RESOLUTION LIMITS OF IN-LINE HOLOGRAPHIC IMAGING SYSTEMS. Joseph Withers. A senior thesis submitted to the faculty of
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1 RESOLUTION LIMITS OF IN-LINE HOLOGRAPHIC IMAGING SYSTEMS by Joseph Withers A senior thesis submitted to the faculty of Brigham Young University - Idaho in partial fulfillment of the requirements for the degree of Bachelor of Science Department of Physics Brigham Young University - Idaho December 2013
2 Copyright 2013 Joseph Withers All Rights Reserved
3 BRIGHAM YOUNG UNIVERSITY - IDAHO DEPARTMENT APPROVAL of a senior thesis submitted by Joseph Withers This thesis has been reviewed by the research committee, senior thesis coordinator, and department chair and has been found to be satisfactory. Date Todd Lines, Advisor Date Evan Hansen, Senior Thesis Coordinator Date Stephen Turcotte, Committee Member Date Stephen McNeil, Dr.
4 ABSTRACT RESOLUTION LIMITS OF IN-LINE HOLOGRAPHIC IMAGING SYSTEMS Joseph Withers Department of Physics Bachelor of Science Methods are developed and carried out to analytically and experimentally determine the resolution limits for in-line holographic imaging systems. Onaxis resolution is limited by the lens numerical aperture, effective aperture and effective pixel pitch. The strongest limit of these three constrains the entire system. Experimental data shows strong agreement with these theoretical limits. Off-axis resolution proves to be more complicated to model. It has the same limitations as the on-axis resolution as well as limits from the detector edge cutting off the airy disk. Thus, resolution at the edges will be less than at the center. This degradation in resolution at the edges, increases with reconstruction distance. With its fast recording and processing times and three-dimensional ability, digital holography has become a popular method for particle imaging. Understanding the resolution limits will enable scientists and engineers to successfully design and build future holographic imaging systems.
5 ACKNOWLEDGMENTS I would like to thank my mentor, Dr. Raymond Shaw, and those in the Cloud Lab at Michigan Technological University, for their guidance and encouragement. I also thank the faculty at Brigham Young University-Idaho for their helpful feedback throughout the semester. I give special thanks to my wife, Meghan, for her continual patience and support.
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7 Contents Table of Contents List of Figures vii ix 1 Introduction Cloud Physics Holographic Particle Imaging In-line Digital Holography The HOLODEC Resolution Limits Theoretical Methods Resolution Derivations Experimental Methods Alignment Data Acquisition Experimental Results 19 5 Conclusions 25 6 Future Research Contrast Transfer Function Signal to noise ratio Roll-off Bibliography 33 A 1951 USAF Test Chart Table 35 B CTF Calculator Manual 37 Index 43 vii
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9 List of Figures 1.1 In-line holography setup Holodec Numerical aperture Effective aperture diagram Predicted resolution Holodec on optical bench Space between benches Holodec arms Ruler Ruler and stage Experiment setup Reconstructed sample volume USAF test chart Coordinate system Test chart wheel Hologram compare Smallest element Predicted resolution with data X axis resolution Y axis resolution Diagonal axis resolution Square wave Gaussian Square wave with gaussian Intensity with contrast Contrast Intensity one Intensity thirteen A.1 Resoltuion test chart table ix
10 x LIST OF FIGURES B B B B B B B B B
11 Chapter 1 Introduction 1.1 Cloud Physics Clouds in the atmosphere serve as a first order boundary between the earth and the sun, which greatly affects climate and weather. The optical properties of clouds cause the suns light to be reflected. These optical properties are dependent upon the droplets from which the cloud is formed of. Cloud particle size and thermodynamic phase affect simplified cloud and climate models. Knowing more about the number density and size distributions of droplets in various clouds types enhances our understanding of the atmosphere and perhaps make our climate models more reliable. Particle imaging provides a way to measure both of these features. 1.2 Holographic Particle Imaging With its fast recording and processing times and three-dimensional ability, digital holography has become a popular method for particle imaging [6]. Three dimensional length scales between particles can be visualized. Particle image velocimetry (PIV), 1
