High Redshift 21cm Line Simulation Data Visualization. with Canny Edge Detection

Transcription

1 University Of California at Santa Barbara High Redshift 21cm Line Simulation Data Visualization with Canny Edge Detection A Dissertation submitted in partial satisfaction of the requirements for the degree Bachelors of Science in Physics by Bang Dinh Nhan Advisor: Siang Peng Oh Associate Professor of Physics UC Santa Barbara Summer 2010

2 The dissertation of Bang Dinh Nhan is approved. Siang Peng Oh August 2010

3 Abstract The 21cm line emission from neutral hydrogen (HI) in the reionization epoch allows us to study structure formation in the early universe. The first generation low frequency radio detectors for 21cm line emission are currently being constructed. At the present time, computer simulations of the reionization epoch are employed to predict possible results from future detections. It demands huge computational power to generate a full radiative transfer simulation. Dr. Andrei Mesinger of Princeton University has developed two efficient semi-numerical cosmological simulations, Deus ex Machina (DexM) [Mesinger and Furlanetto, 7] and 21cmFAST [Mesinger et al., 2010], which are used to study the high redshift universe. The former can efficiently generate realizations of halo distribution and ionization maps at high redshift. The latter has bypassed the halo finder and only focuses on rapid estimation of the 21cm signal. I implement the Canny edge detection operator and other image processing techniques on Mesinger s DexM simulation box in order to develop a unique tool for future collected 21cm data reduction. This edge detection algorithm is expected to perform a lot better for the second generation detectors. For current purpose, I only simulate the low resolution 21cm signal due to the first generation detector s distortion by using anisotropic smoothing on the simulation box. I then apply Gaussian white noise to the distorted signal box as the background noise. The Canny edge detector will aid us to determine the distribution of ionized regions in the Intergalatic Medium (IGM). We have used the traditional 2-dimensional Canny, that is applied on each two dimensional (2D) scan of the box, to process the signal. I am developing a thre dimensional (3D) extended version of the Canny, which will operate on the entire simulation box directly. Here we present the results and parameter choices for each detector to give some general insights on how this imaging technique can process low frequency 21cm data.

4 Acknowledgements I need to thanks Chris Phelp, a former undergraduate at UCSB who worked with Prof. Peng Oh, for starting up the 3D Canny edge detection extension in 7. Chris has also written couple MATLab scripts that are helpful for me to start this project. I sincerely thank my advisor, Prof. Oh, for answering many of my questions and giving me insights to understand this project better. Many hours we have been discussing and tackling problems for the anisotropic noise suppression using the Canny detector. His encouragement is invaluable for me to overcome challenges and to appreciate this scientific research experience.

5 Contents 1 Introduction Motivation The Dark Age and the First Star Formation The Epoch of Reionization Probing The Intergalatic Hydrogen Edge Detector Concept of Edge Detection Available Edge Detection Algorithms First-order Derivative Operator Second-order Derivative Operator Why Canny? Three Dimensional Implementation of the Canny s Algorithm Gaussian Filter Find Magnitude and Direction of the Gradient Non-maximum Suppression Hysteresis Thresholding Pratt s Figure of Merit High Redshift 21cm Simulation Visualization with Canny Edge Detection D Canny Original Simulation Box Low Resolution Signal Box Noise Added Distorted Signal Box Noise Added Original Signal Box

6 4.2 Performance Analysis Canny s Parameters Selection Development of 3D Canny Detector D Gaussian filter D Gradient D Non-maximum Suppression Future Tasks Needed for Optimal Calibration D Canny Improvement Optimal Threshold Selection Improved Pratt s Figure of Merit Conclusion 44 6

7 List of Figures 1 A simplified timeline of the early universe [Miralda-Escude, 3] Spectra of 19 QSOs with redshift 5.74 < z < 6.42 from the SDSS, [Fan, 6]. Starting from redshift of 6, I can see the Lyman-α absorption. This indicates there is a large neutral fraction in high redshift IGM Edges are normal to lines. The line shown here is vertical as the edge direction is horizontal. The image brightness function, i.e transition of black and white, changes along a row in the edge s direction. The line is a collection of the first edge points of the function. [Umbaugh, 5] Noise in an image will affect performance of an edge detector. Tradeoffs are made between sensitivity and accuracy for edge detectors. The top right panel illustrates that the edge detector is too sensitive, many false edges are found due to the noise. The lower left panel shows that the detector is not sensitive enough, many valid edge points have lost. [Umbaugh, 5] The step edge is on the left panel and the ramp edge is on the right. They are used to model the ideal and real edge, respectively. [Umbaugh, 5] Edge model with the ramp edge. The first and second derivative of the edge. [Umbaugh, 5] The strong edges are one has gradient values above the high threshold. The weak edges are one between the low and high threshold. If the weak edges are connected to a strong edge, they are recorded as valid edges. Otherwise, they will be discarded Edge contours for slide No.10 of Andrei s signal box, using thresholds T = [0.05, 0.6] with varying σ Canny from 0.5 to 5.5 in order to determine an optimal value for sigma

