Lab 1-1-D Gradient Operator

Transcription

1 Due Date: 17:00, Friday, October 7, 2011 TA: Yunyue Elita Li Lab 1-1-D Gradient Operator William of Orange 1 ABSTRACT The gradient of a continuous function of more than one variable is a vector quantity. The gradient of a discrete function is normally approximated by finite differences. You will make simple modifications to two Fortran90 programs in order to compute the discrete gradient vector of a topographical image of the San Francisco Bay area. You will then modify a third program to compute the gradient magnitude, which is a scalar measure of relative total steepness and answer a couple of simple questions about the results. PRELIMINARIES If you have not already, log into one of SEP workstations using your class account, use pompei.stanford.edu as your working machine. Visit the GEE class webpage ( then find and download the source code for this lab, Lab1.tar. Save this file in your home directory, then type tar -xf Lab1.tar to create your Lab1 directory. In your Lab1 folder you will find a Makefile (Makefile) a SConstruct file (SConstruct) and a LaTeX file (paper.tex). Edit the paper.tex file where prompted. The Makefile states the targets to create executables, process data sets and construct figures. The SConstruct file contains a target to produce the paper. The paper can only be compiled if all figures needed are present. In the Src folder you will find three Fortran90 source codes (Igrad ns.f90, Igrad ew.f90, Gradmag.f90), and one subroutine (igrad1.f90). 1 orange@juice.com

2 GEE - Lab 1 2 1D Gradient Operator BACKGROUND Let us assume that we have a single-valued function of two variables, h(x, y). The gradient vector of h(x, y) is defined simply as ] h(x, y) = [ h x h y Physically, the gradient vector gives the direction of steepest ascent/descent on the surface h(x, y). Now consider the topographical data shown in Figure 1. The data is the elevation of the earth s surface across a discrete spatial grid. Call it h ij. However, equation (1) is defined for a continuous h(x, y), not for a digital image like Figure 1. Normally, we approximate continuous differential operators by convolution with an appropriate finite difference filter. In the case of the gradient, we use the following two-point filter approximation: (1) Figure 1: San Francisco Bay area topography. Can you locate Stanford on the map?

3 GEE - Lab 1 3 1D Gradient Operator h x h y [ 1 1 ] h ij = D x h ij (2) [ x] 1 1 h ij = D y h ij (3) y Figure 2 shows the impulse responses of the 1-D convolution filters D x and D y. Figure 2: Left: A spike. Center: Impulse response of D x. Right: Impulse response of D y. In practice, the gradient may be clumsy to use, since it is a vector quantity. The magnitude of the gradient vector provides a valid measure of steepness, and has the advantage of being a scalar function: [ ( h ) 2 h(x, y) = + x ( ) ] 2 1/2 h, (4) y Note that we have used the L 2 norm. The gradient magnitude is a classic edge detector, and in the 1970 s, the USGS produced a series of topographical steepness maps using it. In terms of the convolutional filters D x and D y, we can write an analogous expression for the gradient magnitude of a digital image h ij : GM(h ij ) = [ (D x h ij ) 2 + (D y h ij ) 2] 1/2 (5) Figure 3 shows the gradient magnitude, as defined in Equation (5), of a smoothed spike, or a cone, so to speak. We see that the gradient magnitude gives a nearly isotropic measure of the relative steepness of a surface. If directional derivatives were applied to the cone in Figure 3, a shadow would be produced. YOUR ASSIGNMENT In this section, you will modify and execute computer programs that apply the operators D x and D y to the San Francisco Bay area topographical image shown in Figure 1. You will then modify and execute a program that computes the gradient magnitude of this image.

