet to be placed close to the screen. However, in order for
the puppet master to control the puppets easily without tearing the screen, the working
distance, 2 in Figure 3, must be at least 2 cm. The size of the puppets used by the puppet
master is h1, and this dimension can not be changed. The two parameters you can adjust
are: (a) the diameter of the light source, d, and (b) the distance from the light source to
the screen, L. The two parameters you want to achieve are: (a) a bright image, and
(b) a sharp image. These design specifications are summarized in Table I.

                     Table I: Design Specifications for the Shadow Box
                   Room for the puppet       L at least 200 cm
                   master to work            (1/2 meter)
                   Room to work puppets      2 at least 10 cm
                   without too much blur
                   Luminance at the          I at least 150 lux
                   viewing screen            (brightness of a dim room)

        In table I, "blur" refers to the problem of a shadow imaging going out of focus if it
is moved too far from the screen. The puppet master wants to keep the puppets in focus
on the screen, but the screen is delicate and easily torn. The puppet master wants the
puppet to stay in reasonable focus for up to 10 cm distance from the screen. This means
we must determine the maximum amount of blur that is acceptable.

(III) Measuring the Characteristics of the System

        Set up the system illustrated in Figure 3. Choose a source diameter, d, of 10 cm.
Your instructor will provide the test pattern and the viewing screen. The test pattern is
shown in Figure 4. Also shown is an example of the kind of image observed on the
viewing screen when the test pattern is held at some distance, 2, away from the screen.
The two holes in the test pattern simulate eyes on a puppet. The maximum amount of
blurring the puppet master will accept occurs when the square hole and the diamond hold
become JUST BARELY indistinguishable.

                Figure 4: Test pattern used in the experiment and an example
                         of the image shown on the viewing screen.




           h1
                                               h2




                        Test Pattern

                                          Shadow Image Observed on the Screen
         Hold the viewing screen a chosen distance, L in cm, away from the screen. Place
the test pattern directly against the screen so the sharpest possible image is obtained. For
the viewing distance you have chosen, L, make a judgment about the contrast quality of
the image. If L is very large, the screen will be dark and the image will be hard to see. If
L is small, the image is easier to see. For your chosen value of L, make a judgment about
the contrast quality of your image based on the scale shown in Table II. Record your
value of C in Table III.

               Table II: Scale for judging the quality of image contrast, C
           Very easy     Easy to     Difficult   Very difficult Impossible
           to see        see         to see      to see          to see
                 4           3            2             1              0

        Move the test pattern away from the screen and observe that it blurs. Adjust the
distance so the image blurs JUST enough so you can no longer see the difference
between the square hole and the diamond shaped hole. Measure this value of, 2, and
write the value in Table III. Also notice that moving the test pattern away from the
screen increase the size of the image. Measure the image size as the distance between the
two holes, h2, and record this value in Table III.

        Choose several different values of L and of d to see the effects of these variables
on the parameters 2, h2 and C. Values of d should range from 2 cm to 400 cm. L should
range from very small (as close as you can) to as large as can be achieved and still see the
image.
                              Table III: Experimental Data
                 Parameters You Control | Parameters You Measure
                   d in cm      L in cm        2 in cm       h2 in cm C




(IV) Test of the Geometric Theory of Image Blurring

        Figure 5 summarizes the geometric features of the system. Figure 5 shows the
relationship between the puppet size, h1, and the image size, h2. From this diagram, we
can write equation (1).
                                    h 2 h1
                                                                                        (1)
                                     L   1
                         Figure 5: Image and light source geometry




The blurred region of the image, b, is caused by the finite size of the light source, d, and
the finite distance away from the light source, L. This is described in equation (2).

                                      b   d
                                                                                             (2)
                                      2 1


Equation (2) tells us that blurring is caused by three different parameters, d, 1, and 2.
Combining equations (1) and (2) gives equation (3). This does not look like a

                                          d  2  h1
                                     b                                                       (3)
                                             L h1


simplification, but notice the ratio d/L. This is the angle, , of the light source as seen at
the screen.