8 9 Similarly, edge contours for slide No.10 for the distorted Andrei box with FWHM = of pixel size, using thresholds T = [0.05, 0.6] with varying σ Canny from 0.5 to 5.5. Note that the edge contours are closed when using larger sigma for the Canny for this low resolution image Here is the edge contours for same slide number as above but with background noise added to the smoothed box. It is apparent that Canny s Gaussian filter has failed to detect edge when the sigmas are small. Here again I use T = [0.05, 0.6] Each panel is the edge output map of the Andrei box with background noise but no distortion is used. Same thresholds are used as other cases mentioned above FOM vs SNR plots for three combinations of signals Histogram of Andrei s gradient map Histogram of Distorted Andrei s gradient map Histogram of Noise Signal s gradient map Histogram of Non-distorted Noise Signal s gradient map This is a comparison of the 2D and 3D Canny. The top two boxes are edge boxes for the Andrei signal. Instead of the full size box, I am only displaying part of the box for better visualization for the voids that the Cannys have detected. Note that the 3D extension of the Canny requires longer computational time. It is quicker to use the 2D canny for initial visualization and users can invoke the 3D Canny if more detail of the signal are needed

9 1 Introduction 1.1 Motivation In Big Bang cosmology, the early universe is filled with ionized gas. As the universe expands, the gas is cooled for about 400,000 years that the recombination rate (at which electron and proton form neutral hydrogen) becomes higher than the ionization rate of hydrogen. During this epoch, the universe is transparent for photons to stream out. Since there is no any other radiation source right after the epoch of recombination except the streaming photons from the early universe, the period between the surface of last scattering and the formation of the first star is known as the Cosmic Dark Age [Loeb, 8]. The universe is filled with neutral atomic medium that is created from the cooling. Not until the first star is formed that this medium is reheated and ionized again, which leads to further structure formation. Hence the Epoch of Reionization (EoR) starts until the IGM is fully ionized at redshift z = 6 [Fan, 6]. In order to understand the energy source of reionization as well as structure formation during EoR, we can either probe the regions of the universe where stars have already formed hence polluted by heavy elements, or study the pristine neutral medium surrounding the HII regions. The IGM can be probed by the 21cm line emission from neutral hydrogen caused by electrons transitioning between two hyperfine levels of the hydrogen 1s state. However, it is challenging to detect the 21cm signal due to its long wavelength and faintness. The first generation of 21cm experiments, some of them such as the LOw Frequency ARray (LOFAR) in Netherlands, the Murchison Widefield Array (MWA) in Australia, the Percision Array to Probe Epoch of Reionization (PAPER) in U.S/Australia, and Five-hundred Aperture Speherical Telescope (FAST) in China are under construction to search for the faint 21cm signal. Therefore, it is crucial to develop pipelines to reduce the upcoming observational data. The Canny edge detection algorithm is one of the possible image processing techniques to 9

10 be used for this purpose. This paper is organized as follows. The remainder of this section covers a brief introduction for background of reionization physics (sec. 1.3), and IGM probing techniques(sec. 1.4). The next two sections I will present a summary in concepts and techniques for edge detection used in image processing. Finally the last section will be comparison and analysis of two versions of Canny edge detectors used on the Mesinger s DexM simulation box. I will finish the section with a calibration recipe for parameters used with each version of the Canny detector (sec. 4.3). To finish up, I will suggest further calibration for the 3D Canny detector. In addition, alternative image processing technqiues are proposed to be used combining with the Canny s result in order to achieve more desirable data reduction for future 21cm signal. 1.2 The Dark Age and the First Star Formation After atoms starts forming at z 1 with temperature about 3000 K, photons from the hot and dense early universe are no longer scattered with the cooled plasma. Hence they are able to stream through space coming toward us. This isotropic Cosmic Microwave Background (CMB) fills the universe with blackbody radiation which is now red shifted to the infared range and has temperature about 2.73 K. It takes about 400 million years after the last scattering surface before the first star is formed. CMB photon is the only light source in the universe at the time. Hence this period is known as the Dark Age of the universe. 1.3 The Epoch of Reionization When the first star is formed, it is the primary UV inonizing photon source. Most stellar inonizing photons have energy slightly above the 13.6 ev ionization threshold of hydrogen. They are absorbed easily when they reach large HI regions. This allows the IGM during EoR to be a two-phase medium. The highly ionized regions are separated from the neutral 10

11 Figure 1: A simplified timeline of the early universe [Miralda-Escude, 3] regions by sharp boundary (see Figure 1). As more stars are formed, most of the HI are reionized. The reionization process will stop when the rate of ionization of HI is surpassed by the recombination rate. From there, further galaxy clustering occurs. One way to determine the end of the Cosmic Dark Age is Gunn-Peterson trough. The trough is caused by supression of photon emission from quasars at wavelengths less than that of the Lyman-α at the emitted redshift. According to Sloan Digital Sky Survey (SDSS), the Gunn-Petterson trough is detected by comparing the hydrogen absorption profile of 19 quasars at different redshift. A rapid change in neutral content around redshift of 6 indicates the end of the EoR (see Figure 2). Here I only present a brief description on the EoR. For more detail EoR physics, please reference to section 8 and 9 in [Furlanetto et al., 6]. 1.4 Probing The Intergalatic Hydrogen One of the techniques to study neutral hydrogen regions in IGM is trhough detection of the 21cm emission line of HI. Hydrogen is the most adbundant element in IGM. For neutral hydrogen (HI), when the spins of the proton and electron change from parallel to antiparallel, the hydrogen 1s state will have hyperfine splitting. A photon is emitted from this energy transition with frequency, 11