12 2 Chapter 1 Introduction which requires the recording of many holograms, is now quite feasible with digital processing. Thus holography is a strong candidate for imaging in situ cloud droplets. 1.3 In-line Digital Holography While there are variations to holographic setups, e.g., reflection and transmission, the basic principles are the same. An interference pattern, which contains information about the phase and amplitude, is recorded. In figure 1.1, an in-line holography setup is shown. A coherent laser source is expanded and collimated to produce plane waves. The waves impinge upon the object, which scatters some of their light. The scattered waves, or object beam, interact with the reference beam and create an interference pattern that can be recorded. Figure 1.1 An in-line holography setup is shown. The beam is expanded and collimating to create plane waves. Object beam and reference beam interfere with each other and a pattern is recorded containing phase and amplitude information. Originally, interference patterns were recorded by photographic plates. Holograms were then reconstructed by illuminating the recorded hologram with a coherent source. As the wave interacted with the pattern it would interfere and form an image of the object. The advances in computer technology in the past twenty years have enabled holographic particle imaging to go from film to digital recording [4]. Digital holograms, recorded onto a charge-coupled device (CCD), can be numerically recon-
13 1.4 The HOLODEC 3 structed using Fourier transforms [1]. 1.4 The HOLODEC The Holographic Detector for Clouds (HOLODEC) is an in-line holographic instrument that images cloud droplets and ice crystals. Cloud water droplet sizes range from 5 to 30 µm and ice crystals range from 50 to 2000 µm. The HOLODEC was designed to image a range of particle types, e.g., liquid and ice; the smallest particle diameter 5 µm and a sample volume of 20 cm 3. In order to know how small of particles can be detected by the optics in the HOLODEC and correctly analyze data obtained from it, a confident understanding of its resolution limits must be ascertained. Figure 1.2 HOLODEC attached to wing of a C-130. HOLODEC is on right with the black tips. Previously, the resolution of the HOLODEC was determined through a process of recording holograms of a resolution test chart [5]. An algorithm was used to flag pixels above a threshold intensity. The flagged pixels were set to a maximum intensity, while all others were set to zero. By inspection, the smallest resolvable bar pattern on the test target was identified, and the resolution limit was determined by looking up the bar patterns width on a data table. While this method does provide a quantitative measure of resolution, it does not tell us anything about the actual width of the
14 4 Chapter 1 Introduction bar pattern compared with the detected width of the object being imaged. As the resolvable limit drops below the size of an object, the image will begin to lose pixels. Consequently, the object width detected is smaller than actual size. This resolution test shows that the HOLODEC can detect objects as small as 6 µm at a distance of 13 cm, it does not indicate the detected width. A resolution test that determines the smallest resolvable bar width at its true size in the presence of noise will give an accurate and operational resolution limit. 1.5 Resolution Limits Digital holography comes with some strong resolution limits that must be taken into account when designing imaging systems, namely pixel size, lens numerical aperture, and reconstruction distance. The CCD introduces a finite pixel size that sets a limit on particle imaging and sizing. As in all optical setups, the system is only as good as the lens, thus the numerical aperture of the lens sets another limit. An interesting feature of in-line holography is the ability to reconstruct down the z axis, the axis going from the center of the lens on the camera side to the center of the lens on the laser side (see figure 3.3). The greater the reconstruction distance, the smaller the opening angle (θ rec in figure 2.2) from the particle to the lens. Thus an effective aperture is manifested, which further constrains the imaging system [3]. The strongest resolution limit of these three constrains the entire system.