4 GEE - Lab 1 4 1D Gradient Operator Figure 3: Left: An isotropic cone, or smoothed spike. Right: Gradient magnitude of the smoothed cone. At the command prompt, type make bay grad ns.h, followed by Grey pclip=100 < bay grad ns.h Tube. What you ve just done is executed a program, Igrad ns.x, which applies the operator D x to the topographical image, then used the Grey and Tube programs to display the result to the screen. Does something look funny? It should, because the program Igrad ns.x isn t working yet, and it s your job to fix it! With your favorite text editor, open the Fortran90 file Igrad ns.f90. You must fix the single line of the program that is marked. To check your results, repeat the process given above. Hint: Think about what the program should do. It is applying a 1-D derivative filter in the north/south direction, i.e., along the 1-axis. Since the filter is 1-D, you can apply it separately to individual data traces. In Fortran90, we can refer to individual data traces in a 2-D array with the following syntax: array(i,:) extracts all points on the 2-axis with 1-axis index i, whereas array(:,j) extracts all points on the 1-axis with 2-axis index j. Similarly, modify the program which computes the east/west derivative, Igrad ew.f90, then test your results by typing make bay grad ew.h, followed by Grey pclip=100 < bay grad ew.h Tube. Finally, modify the gradient magnitude program, Gradmag.f90, then test your results by typing make bay gradmag.h, followed by Grey pclip=100 < bay gradmag.h Tube. Note that within this program, you have access to one 2-D array that is the x-component of the gradient, and another that is the y-component. Following the definitions given above, this should be straightforward. Hint: There is a one-line solution and a longer one. In Fortran90, the following two code fragments perform identical tasks. First the long one: do i=1,n1 do j=1,n2 array1(i,j) = array1(i,j)**2 array2(i,j) = array1(i,j)*array2(i,j) end do end do...and now the short one:

5 GEE - Lab 1 5 1D Gradient Operator array1 = array1**2; array2 = array1*array2 When you are satisfied that your results are correct, type make read to see what your final document will look like. Figures 4-6 show the north/south derivative, the east/west derivative, and the gradient magnitude of the topographical image. To see a movie of all your results, type make bay.view. Questions Figure 4: North/South derivative. 1. Give a simple everyday analogy to what you see in Figures 4 and 5. Hint: Your answer should include something like flashlight or light source. 2. Although we have not analyzed the spectra of the topographical image explicitly, the filters we have used have interesting effects on the spectrum of the data they are convolved with. If we consider a 1-D data trace, how does convolution with the 1-D gradient ( [-1 1] ) affect the zero frequency, i.e. constant values? How

6 GEE - Lab 1 6 1D Gradient Operator Figure 5: East/West derivative.

7 GEE - Lab 1 7 1D Gradient Operator Figure 6: Gradient magnitude.

8 GEE - Lab 1 8 1D Gradient Operator about the high frequencies? Now consider the simple two-point averaging filter ( [1 1] ). How does convolution with this filter affect the low and high frequency components of the 1-D data trace? Roughly sketch the frequency responses of both filters on the blank axes provided below, Figure 7 1-D Gradient: [-1 1] 2-Point Average: [1 1] Filter Response 1 Filter Response 1 0 Frequency 0 Frequency Figure 7: Fill in the approximate frequency response of the 1-D gradient ([-1 1]) and two-point averaging ([1 1]) filters. The operator in igrad1.f90 implements the gradient [-1 1]. Copy Igrad ns.f90 and igrad1.f90 to other programs and implement the linear operator [1 1]. Apply this new operator to the Bay Area data and include your result in another figure analogous to Figure 4. Is the result what you have expected? Modifying the Makefile The Makefile contains two hints on which rules to duplicate and modify. Include the new result also in the default target rule, such that a simple make command builds all pdf figures. 3. The filter [-1 1] implements an asymmetric 1-D first order derivative. A centered second order 1-D derivative can be implemented using the filter [-1 0 1]. Copy Igrad ns.f90 and igrad1.f90 to other programs and implement the linear operator [-1 0 1]. Apply this new operator to the Bay Area data and include your result in another figure analogous to Figure 4. Is the result what you have expected? Compare the result in your new figure to the one in Figure 4. Extra Credit 4. Who was William of Orange.

9 GEE - Lab 1 9 1D Gradient Operator 5. Can you mark San Francisco, San Jose, Stanford and Berkeley on Figure With a green pen, correct all grammar and spelling mistakes on this Lab. 7. Explain the topic of this lab (gradients of a model space) in the context of inverse theory. Hint: Think about regularization and preconditioning. DONE When you are all finished modifying the latex file, compile it into a pdf using the scons command. Print your paper on the SEP s 4th floor printer (do not forget to draw in Figure 7). Hand your copy in to your TA. Clean up your directory by typing make clean, note that scons should still compile your paper. In SEP, we highly value the reproducibility of our research. Type make clean, make burn, make and scons at the end to make sure all your results in the paper are reproducible.