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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 = (Bh1)/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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