15 Chapter 2 Theoretical Methods 2.1 Resolution Derivations As previously noted, assuming all other aberrations are negligible, the resolution of a holographic imaging system is limited by three factors: the numerical aperture NA lens, imposed by the lens itself (which is a property of the lens design), the resolution limit from diffraction and reconstruction aspects of in-line holography NA rec, and the effective pixel size D pixel,rec. Each of these limits are analytically determined. Calculating the resolution of a diffraction limited system begins with the Airy disk, or Poisson s spot. In Fresnel diffraction, constructive interference occurs at the center of the interference pattern from a circular aperture. At a minimum, the airy disk needs to be detected. The Rayleigh criterion says the smallest resolvable angle in a diffraction limited system is sin θ 1.22λ D, (2.1) where D is the diameter of a circular aperture, λ is the wavelength, 1.22 comes from the first zero of the first order Bessel function, and θ is the angular resolution. Recall that the numerical aperture, which is a measure of the acceptance angle of a 5
16 6 Chapter 2 Theoretical Methods lens, is given by NA = n sin θ, (2.2) where θ is half of the opening angle. Since the medium is air, n 1, thus NA sin θ, (2.3) and Solving for D we get NA 1.22λ D. (2.4) D 1.22λ NA. (2.5) This equation determines the smallest resolvable diameter as a function of the wavelength and the lens numerical aperture [3]. The HOLODEC has a numerical aperture of and a magnification of 2.5. Numerical apertures typically range from 0.07 for low-power lenses to 1.4 or so for high-power (100X) ones [2]. Figure 2.1 Numerical aperture for a lens is a function of the half angle from the from the focal length. The lens numerical aperture is defined from the focal length (see Figure 2.1). In holography, objects can be imaged from a range of distances from the lens. Imaging particles at varying distances from the lens imposes an effective aperture, NA rec, due
17 2.1 Resolution Derivations 7 to the varying opening angle. For far field diffraction sin θ θ, (2.6) and thus NA rec θ rec. (2.7) θ rec arises from a geometrical derivation, and is Figure 2.2 D eff, shown in the figure, arises from the size of the camera detector and magnification of the system. The half angle from the axis to the edge of the detector is a further constraint from the numerical aperture of the lens. The wider angle in the figure is for the numerical aperture. The smaller angle is for the effective aperture. As particles are imaged at various distances from the camera in the sample volume, the opening angle, θ rec, changes. And thus the effective aperture changes. The series expansion for the inverse tangent is θ rec = arctan D eff/2. (2.8) z arctan x = x x3 3 + x5 5 x (2.9) 7 Because z >> D/2, a first order approximation is sufficient. (2.10)
18 8 Chapter 2 Theoretical Methods and thus NA rec D eff 2z rec. (2.11) D eff is the geometrical mean of the diameter of the effective sensor size. The effective sensor size is obtained by dividing the actual dimensions of the sensor by the magnification. For the HOLODEC D eff = 9.66mm 14.4mm. (2.12) Magnification effectively makes the pixels smaller in order to make the object larger. According to the Nyquist sampling theorem, at least two pixels are needed to resolve the smallest feature. This means that the pixel size on the object side must be at most half of the optical resolution to avoid further constraining the resolution [3]. The limit from the effective pixel size is found by dividing the pixel size on the object side by the Magnification multiplied by a factor of 2: D res,pixel = 2D pixel,obj M. (2.13) The pixel pitch on the object side for the HOLODEC II is 7.4 µm, thus D res,pixel = 2.96 µm. A plot of these resolution constraints on the HOLODEC is shown in figure 2.3. The resolution is based on objects being on the z axis. The resolution of the entire system will be limited to the greatest resolution constraint. At distances z < 80 mm, the resolution is limited to D res,pixel. At z > 80 mm, the resolution is limited to D res,rec, which is linear in z.
19 2.1 Resolution Derivations 9 Figure 2.3 Plot of predicted resolution limits as a function of reconstruction distance z. Limits from the numerical aperture D res,lens and the pixel size D res,pixel are constant in z and constrain the system up to z = 80 mm. The effective aperture from reconstruction linearly limits the system in z and is the stronger constraint past z = 80 mm.