12 Figure 2: Spectra of 19 QSOs with redshift 5.74 < z < 6.42 from the SDSS, [Fan, 6]. Starting from redshift of 6, I can see the Lyman-α absorption. This indicates there is a large neutral fraction in high redshift IGM. ν 10 = 8 3 g I ( me m p ) α 2 (R M c) MHz (1) where g I is the nuclear g for proton, α = e 2 /( c) 1/ is the fine structure constant, and R M c is the hydrogen Rydberg frequency. The HI hyperfine splitting frquency corresponds to a wavelength of about 21 cm. This 21cm line radiation will help us to map out the ionized regions contrasting to the 21cm background (after I have subtracted the CMB, our galaxy s foreground, and other emitting sources). If the hydrogen spin temperature T S is higer than the CMB temperature, I will have the emission line. The T S is defined as following (see [Loeb, 8]), 12

13 { n 1 = 3 exp T } n 0 T S (2) where n 0 and n 1 are respectively the singlet and triplet hyperfein levels in the atomic ground state, and T = K with k B T = E 21 = ev is the 21 cm transition energy. Including the redshift, this line will be observed at frequencis between 9 to 20 MHz on Earth. Hence it is challenging to detect this long wavelength and faint signal. 2 Edge Detector 2.1 Concept of Edge Detection Edge detection operators are implemented with convolution masks that are based on differential operators. Differential operators measure the rate of change in a function, i.e. the image brightness function. An abrupt change in image brightness over a short spatial distance implies presence of an edge or edge point. Most edge detectors will return information about the edge s orientation as well as the location of each edge point. In order to find complex object boundaries, the edge detector will mark potential edge points and then connect them together to form lines representing an object s outlines. There is distinction between an edge and a line. An edge is where a sudden change occurs. A line is a collection of connected edge points along a certain direction. An edge occurs at a point and its direction is normal to the line s direction (see Figure 3). Due to the nature of differential operators, noise is the main problem causing false edge points for an image. There are tradeoffs between sensitivity and accuracy of an edge detector when noise is present in the signal. Suppose the detector s parameters are set to be less sensitive, it may miss valid edges. If its sentivity is increased, false edges due to noise are detected (see Figure 4). Hence parameter callibration is crucial for different edge detection algorithms. Two most common parameters are the size of the edge detection mask and the 13

14 Figure 3: Edges are normal to lines. The line shown here is vertical as the edge direction is horizontal. The image brightness function, i.e transition of black and white, changes along a row in the edge s direction. The line is a collection of the first edge points of the function. [Umbaugh, 5] value of the gray level threshold (or simply threshold). Edge detection algorithms are formulated based on the concept of determining relationship between a pixel and its neighbors. If a pixel s gray level coincides with those surrounding it, no edge point will be recorded. When the detectors find a pixel with widely varying gray level neighbors, it may be recorded as an edge point. A discontinuity in gray level values of a pixel results an edge. However, the edge line we have been discussing is the ideal edge line, which separated two regions of an image distinctively. Such an ideal edge is also called a step edge. In one dimension, it is represented by the step function with height h. This kind of edge exits only in artificially generated images. In practice, a real edge of an image changes more gradually due to blurring caused by imaging device, lens or lighting effect. A real edge can be modeled by a linear image value function which increases in image brightness from low to high level. This is called a ramp edge. One dimensional continuous edge models are illustrated in Figure 5. 14

15 Figure 4: Noise in an image will affect performance of an edge detector. Tradeoffs are made between sensitivity and accuracy for edge detectors. The top right panel illustrates that the edge detector is too sensitive, many false edges are found due to the noise. The lower left panel shows that the detector is not sensitive enough, many valid edge points have lost. [Umbaugh, 5] 2.2 Available Edge Detection Algorithms There are two main differential operators are used for edge detection: the first- and the second-order derivative operator First-order Derivative Operator The gradient of an image function F (x, y) is given by: G(x, y) = F (x, y) = F (x, y) ˆx + x F (x, y) ŷ = G x (x, y)ˆx + G y (x, y)ŷ (3) y 15

16 Figure 5: The step edge is on the left panel and the ramp edge is on the right. They are used to model the ideal and real edge, respectively. [Umbaugh, 5] The magnitude of the gradient is given by G(x, y) = G x (x, y) 2 + G y (x, y) 2 (4) The orientation of the spatial gradient with respect to the x-axis is θ = arctan [ ] Gy (x, y) G x (x, y) (5) Many existing gradient masks are used to approximate the gradient operator. Below are the common used approximations for the gradient. 1. Roberts Operator. This is a simple approximation to the first derivative and only records edge points but not the direction of the gradient. Ths edge detector performs 16

17 well for binary images. The detector consists of a pair of convolution kernels. Below are the 3 3 masks The kernels respond best to edge points going in the 45 o. Each kernel will measure the gradient component in each direction [the left one for row grdient (G y ) and the right one for column gradient (G x )] The magnitude of the gradient can be calculated by eq.(4) But it is more common to use the approximation as sum of absolute value of the gradient component instead of square root of the squared for computational efficiency, G = G x + G y (6) 2. Sobel Operator. The convolution masks for this operator are designed to detect horizontal and vertical edges. One of the kernel is just the transpose of the other if we approximate them in 3 3 masks, Similarily to the Roberts operator, each of the Sobel kernal will ouput a gradient component and the magnitude of the gradient canbe calculate using metric eq.(4). However, this operator does provide the information for the edge orientation which is calculated from eq.(5). 17

