Assignment Matrix Decomposition

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[WS14/15] Mathematics for Robotics and Control:Assignment 004 - Matrix decomposition

First we will setup this notebook so that figures and plots can be shown in the notebook page.

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Hint: Before you start solving the assignment, you might want to check the following numpy functions:

numpy.linalg.svd

Covariance matrix properties

Write a mathematically sound proof for each of the following properties of the covariance matrix:

1) That for a given matrix , where is the number of observations and is the number of

variables, the expression

yields the covariance matrix.

2) Under which conditions taking does the eigendecomposition of a matrix yield an orthonormal basis,

and for which vector space.

X

( ) ( )

1

N 1

try: shell = get_ipython() shell.enable_pylab("inline")exceptNameError: passimportnumpy.linalg asLA

importnumpyimportmatplotlibfrommatplotlib importpylab, mlab, pyplotnp = numpyplt = pyplotfromIPython.display importdisplayfromIPython.core.pylabtools importfigsize, getfigsimportIPythonfrompylab import*fromnumpy import*


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3) When applying Principal Components Analysis, each eigenvector points into the direction of one the

dataset's principal components and how much do the component contribute to the overall variance is

related to each eigenvector's eigenvalue.

Fitting lines to point clouds

Read this article about least squares fitting (http://mathworld.wolfram.com/LeastSquaresFitting.html).

Read this lecture about linear least squares and matrix decompositions

(http://classes.soe.ucsc.edu/cmps290c/Spring04/paps/lls.pdf). For all of the following fitting tasks,

use a singular value decomposition to fit a pair of lines to each of the given point clouds.

Detecting a hallway

The robot is driving in a hallway when it starts its Kinect obtaining the following ploint cloud.
http://classes.soe.ucsc.edu/cmps290c/Spring04/paps/lls.pdfhttp://mathworld.wolfram.com/LeastSquaresFitting.html


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importIPythonIPython.core.display.Image("images/hallway.png", embed=True)


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Your task is to fit a pair lines based on the points generated by the robot's Kinect.

%matplotlib inlineimportmatplotlib.pyplot asplt

defmatrix_calculation(filename):

line = np.load(filename)

x= line[:,0] y = line[:,1] A = np.column_stack((x, np.ones(len(x)))) m,c = svd_solver(A,y) returnx,m,c # x, y, m, c = matrix_calculation('data/Q1.npy') # print y

defplotting(line, halt): # to plotting the data from matrix calculation

plt.title('Scatter plot') #plotting the values of x, m, c plt.plot(line[0], line[1]*line[0] +line[2] , 'r', label = ' Fitted line d #plt.plot(x2, m1*x2+c1, 'r', label = ' Fitted line data')

plt.hold(halt)

ifnothalt: plt.legend() plt.show()

defsvd_solver(A, b): U,s,V= (LA.svd(A)) #print U.shape, s.shape, V.shape

#U = np.matrix(U),V inv = np.dot((V.T).dot(np.diag(s**(-1))), U[:,:2].T) #print inv x = np.dot(inv,b) #print x

returnx


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Detecting a corner

At the end of the hallway the robot detects a dramatic change in the point cloud.

defcompute_parallel_lines(first_line, second_line): plotting(first_line, True) plotting(second_line, False)

first_line=matrix_calculation('data/P1.npy')

second_line = matrix_calculation('data/Q1.npy')compute_parallel_lines(first_line,second_line)


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Your task is to fit a pair lines based on the points generated by the robot's Kinect.

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importIPythonIPython.core.display.Image("images/corner.png", embed=True)

defcompute_perpendicular_lines(first_line, second_line): plotting(first_line, True) plotting(second_line, False)


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Detecting a table

When the robot turns at a table near by the point cloud it receives is the following.

first_line=matrix_calculation('data/Q1.npy')

second_line = matrix_calculation('data/Q2.npy')compute_perpendicular_lines(first_line,second_line)


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In [101]:

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importIPythonIPython.core.display.Image("images/rectangle.png", embed=True)

defcompute_rectangle_lines(first_line, second_line, third_line, fourth_line): plotting(first_line, True) plotting(second_line, True) plotting(third_line, True) plotting(fourth_line, False)


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In [103]:

first_line=matrix_calculation('data/P2.npy')second_line = matrix_calculation('data/S2.npy')third_line = matrix_calculation('data/R2.npy')fourth_line=matrix_calculation('data/Q2.npy')

compute_rectangle_lines(first_line, second_line, third_line, fourth_line)

Assignment Matrix Decomposition

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