20 10 Chapter 2 Theoretical Methods
21 Chapter 3 Experimental Methods 3.1 Alignment Reconstructing many holograms can be quite tedious. This is especially true for large sample volumes, so knowing how far to reconstruct saves time. An alignment method was therefore created. A three-dimensional coordinate system is used between the HOLODEC arms. As previously noted, the z axis is the line from the camera lens to the laser lens, and z = 0 at the camera lens outer surface. The plane perpendicular to the z axis is the x, y plane. Due to the sizes of optical benches that were available, the HOLODEC and the optical mount are on two different optical benches. This presents a complication in ensuring proper alignment. There is a slider between the HOLODEC arms that needs to be parallel to the z axis of our coordinate system. The slider is fixed to the optical bench not containing the HOLODEC. Thus the optical bench needs to be aligned with the HOLODEC. The HOLODEC is supported by a mount that is bolted to the optical bench. It was assumed the precision of this mount was sufficient for alignment purposes and that the HOLODEC is therefore in-line with the optical bench. Thus in theory, only the optical benches need to be 11
22 12 Chapter 3 Experimental Methods aligned with each other, and then the HOLODEC and optical mount will be aligned. Once the optical benches were aligned, further tuning was accomplished by recording holograms at each lens, reconstructing, and then using holoviewer to locate the object positions on the x, y plane. Once the object has the same x, y position at both lenses, the mount and slider are aligned. Figure 3.1 HOLODEC shown mounted to optical bench. Once the mount and HOLODEC are aligned there needs to be a way to know approximately where the object is along the z axis. One way to do this is to simply measure it. This is tedious to do every time, so a systematic way was created. An object was placed 5 cm from the lens on the camera side. This distance was measured with a ruler and a caliper. Once the position of the object was determined with confidence, a ruler was attached to the optical bench next to the slider (slider and mount shown in figure 3.5). Then a piece of thin polyimide tubing was attached to the mount so that it would move with the mount along the z axis and mark the z position. Once the hologram has been reconstructed to the estimated position based on the system above, the hologram can be fine-tuned for better focus. The off-axis, meaning off the z axis, or on the x, y plane perpendicular to the z axis, alignment of the object is based on positioning noted on axes in holoviewer (a
23 3.2 Data Acquisition 13 Figure 3.2 Alignment between optical benches aligns HOLODEC and mount. Dual stage mount is shown on slider between HOLODEC arms. hologram reconstruction software). 3.2 Data Acquisition Reconstructed sample volumes taken by the HOLODEC show a trend of decreasing detected number of particles near the outer edges, especially the farther down the z axis they are. This is particularly true of the corners as shown in figure 3.7 It is assumed that the decrease in detected particle count is due to a decrease in resolution, i.e., the actual particle density does not decrease. There is an uncertainty that what is being detected are actual particles and what their sizes are. The laser intensity is greatest in the center, or on the z axis, of the sample volume, and drops radially outward from it. The resolution may be lower from this decreasing intensity of light hitting the CCD at the outer areas. Particles on the outer edges of the sample
24 14 Chapter 3 Experimental Methods Figure 3.3 The z axis shown between the Arms of HOLODEC. The laser and camera are in the left and right arms, respectively. Figure 3.4 Ruler used to calibrate z axis measurement. Object is measured 5 cm from lens on camera side. volume will have parts of their diffraction rings cut off. Theoretically this would lower resolution at the edges. This would be especially true as particles are imaged farther from the camera lens as their diffraction patterns would be larger. Resolution data was obtained using a 1951 USAF resolution test chart. The USAF chart contains bar patterns organized by size into groups and elements. This test chart is ideal due to its range of bar widths, from µm. The resolution test chart was placed in the sample volume of the HOLODEC along the z axis and individual holograms were recorded at z = 10, 20, 30,130 µm. To have good representative data, this process was repeated five times. Each of the holograms were then reconstructed to the object distance, where
25 3.2 Data Acquisition 15 Figure 3.5 The slider is what is attached to the optical bench. The dual stage mount is on top of the slider. A is attached to the optical bench to provide an approximate reference for the object. Polyimide tubing is used as a marker to indicate the approximate z distance of the object in the hologram. the image had the optimal focus. The resolution of a given image was determined by identifying the smallest complete bar pattern. The vertical and horizontal components needed to be resolvable and complete. Each of the five resolution values for a given reconstruction distance were averaged. The resolution perpedicular to the z axis can be obtained using the same 1951 USAF resolution test chart. Holograms of the test chart were recorded at the same z distances aforementioned. For each z distance, roughly 20 holograms were recorded along the x axis, 30 along the y axis, and radially outward following the diagonal line in figure 3.9. An optical mount was built to hold the test chart. The optical mount is on a slide, which can move laterally (along z) the entire length of the sample volume. The mount itself has two stages allowing for more accurate lateral movement along z as well as perpendicular movement (along x). The final piece to the mount is a variable vertical post which allows for accurate vertical adjustments.