18 3. Prewitt Operator. This is a variant of the Sobel. The kernels are similar to the Soble s and are used to find edges in bot hthe horizontal and vertical directions. However, these two operators do have their own advantage. As pointed out by Scott Umbaugh [Umbaugh, 5], the Sobel is designed to weight on pixels that is closer to the center, when the Prewitt is easier to implement because of its unity coefficients Canny Operator Based on similar principle of the first-order derivative operator, John Canny in 1986 developed an advanced edge detection algorithm which acheive minimal error for the edge output. It is still considered a state-of-art edge detector in present time. So far, no other edge detector has surpassed the Canny s performace (see sec.2.3). In later sections of this paper will focus on calibrating this operator for the high redshift simulation signal Second-order Derivative Operator A maximum of the first derivative will occur at the zero-crossing of the second derivative. In 2D Cartesian coordinate, the second derivative in both x, and y direction is the Laplacian of F (x, y) G(x, y) = 2 F (x, y) = 2 F (x, y) x F (x, y) y 2 (7) Three masks below represent different 3 3 approximations of the Laplacian. 18

19 Laplacian Operator In contrast to the first-order gradient operators, the Laplacian is linear and rotational symmetric. To implement this operator, we normally will smooth the object image with a Gaussian filter and take the second derivative of the convolved image. Consider the Gaussian function h(x, y) = e (x2 +y 2 ) 2σ 2 (8) The Laplacian of this function is [ ] x y 2 σ 2 h(x, y) = e (x2 +y 2 ) 2σ 2 (9) σ 4 To simplify the computational process, we can take the Laplacian of the Gaussian then convolve it to the image function F (x, y) because Laplacian is a linear operator. The new operator is called Laplacian of Gaussian (LoG) 2 (h(x, y) F (x, y)) = 2 h(x, y) F (x, y) (10) In the Figure 6, we use the ramp edge model and compare its first and second derivative to illustrate the difference between those operators. 19

20 Figure 6: Edge model with the ramp edge. The first and second derivative of the edge. [Umbaugh, 5] 2.3 Why Canny? The Canny edge detection algorithm was developed by John Canny in It is an optimal edge detection algorithm based on the step edge model with Gaussian noise corruption. Canny s one dimensional continuous domain model consists of a step edge and additive Gaussian white noise with standard deviation σ n. The edge detection is carried out by convolving a one dimensional continuous domain noisy edge signal F (x) 20

21 centered at x = 0 with an anti-symmetric impulse response function (IRF), G(x), of range [ W, W ]. An edge is marked at the local maximum of the convolution integral H G = F (x) G(x) = W W F ( x)g(x)dx (11) The Canny detector is designed to satisfy three criteria [Canny, 1986]: Good detection. The detector should mark as many edges in the image as possible. This means the detector should obtain a low probability of missing the real edge points and low probability of falsely marking nonedge points. As Canny points out [Canny, 1986], signal-to-noise ratio (SNR) is required to maximized in order to acheive this criteria due to the monotonically decreasing probability functions of the output SNR. With the amplitude of noise or root-mean-squared response to noise only is H n = σ n [ W W [F (x)] 2 dx] 1/2 (12) Then the SNR is defined as SNR = H G H n = W F ( x)g(x)dx W [ ] 1/2 (13) W σ n [F W (x)]2 dx Good localization. An edge detected should be as close to the center of the true edge as possible. The localization [Canny, 1986] is defined as LOC = W F ( x)g (x)dx W [ ] 1/2 (14) W σ n [F W (x)] 2 dx where F (x) is the derivative of F (x). 21

22 Single response. A single response is needed for each true edge. In presence of noise, false edge should not occur. In order to eliminate multiple responses, the distance between peaks of the gradient in the noise response, x max, is set to some fraction k of the operator width W x max = kw (15) The expected number of noise maxima in a region of width 2W is N n = 2W x max = 2 k (16) Canny has combined these three criteria by maximizing the produt of SNR and LOC then apply the contraint of eq.(16). According to Pratt [Pratt, 7], the formulation is too complex to be solved analytically, so variational approach is used. Pratt also points out that from the study of Tagare and defugueyredo [Tagare and defigueiredo, 1990], gradient edge detection of step edges with first derivative of Gaussian impulse response function is optimal, which further shows that Canny s algorithm is best suited for signal with Gaussian noise. 3 Three Dimensional Implementation of the Canny s Algorithm The Canny edge detector is an implementation of the Canny gradient operator. The algorithm involves four stages [Canny, 1986, Umbaugh, 5]: (a) Noise reduction using Gaussian filter. (b) Highlighting regions with high spatial derivatives using image gradient. (c) Suppress non-maximum pixels. 22