                                          d
                                     θ                                                       (4)
                                          L

                                          θ  2  h 2
            Therefore,               b                                                       (5)
                                               h1

Equation (5) can be further simplified by considering the geometry of blurring, b, as
described in Figure 5 and the blurring observed on the screen. The degree of blurring
that is considered acceptable by an observer is not the absolute blurring distance, b, of
Figure 5. It is the distance b relative to the size of the image. We will use the distance
between the holes in the test pattern as our measure of the image size. Thus, according to
theory, visual blur is the ratio b/h2.

                                           b
                                     B                                                       (6)
                                          h2

Combining equation (6) with equation (5) gives equation (7), which is a useful equation
to describe the practical, visual blurring of a puppet image, B.
                                             θ  2
                                       B                                                 (7)
                                              h1

        You can test the theory of blurring shown in equation (7) by examining your
experimental data. You collected the data at the same degree of blur. In theory, then, the
value of B is a constant for all of your experimental data. If this is so, then equation (7)
can be re-arranged as follows.

                                             B h1
                                      2                                                 (8)
                                               θ



         Equation (8) says the practical working range between the screen and the puppet,
2, is dependent on three things: The amount of acceptable visual blur, B, the size of the
puppet, h1, and the angle subtended by the light source, . To test this theory, plot your
measured values of 2 versus values of h1/. Use your experimental values of L and d to
calculate . You should observe a graph similar to the one shown in Figure 6.

              Figure 6: Example of experimental results for 2 versus h1/.
                            20
                                  B=(20/120) = 0.20

                      2
                      in cm
                                                          120

                              0
                                  0             50     100         150
                                              (h1/) in cm
If the experimental data is reasonably well described by a straight line, then the slope of
the line is our experimental estimate of the visual blur, B, that is acceptable for the
shadow box. Sketch a straight line through your data and estimate the value of B as
illustrated in Figure 6. Record this value of B in Table IV.

                               Table IV: Design Results
                   Maximum allowable blur factor                    B=
                   Maximum allowable size of light source           d=      cm
                   Power of light bulb used in the test             Wo      watts
                   Lux reading behind the screen                    Io      lux
                   Minimum required power of the light source       W=      watts
(V) Design the Maximum Diameter of the Light Source, d

        The maximum acceptable blur, B, and the size of the puppet, h1, are both fixed.
Therefore, the only thing that can be controlled in designing the shadow box for the
puppet master is the value of the angle, . As shown in Table I, the puppet master
requires a space of 2 = 2 cm to work the puppets. Solve (7) for  = (Bh1)/2, and
calculate the maximum allowable value for .

      The puppet master also requires at least L = 50 cm in order to have room to work
between the light source and the screen. Using equation (4), we can calculate the
maximum allowable size of the light source, d = L.

      At this point your data should have shown that the geometric theory of the
shadow imaging system is reliable and useful, and you should have a design value for the
maximum dimension of the light source, d. Record this value in Table IV.

(VI) Design the Required Power of the Light Source

       Place your screen a distance L = 200 cm in front of an ordinary light bulb. Write
down the power, Wo, of the light bulb in Table IV. Use a lux meter to measure the
brightness of the screen, Io, and record this value in Table IV. In order to produce the
minimum required brightness of I = 200 lux, a light source of power W is required. Use
equation (9) to record the required value of W and record this value in Table IV.

                                           I
                                   W  Wo  o                                             (9)
                                             I



(VII) Your Report

       Your report to the puppet master should include the results shown in Table IV. In
addition, you should provide answers to the following questions.

1. The values of 2 and h1 are fixed by the requirements of the puppet master. What is
the single parameter that controls visual blur, B, in the image? Which equation tells you
this?

2. You estimated the value of B by sketching a straight line as illustrated in Figure 6.
Sketch additional lines to estimate the maximum and minimum value of B you believe
might be justified by your data. Use these limiting values of B to estimate the upper and
lower range of your estimate of the required source diameter, d.

				
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