26 16 Chapter 3 Experimental Methods Figure 3.6 These images show the HOLODEC, optical benches, and optical mount configuration. Figure 3.7 Reconstructed sample volume from HOLODEC. Particle number density decreases at the edges. This decreases appears to be stronger at larger z.
27 3.2 Data Acquisition 17 Figure USAF Resolution Test Chart. Bar pattern are organized into groups and elements. Groups are in columns and individual elements in a group are in the rows. Each trio of bars is an element [7]. Figure 3.9 This diagram shows a side view of the leading edge of the HOLODEC. Coordinate system perpendicular to the z axis is shown. The z axis goes into the page. Holograms were recorded along the x, y and diagonal axes.
28 18 Chapter 3 Experimental Methods Figure USAF resolution test chart wheel that was used.
29 Chapter 4 Experimental Results Once the necessary holograms were recorded along the z axis and radially outward along the x and y axes, they were reconstructed to determine the resolution. A MATLAB program was used to view and reconstruct the holograms. The code, HOLOSUITE [1], utilizes a user interface that allows one to pick a desired hologram and reconstruction distance. The appropriate reconstruction distance is found when the objects in the hologram are at the greatest focus. Selecting the distance of best focus is up to the viewers discretion. Objects will generally have the smallest size and sharpest edges at their best focus. Figure 4.1 On the left is a raw unconstructed hologram. On the right is the same hologram reconstructed to the distance the object appears in focus, i.e., the object distance. 19
30 20 Chapter 4 Experimental Results Once the hologram has been reconstructed to the desired distance, the resolution is found from the USAF resolution chart [3]. The smallest complete bar pattern in the chart gives the smallest resolvable diameter. In order for a bar pattern to be complete, the horizontal and vertical bars must be relatively whole. The actual width of bars is found by looking up the corresponding group and element numbers on a table (see appendix A.1). Figure 4.2 shows a circle around the smallest complete bar pattern. Its group and element numbers are 5 and 4 respectively. The width of this element is µm. Figure 4.2 Smallest complete element is circled. This process was repeated for each recorded hologram. Figure 4.3 shows a plot of the predicted resolutions as well as the USAF test chart resolution data along the z axis. Each resolution data point is taken from an average of 5 separate holograms each recorded at the same distance. The experimental data shows agreement with the predicted resolutions. That is, the resolution is limited by the numerical aperture of the lens, the effective aperture introduced from reconstruction and the effective pixel size. The data also agrees with the most limiting factor being the limit of the entire system. Figures 4.5, 4.6 and 4.7 show the results of radial resolution. These plots show the resolution limit versus distance from the hologram center for a given z. Figure 4.5 shows resolution along the x axis. Data for z = 1, 7 and 13 cm are shown. As
31 21 Figure 4.3 Plot of predicted resolution limits with USAF resolution test chart data along the z axis. Each data point represents an average taken from 5 separate holograms at that reconstruction distance. Experimental data shows agreement with predicted resolution. expected, the resolution decreases with reconstruction distance. The first 2500 µm from the center shows relatively constant resolution. This likely because the pixel limit, which is constant, is the strongest resolution limit in this region. The closer to the edge of the detector the object is, the closer the airy disk will be as well. When the airy disk starts to get cut off by the edge of the detector resolution decreases. On the x axis, for z = 1 cm, the cutoff appears to begin around 4000 µm from the center, or 500 µm from the edge. As the object moves farther from the detector, its airy disk will be larger. This should cause the resolution to degrade sooner, i.e., closer to the center. The larger airy disk will reach the edge sooner. On the x axis plot, for z = 13 cm, the resolution starts degrading just after 2000 µm from the center. The plot for the x axis shows the decrease in resolution happens closer to the center of the hologram at greater reconstruction distances. Figures 4.6 and 4.7, resolution plots for the y and diagonal axes, show a similar
32 22 Chapter 4 Experimental Results trend as observed in figure 4.5 for the x axis. Resoultion decreases radially outward, and with greater z. Figure 4.4 Plot of resolution versus distance from the center of the hologram along the x axis. Resolution decreases with distance from center. The further in z the test charts were, the faster the resolution decreased.