23 (d) Hysteresis thresholding with two threshold values to find connected sets of edge points. 3.1 Gaussian Filter Canny has invoked the Gaussian filter in order to smooth out sharp edges that may have been caused by noise. The Gaussian spatial filter will convolve over the image of interest to smear out the noise pixels. Ideally, the Gaussian function is nonzero everywhere, thus resulting an infinite convolution kernel. Since the image is stored as a finite collection of pixels, the filter mask will also be a discrete approximation of the Gaussian. It is more practical to approximate the Gaussian out to three standard deviations from the mean value. Anything more than three sigmas is effectively zero. When dealing with 2D image, the 2D Gaussian mask is approximated up to three sigmas. For 3D object, such as our simulation signal box, we use the 3D Gaussian sphere for the convolution. However, due to the anisotropic distortion caused by the first generation detector s limited resolution, I modify the spherical Gaussian to an asymmetric filter. The modified filter has the same equatorial radii (σ x and σ y along the x and y axes) as the 2D mask but with a smaller polar radii (σ z along the line of sight or the z axis). Due to the Gaussian white noise nature from the 21cm radiation, Gaussian filter will handle the detector s distortion and background noise effectively. This is one of the main reasons we set out to utilize the Canny detector to process the EoR data. 2D Canny. deviation σ In 2D Cartesian coordinate, I use a Gaussian kernel with standard h 2D (x, y) = 1 x 2 +y 2 2πσ 2 e 2σ 2 (17) For the 2D version of the Gaussian, we approximate the Gaussian mask up to three 23

24 standard deviations. The mask size is a function of sigma: Std Mask Size σ = σ = σ = σ = Table 1: Samples of a given standard deviation corresponds to a mask size. 3.2 Find Magnitude and Direction of the Gradient The edge points will be marked by calculating the local gradient at each point using the gradient of the Gaussian. The edge direction will also be computed based on the gradient of the Gaussian. 2D Canny. For 2D operator, we can use the Roberts s, Sobel s, or Prewitt s mask mentioned earlier to approximate the derivative of the Gaussian. However, to be consistent, I calculate the first derivative of the 2D Gaussian and approximate up to the mask size of the Gaussian we used for smoothing. The local gradient value is g(x, y) = G(x, y) = G x (x, y) 2 + G y (x, y) 2 (18) The edge direction is θ(x, y) = arctan [ ] Gy (x, y) G x (x, y) (19) 24

25 3.3 Non-maximum Suppression The goal of edge detection is to find a thin edge that is as close to the ideal edge as possible. The edge points determined in the second stage give rise to ridges in the gradient magnitude image. Canny detector will compare those ridges with its neighbors in the edge direction. This is acheived by using a small mask (i.e. 3 3 for 2D) to compare the center value to its neighboring pixels in the gradient direction found in the previous step. If the center value is larger than its neighbor, it is registered as local maximum and becomes a valid edge point. Otherwise, it will be set to zero. Then the edges points will be linked to other edges along the gradient direction to form into lines by 8-connectivity. 8-connected pixels are neighbors to every pixel that touches one of their edges or corners. These pixels are connected vertically, horizontally, and diagonally. This process is called non-maximum suppression (NMS). This algorithm will iterate through the whole image and selects the best point along a particular direction to produce thinner lines in the output. Here is an example showing how the center pixel is chosen to be an edge [Umbaugh, 5]. A 3 3 subimage is considered with different valued pixels The arrows indicate the gradient direction. The center pixel is with horizontal edge direction. It is compared to 40 and 91. Since it is larger than these both, it is registered as a local maximum. If it was less than either the left or right pixel, it will be set to zero. 25

26 3.4 Hysteresis Thresholding As we have mentioned above that there are tradeoffs between sensitivity and accuracy of an edge detector. Thresholding will help the detector to eliminate insignificant edges, such as false edges caused by noise. The larger the intensity gradient is, the more likely it corresponds to an edge point. However, it is less flexible for just using one threshold to determine the change in intensity gradient. According to Canny [Canny, 1986], edge detectors with one threshold tend to have the streaking effect or the breaking up of an edge contour. This is caused by the output fluctuating about the threshold along the length of the contour. Canny introduced the hysteresis thresholding to eliminate streaking since now it is harder for the output to fluctuate above a high threshold and below a low threshold. Any edge pixel of the contour that is larger than a high threshold will be a strong edge point. An edge that is connecting to the strong edges and has value between a high and a low threshold will become a weak edge point. Anything below the lower threshold is set to zero. This process will be continue until it reaches a pixel below the low threshold. Figure 7: The strong edges are one has gradient values above the high threshold. The weak edges are one between the low and high threshold. If the weak edges are connected to a strong edge, they are recorded as valid edges. Otherwise, they will be discarded. 26

27 In general, it is hard to set a generic set of edge detecting thresholds for all image. One of the systematic ways to do it is to set the thresholds according to an image s gradient histogram. The gradient histogram is a plot of gradient magnitude versus the number of local maximal pixels determined in the NMS stages. For 2D image, we will have edges connected with eight connectivity as discussed in previous part. For method of choosing the thresholds, please refer to subsection Pratt s Figure of Merit With the fine tunning of the mask size and threshold levels, the Canny detector should performs at optimal level. However, three major errors associated with edge determination are generic for all edge detection algorithms [Pratt, 7] (a) Missing valid edge; (b) Failure to localize edge points; (c) Classification of noise fluctuations as edge points. Although I have used the optimal detector, it is still challenging to quantify the detector s performance due to the complexity of underlying features embedded in the signal, especially for processing the faint 21cm line data. In practice, we are uncertain of how the reionized regions distribute. The bubbles we detected may be false signals due to instrument s limitation and background noise. It is neccessary to investigate the effectiveness of the edge detector, especially the Canny detector, in a methodological way. With the assumption that the underlying features are step edges with Gaussian white noise, according to William Pratt [Pratt, 7] couple criteria are considered: (a) The probability of false detection; (b) The probability of missing edges; 27

28 (c) The error in estimating the edge gradient; (d) The mean square distance of the edge estimate from the true edge; (e) Tolerance to nonlocalization and multiple response. Pratt introduces a figure of merit (FOM) that takes those criteria under consideration and balances three major errors above to quantify the efficiency of an edge detector. The FOM is defined as FOM = 1 I N I A i= αd 2 i (20) I N = max{i I, I A } I I = the number of ideal edge points in the image I A = the number of actual detected edge points by the detector α = a scaling constant that adjust the penalty for offset edges, varies from detector to detector d i = the distance of a found edge point to an ideal edge point It is inconclusive on how to select a penalty factor for different types of edge detector. It is depended on the drawbacks of each edge detection algorithm. For the Canny edge detector, Pratt recommends to set the α penalty factor for unmatched edges to 1/9 (see pg. 516 of [Pratt, 7]). 4 High Redshift 21cm Simulation Visualization with Canny Edge Detection Edge detection is one of the crucial tools in analyzing medical imaging such as MRI or CT scans. In quality control, edge detectors are used to inspect X-ray images 28

29 sampled products. The primary objective of this project is to implement the Canny edge detection technique on astronomical visual simulation. Due to the low resolution of the first generation 21cm line detectors, the low SNR data will prohibit naked eyes to recognize the high redshift structures. With the help of image processing tools, namely the edge detector, we hope to achieve higher level of visualization on reionized regions in the EoR. So far the fundamental mechanisms of the Canny edge detector are discussed. To illustrate the effecacy of this algorithm on the numerical simulation, following tests and comparisons are carried out D Canny Original Simulation Box I have utilized Mesinger s DexM to generate a reionization simulation box with reionization factor x HI 0.82 at z = 8. The simulation box is an array of two hundred by matrices. Each layer of matrix has width of one pixel size. The box length is scaled to represent a Mpc volume. Each pixel stands for 0.5 Mpc. Practically, actual data is collected without knowledge of the actual structures. It is insufficient to make much assumption about the actual reionization structure. In following parts, I will discuss how I generate a more realistic signal simulation box as signal recieved by future telescopes. I distort Andrei s signal box with a Gaussian filter and apply background noise to it. The 2D Canny is an operator applied on each slice of the box. On each layer of the orignal simulation box, the edge finder will locate edge points and trace out edge lines of the signal as discussed in previous sections. The process will be carried on iteratively until the last layer of the simulation is processed. We runs the detector using the 2D Canny in MATLAB s function edge.m [Gonzalez et al., 4]. Each layer of the 29

30 Andrei s box will have edge detected by the edge.m. All these 2D edge scans are recombined in the right order to output a binary edge box. Figure 8 illustrates how the 2D Canny edge detector work on the original Andrei simulation. σ = 0.5 σ = 1 σ = 1.5 σ = 2 σ = 2.5 σ = 3 σ = 3.5 σ = 4 σ = 4.5 σ = 5 σ = 5.5 RMS= Figure 8: Edge contours for slide No.10 of Andrei s signal box, using thresholds T = [0.05, 0.6] with varying σ Canny from 0.5 to 5.5 in order to determine an optimal value for sigma. 30

31 Each panel of Figure 8 is a scan of the binary edge output box for Andrei s signal with a different Canny s Gaussian Smoothing filter sizes. The smaller the sigma for the filter, the finer details are detected by the edge detector. It is useful to use small sigmas for details in the original box. However, when we try to rescale our simulation data down to the first generation detectors resolution, the fine details will disappear. Only large ionized bubbles are seen from the detection. Larger sigmas are needed in order to pick out outline for these large regions Low Resolution Signal Box In order to predict what signal the first generation low frequency arrays may detect, I blur the Andrei signal with a Gaussian filter with FWHM comparable to the actual low frequency s beam width, FWHM = of pixel size. The filter will distort individual signal scan along the direction of line of sight. My objective is to apply 2D Canny to this smoothed 1 signal and adjust parameters such that the detector will outline boundaries of the large reionized regions from the IGM, normally has size between 30 to 40 Mpc [Furlanetto et al., 6]. Figure 9 is an example of how the 2D Canny performs on the smoothed signal box Noise Added Distorted Signal Box To fully simulate the actual signal for the future telescopes, I introduce background noise into the smoothed box. I generate a random white noise box and recalibrate it to the same resolution of the smootehd box by smoothing it a Gaussian filter of similar FWHM. The noise box is added to the smoothed box as result of the final detected signal on the ground. In principle, 2D Canny detector will detect outlines of 1 I will refer smoothed signal as the distorted signal box that collected by the low resolution telescopes. 31

32 σ = 0.5 σ = 1 σ = 1.5 σ = 2 σ = 2.5 σ = 3 σ = 3.5 σ = 4 σ = 4.5 σ = 5 σ = 5.5 RMS= SNR = Figure 9: Similarly, edge contours for slide No.10 for the distorted Andrei box with FWHM = of pixel size, using thresholds T = [0.05, 0.6] with varying σ Canny from 0.5 to 5.5. Note that the edge contours are closed when using larger sigma for the Canny for this low resolution image. substructures embedded in the white noise and gives hints on possible HII regions to astronomers. 32