33 Figure 4.5 Plot of resolution versus distance from the center of the hologram along the y axis. Resolution decreases with distance from center. The further in z the test charts were, the faster the resolution decreased. 23
34 24 Chapter 4 Experimental Results Figure 4.6 Plot of resolution versus distance from the center of the hologram along the diagonal axis. Resolution decreases with distance from center. The further in z the test charts were, the faster the resolution decreased.
35 Chapter 5 Conclusions Methods were developed to analytically and experimentally establish the resolution limits of the HOLODEC. On-axis resolution is limited by the lens numerical aperture, the effective aperture from reconstruction, and the effective pixel pitch. Experimental data from the HOLODEC shows strong agreement with these limits. These three limits must be taken into account in the design of in-line holographic imaging systems. Off-axis resolution of the HOLODEC proves to be more complicated. It has the same limitations as the on-axis resolution as well as limits from the detector edge cutting off the airy disk. Due to limits at the edges, the HOLODEC, or any other in-line holographic system, will have less resolution at the edges than at the center. This degradation in resolution at the edges increases the further in z (further from the camera side) the objects are, thus the cone shaped profile of particle number density observed in figure 3.7. The attained resolution data will serve as a guide when analyzing in situ particle data. Effectively determining the resolution along the z axis allows a way to compare experimental to predicted resolution. Experimental agreement with predicted resolution demonstrates the utility of the three aforementioned resolution limits. 25
36 26 Chapter 5 Conclusions
37 Chapter 6 Future Research 6.1 Contrast Transfer Function A common method for determining the resolution of an optical system is to measure its contrast transfer function. As the light from an object passes through a lens, the edges of the image are blurred. As an example, take the initial intensity profile of a single bar from the 1951 USAF resolution test chart to be figure 6.1. Figure 6.1 Initial intensity profile of a single bar form a 1951 USAF resolution test chart After the light from the bar passes through a lens the intensity profile looks like figure
38 28 Chapter 6 Future Research Figure 6.2 Intensity profile of a single bar after passing through a lens. If the before and after profiles are superimposed the result would be something similar to figure 6.3. Figure 6.3. Initial and final intensity profiles are superimposed. The contrast, or roll-of of the image is defined as Contrast = max min max + min, (6.1) where the max and min values are the upper and lower bounds of the intensity. Contrast, or the depth of the intensity well, is defined as the ratio of the intensity amplitude over the level of bias. A useful way to use the contrast is to plot the contrast as a function of the number of lines per unit length. It is expected that the contrast would decrease as the number of line pairs per unit length increases. Contrast as a function of frequency is called the contrast transfer function (CTF). A common measure of the performance of an optics system is to plot the normalized
39 6.1 Contrast Transfer Function 29 CTF verse frequency. To normalize it you simply divide the equation by the largest value of contrast. Figure 6.4 shows an example of the intensity, contrast and CTF for a bar pattern of increasing frequency. Figure 6.4 Intensity, contrast and CTF for a bar pattern of increasing frequency [8]. In the plot in figure 6.4, it can be seen that the contrast decreases with higher frequencies, or smaller objects. The resolution is determined by identifying the frequency associated with the intersection of the CTF curve and the average noise contrast. This method of determining the resolution would be a viable and useful for the HOLODEC. The CTF method does not have the user bias that the method outlined in this paper has; however, there are some difficulties. It turns out the HOLODEC s laser profile is not uniform and the CTF method depends on a uniform background. As the contrast is a function of the minimum and maximum intensities, a varying background will alter the contrast. Figure 6.5 shows 2 CTF curves for the HOLODEC, at a reconstruction distance of 1 cm and 13 cm. Figure 6.6 and 6.7 shows the intensity pattern of the bar pattern at the two reconstruction distances.
40 30 Chapter 6 Future Research Figure 6.5 Two CTF curves shown for the HOLODEC at z = 1 cm and z = 13 cm. CTF curves are normalized and the units of the x axis are in µm. Figure 6.6 Bar pattern intensity at z = 1 cm. Observing the intensity pattern, one can readily see the effects of a uninform background. In order for the CTF method to be used on the HOLODEC, the laser profile must be made uniform, or a way to compensate for the uneven background is needed. 6.2 Signal to noise ratio Another method to determine the resolution is to calculate the signal to noise ratio. Similar to the contrast yransfer function, the signal to noise ratio reflects the contrast between the signal and noise of the imaging system. The noise can be calculated by averaging the intensity of the hologram at the current reconstruction distance. A ratio would then be taken between the intensity
41 6.3 Roll-off 31 Figure 6.7 Bar pattern intensity at z = 13 cm. of the bar pattern and the averaged noise. The greater the ratio the greater the contrast. 6.3 Roll-off The optics and electronics of an imaging system transfer the sharp edges of a square signal into a roll-off as seen in figure 6.3. For large objects, i.e., greater than 40 µm, the roll-of accounts for less than 5% of their width. For small objects, i.e., less than 10 µm, which is typical for cloud particles, this roll-off accounts for significant portion of their width. It becomes difficult to accurately determine the sizes of small objects. Researching a method to quantify the size of an object in the presence of a roll-of would improve cloud particle sizing.
42 32 Chapter 6 Future Research
43 Bibliography [1] Fugal, Jacob P., Timothy J. Schulz, and Raymond A. Shaw. Practical Methods for Automated Reconstruction and Characterization of Particles in Digital In-line Holograms, Meas. Sci. Technol., Vol. 20 (No. 7), [2] Hecht, E., Optics. (Addison-Wesley, 2002), p [3] Henneberger, J., Fugal, J. P., Stetzer, O., and Lohmann, U. HOLIMO II: a digital holographic instrument for ground-based in situ observations of microphysical properties of mixed-phase clouds, Meas. Sci. Technol. 6, , [4] Meng, H., Pan, G., Pu, Y., and Woodward, S. Holographic particle image velocimetry: from film to digital recording, Atmos. Meas. Tech. Vol. 15 (No. 4), [5] Spuler, S., Fugal, J., Design of an in-line, digital holographic imaging system for airborne measurement of clouds, Appl. Opt. 50, (2011). [6] Yang, W., Kostinski, A., Shaw, R., Phase signature for particle detection with digital in-line holography, Opt. Lett. 31, (2006). [7] 1951 USAF resolution test chart image. closeup 780px.jpg. 33
44 34 BIBLIOGRAPHY [8] Contrast transfer function curves. P9-12 f1.11.jpg.