33 σ = 0.5 σ = 1 σ = 1.5 σ = 2 σ = 2.5 σ = 3 σ = 3.5 σ = 4 σ = 4.5 σ = 5 σ = 5.5 RMS= SNR = Figure 10: Here is the edge contours for same slide number as above but with background noise added to the smoothed box. It is apparent that Canny s Gaussian filter has failed to detect edge when the sigmas are small. Here again I use T = [0.05, 0.6]

34 4.1.4 Noise Added Original Signal Box For comparison, the orignal simulation box is not downgraded by the smoothing Gaussian. Instead, I only superimpose background noise over Andrei s original box. This will gives us indication on Canny s performance with only white background noise. With the same thresholds, Figure 11 illustrates how 2D Canny is able to determine the 21cm signal from the noisy background. 4.2 Performance Analysis Based on panel plots above, it is clear that the 2D Canny detector is able to detect edge line of the original signal to a very good extense. Small details from the simulation is easily picked out if the sigma used for the Gaussian smoother is small enough. As soon as I introduce distortion to the signal, most of the small details are suppressed and leaving large reionized regions that are blurred together. In that case, the 2D Canny has indentified boundaries for those larger regions. If we only compare the smoothed signal (Figure 9) with the original (Figure 8), we are more certain of the distribution of reionized regions from signal collected with the low resolution radio arrays. However, when the background noise is included in the simulation the 2D Canny now pick out regions that were not supposed to be reionized. The background noise has introduced false signals which may be interpretted as real signal. In Figure 10, most of the edge lines for the supposed HII regions do not coincide with the edge lines of the actual simulation data. The final signal has deviated a lot from the original datas. The performance of the 2D Canny has dropped dramatically so that it is impossible to locate HII regions from the signal with high condifence. Most of the limitations are fro mthe detectors. We expect that the Canny technique will adapt to second generation detectors better and helps us visualize the EoR in greater detail. 34

35 σ = 0.5 σ = 1 σ = 1.5 σ = 2 σ = 2.5 σ = 3 σ = 3.5 σ = 4 σ = 4.5 σ = 5 σ = 5.5 RMS= SNR = Figure 11: Each panel is the edge output map of the Andrei box with background noise but no distortion is used. Same thresholds are used as other cases mentioned above. Equation 20 give us an idea how the 2D detector work with different scenarios I have considered. I calculated the FOM (see subsection 3.5) between: orignal Andrei vs 35

36 smoothed signal, orignal vs dirty 2 signal, and smoothed vs dirty signal FOM as a Function of SNR 10 0 FOM as a Function of SNR FOM of original vs dirty FOM of original vs smoothed SNR (a) Original vs smoothed box SNR (b) Original vs dirty box 10 0 FOM as a Function of SNR FOM of smoothed vs dirty SNR (c) Smoothed vs dirty box Figure 12: FOM vs SNR plots for three combinations of signals. 4.3 Canny s Parameters Selection Parameters need to be considered: 2 From now on I will refer dirty signal as the signal box that has been distorted and added to the background noise. 36

37 (a) Choices of Sigmas for Canny s Filter. Since I only approximate the Gaussian up to three standard deviations, size of the Gaussian filter is dependent on choices of sigma. For the 2D Canny, it is better to use σ in range of [0.5, 1.5] for the Andrei box if users want to resolve smaller details. Range between 3.0 and 5.0 is ideal for large regions for all the original, smoothed and dirty signal. Since our main concern is the distribution of the large bubbles, I will focus only on locating the larger HII regions hence choosing a larger sigma for the task. (b) Thresholds Choices. The thresholds are ranging between 0 and 1. It is useful to look at the histogram of the gradient output of the image. The histogram displays the number of pixels versus the value of the the local maxima. Throughout the simulation, I have used 0.05 as the lower threshold and 0.6 as the higher one. If I increase the lower threshold, the small details will not be able to detect. That may be another alternative to use 2D Canny to detect the large regions. The following figures display the input signal and its gradient output as well as the histogram of the gradient image. For using 2D Canny for high reshift structure signal anaysis, users may want to select threshold values such that the resulting edge maps will contain most of large structures. 37

38 Scan 10 Gradient of the Input Signal Histogram of the Gradient Number of Pixels 0 RMS Signal = RMS Grad = Figure 13: Histogram of Andrei s gradient map. Scan 10 Gradient of the Input Signal Histogram of the Gradient RMS Signal = RMS Grad = Number of Pixels Figure 14: Histogram of Distorted Andrei s gradient map

39 Scan Gradient of the Input Signal Number of Pixels 0 Figure 15: Histogram of Noise Signal s gradient map. 0 Histogram of the Gradient RMS Signal = RMS Grad = Scan Gradient of the Input Signal Number of Pixels Figure 16: Histogram of Non-distorted Noise Signal s gradient map. 0 0 Histogram of the Gradient RMS Signal = RMS Grad =