45 Appendix A 1951 USAF Test Chart Table Figure A.1 Table shows width of bar patterns from the 1951 USAF resolution test chart. Bar widths are in µm. 35
46 36 Chapter A 1951 USAF Test Chart Table
47 Appendix B CTF Calculator Manual A 1951 USAF resolution test chart is typically used in calculating the contrast transfer function (CTF) of an electro-optical imaging system. The purpose of this manual is to instruct the user on how the ctf.m code works and how to use it in MATLAB. 1. Place the image file and ctf.m code in the current folder in MATLAB. 2. Call function from command window. The inputs for the function are the filename and the association name of the variables the function will save in the workspace. Generally the name of the variable association name is one which ties the variables to the image. If you are working with the image testchart.png and you want the variables association name to be testchart, you would type >> ctf(testchart.png, testchart) Once the function executes and the figures open it will look like figure B The red box defines what pixels are being plotted and averaged. Plot 1 in figure B.1 shows the intensity averaged over the columns within the red box. Plot 2 shows the intensity averaged over the rows within the red box. The red box 37
48 38 Chapter B CTF Calculator Manual Figure B.1 Plot 1 is the top right, and plot 2 is just below it. will be set at some default location and will need to be moved to the desired location using the sliders and/or edit text boxes shown in figure B.2. Figure B.2 There are two ways to make sure the red box is in the correct location: 1) look at the image, and 2), look at the intensity plots. If you want to analyze a vertical bar pattern you will want the left and right edges to be just beyond the outer edge of the left and right hand bars. You will want the top and bottom edges of the red box to be a pixel or two inside the bar pattern. For looking at horizontal bar patterns, its just the opposite. You want the top and bottom just outside the upper and lower bar edges and the left and right edges just inside the left and right edge of the bar pattern. 4. Once you have the red box in the desired place, you will select the group and
49 39 Figure B.3 element number of that bar pattern. Figure B.4 An outline of the actual bar widths will appear in the intensity plots as in figure B.5. Figure B.5 5. The data points within the dashed bars will be averaged to find the maximum intensity, I max, in the contrast equation. The data points in between the dashed bars will be averaged to find the I min. Sometimes there will be data points on the boundaries that may be undesirable to include in the calculations, since its value will go into the average. A way around this is to shrink the dashed lines using the line cut edit box. Each value entered represents how many data points on each end of each bar is cutoff. The dashed bar pattern may need to
50 40 Chapter B CTF Calculator Manual be shifted to the left or right. Use the vertical line shift slider and edit box, shown in figure B.2, if you are using the vertical intensity plot, and horizontal if otherwise. Figure B.6 Select the desired number of data points to cut out. Although it will not appear on the plot, the data points between the dashed lines will be cut on each side by the amount the user specified. 6. Once you know the correct data points will be used for the CTF calculation you need to add the vertical or horizontal vector to the CTF vector. The vertical and horizontal vectors are are the data points shown in plots 1 and 2 in figure B.8. The CTF vector is the vector containing the contrast and corresponding frequency data points. This button will cause the contrast to be calculated based off the averages of the low and high data points in the plot. It will then store this data with the corresponding frequency from the group and element number. Figure B.7 When you add the vector, the portion you added will show up in the third plot
51 41 of figure two. This plot builds as the user adds more data. Figure B.8 7. Once the contrast has been calculated for the given region in the red box, the user must move the red box as noted in step 3 to the next bar pattern. And the steps 4-6 are repeated. 8. When the user has stored all the contrast data they want, they must click Calculate CTF. This will plot the CTF verses line frequency as well as store the contrast, minimums, maxes, and frequencies in a matrix in the workspace. CTF plot shown in figure B.9.
52 42 Chapter B CTF Calculator Manual Figure B.9
53 Index Index 1951 USAF resolution test chart, 14 Airy disk, 5 Alignment, 11 Climate models, 1 Contrast transfer fuction, 27 Contrast transfer fuction, calculation, 28 Contrast transfer fuction, calculator manual, 37 Coordinate system, z axis, 14 Coordinate system, off-axis, 17 Data aquisition, 13 Effective aperture, 7 HOLODEC, 3 In-line digital holography, 2 Magnification of the HOLODEC, 6 Numerical aperture, 5, 6 Particle imaging velocimetry, 1 Pixel resolution, 8 Rayleigh criterion, 5 Reconstruction code, 12, 19 Resolution derivations, 5 Resolution, radial, 15 43
Be aware that there is no universal notation for the various quantities.
Fourier Optics v2.4 Ray tracing is limited in its ability to describe optics because it ignores the wave properties of light. Diffraction is needed to explain image spatial resolution and contrast and
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