40 4.4 Development of 3D Canny Detector Since I am dealing with 3D signal sources like the simulation boxes generated by DexM or 21cmFAST, I am losing information between layers if I only process the data layer by layer using the 2D Canny. The positions of edge points on the previous scan are not correlated to ones on the following scan. Each 2D edge image is a standalone structure. It is crucial to develop a 3D operator that can be applied on the signal box directly instead of recombining individual edge image into a box. To resolve this discrepancy, I modify the 2D Canny algorithm to 3D adaptable. The main difference between the 3D extension and 2D Canny are alternations on the Gaussian filter, the directional gradient and the non-maximum suppression. These differences have increased the computational time for the edge finding process. However, the 3D Canny is supposed to output better edge image. Figure 17 is a comparison of two versions of Canny D Gaussian filter One trivial 3D extension for the smoothing filter is a spherical Gaussian. The spherical Gaussian kernel is h 3D (x, y, z) = 1 ( ) 3 e x 2 +y 2 +z 2 2σ 2 (21) 2πσ However as soon as the signal is distorted by anisotropic noise due to the telescope resolution constraint, the spherical Gaussian fails to suppress the asymmetric noise. To overcome this challenge, an ellipsoidal Gaussian filter is employed. By using a smaller standard deviation in the line of sight direction (σ z ) comparing to other two directions (σ x, σ y ), the Gaussian now will adapt to the anisotropic noise better. Instead of sharing 40

41 Figure 17: This is a comparison of the 2D and 3D Canny. The top two boxes are edge boxes for the Andrei signal. Instead of the full size box, I am only displaying part of the box for better visualization for the voids that the Cannys have detected. Note that the 3D extension of the Canny requires longer computational time. It is quicker to use the 2D canny for initial visualization and users can invoke the 3D Canny if more detail of the signal are needed. a same standard deviation, we now have three different sigmas h e (x, y, z) = ( 1 ( ) 3 e 1 2 2π σx σ y σ z x 2 σ 2 x ) + y2 σy 2 + z2 σz 2 (22) 41

42 In order to preserve correlating information of the edge points on different layers of the simulation box. It suffices to use a nonzero σ z Gaussian spatial filter D Gradient The gradient of the 3D Gaussian is the gradient of an image function F (x, y, z) is given by G(x, y, z) = F (x, y, z) = F (x, y, z) ˆx + x F (x, y, z) ŷ + y F (x, y, z) ẑ z = G x (x, y, z)ˆx + G y (x, y, z)ŷ + G z (x, y, z)ẑ (23) The local gradient value is G 3D (x, y, z) = G x (x, y, z) 2 + G y (x, y, z) 2 + G z (x, y, z) 2 (24) D Non-maximum Suppression Similar to the NMS of the 2D Canny, this stage of the 3D edge detection algorithm will compare ridges detected from the 3D gradient with its neighbors in three directions. A small mask (i.e ) is used to compare the center value to its 26 neighbors. If the center value is larger than the neighbors, it is considered as a local maximum. Edge points that are next to each other in the gradient direction are linked together to form a thin edge line. In stead of only connecting valid edge points on a 2D plane using 8-connectivity, all 26 neighbors of a pixel will be review and connected if there is an edge in that range. 42

43 5 Future Tasks Needed for Optimal Calibration 5.1 3D Canny Improvement In this paper, I have focused on the development and calibration for the 2D Canny algorithm. As I have pointed out, by just making edge output from signal scan one by one, information will lose. Hence a single 3D Canny operator is needed to overcome that. Due to the anisotrpic distortion of the signal, a 3D spherical Guassian is not practical. An ellipsoidal Gaussian is needed for the task. In addition, further testing is needed for the 3D Canny extension before we can invoke it to actual data processing. We will need to constraint the range for σ z to squish the 3D filter. 5.2 Optimal Threshold Selection Hysteresis thresholing is one of the main features of the Canny algorithm. However, I have not been able to come up a systematic way to estimate the threshold values for different input signals. One of the suggestive way to acheive this is to look into the histogram of the signal and combining with information of the gradient magnitude of the signal, users should be able to determine the thresholds accordingly. 5.3 Improved Pratt s Figure of Merit The Pratt s Figure of Merit is one of the useful way to quantify the performance of an edge detector. By plotting the FOM with different parameters on the simulation signal, we can understand how the Canny edge detector response to different signals. By constrainting the penalty factor in the FOM formula, the optimal FOM can be acquired. Hence, more tests are needed in order to determine this penalty scaling constant. 43

44 In addition to Pratt s FOM, [Pinho and Almeida, 1995] introduce a new figure of merit based on Pratt s for edge detection using neural network filter: ( 1 FOM improved = I I I I i= αd 2 i ) ( β I F II ) (25) I I = the number of ideal edge points in the image I F = the number of falsely detected edge points by the detector α = a scaling constant that adjust the penalty for misplaced edges. β = a scaling factor used to adjust the penalty for false edges. d i = the distance of a found edge point to an ideal edge point The first term in parentheses of eq.(25) is a modified version of Pratt s FOM, eq.(20), differing only in direction of matching. The former is designed such that each actual edge point looks for one, and only one, actual edge while the latter matches each actual edge element to one ideal edge point. [Pinho and Almeida, 1995] give evidence that the neural network edge detector can perform better than the linear optimal filters. In their simulation, they have used α = 1/9 and β = 1.0 such that the second term in the parentheses will be 0.5 if the number of false edges is equal to the number of ideal edges. So this new quantifying quantity may be implemented for future performance tests on the edge detection to be used for simulation processing. 6 Conclusion So far, I have been able to implement the tradiational 2D Canny edge detector from 2D image to a 3D signal box. Although the method is not the optimal way to retain all the edges as a single 3D edge detector would have, the final edge output is reasonable. 44