# GOMP0114 Inverse Problems in Imaging. Coursework 2 Hints

GOMP0114 Inverse Problems in Imaging. Coursework 2 Hints

Introduction

This document is designed as support for Coursework 2, containing programming hints for Matlab and Python.

——————— HINTS ——————— 1. Convolution and deconvolution

a.) Matlab: You can use the function imread to read an image from file. To convert the data type to double precision float, one may use the function double. One can display it with the function imagesc and a gray colormap.

Python: We require the library numpy and matplotlib:

import numpy as np

import matplotlib.pyplot as plt

import matplotlib.image as mpimg

Use mpimg.imread to read an image from file. The function np.float32 converts the data type to double precision. You can display the image with a gray colormap via

imgplot = plt.imshow(im,cmap=’gray’)

plt.colorbar()

plt.show()

b.) Matlab: The forward convolution can be done in several ways, but it’s best if one writes a function with some parameters that one can input, such as:

BlurredIm = imblur(im, OtherParameters).

A possible way to construct imblur is first to generate a Gaussian kernel using fspecial then apply the kernel on the image using imfilter. We recommend an odd kernel size for the Gaussian kernel that ensures the transposed operator is the same as the original operator.

Note that above is only the convolution step. The noise should be added separately to make a particular data instantiation:

g = imblur(f,OtherParameters)

g = g + theta*randn(size(g));

Python: You can use the function scipy.ndimage.filters.gaussian filter to perform Gaussian convolution with blur width σ and add noise as follow:

g = scipy.ndimage.filters.gaussian_filter(f,sigma)

w,h = g.shape

g = g + theta*numpy.random.randn(w,h)

c.) Matlab: Note that in our case Af=imblur(f) and since convolution is self-adjoint, i.e. A⊤ = A, we have A⊤g=imblur(g). Then the call looks like this (similarly for calling gmres)

fa = pcg(@(x)ATA(x, alpha), ATg).

When calling the functions pcg, gmres and even lsqr, one should pay attention to the dimension of the input variables. To be concrete, these solvers are solving a linear equation, say Hx = y, and expecting both x and y of vector form. As a result, one has to reshape the images from matrices to vectors before passing them to the solver. And one may also have to reshape the images from vectors to matrices before blurring them by functions.

Python: You can use scipy.sparse.linalg.gmres to call gmres in Python. The imple- mentation of gmres requires function scipy.sparse.linalg.LinearOperator. You can first use lambda function to create the function handle for ATA, then define the Linear Operator as follow:

A = scipy.sparse.linalg.LinearOperator((M,N),ATA)

where M and N are the size of the M-by-N matrix of the linear system. Notice that gmres in Python requires the right hand side of the linear system to be in the vector form. Hence, you need to vectorize ATg before putting into gmres. You are able to call the gmres as follow:

gmresOutput = scipy.sparse.linalg.gmres(A, ATg)

d.) Matlab: You can define a function handle

AaugHandle = @(f, transposeFlag)Aaug(f, A, , alpha, transposeFlag),

where Aaug implements the application of the matrix in the augmented equation above. You need to define a transpose and non-transpose. You can do this as follows:

function z=Aaug(f,A,alpha,transposeFlag)

switch transposeFlag

case ’notransp’

% implementation of the augmented matrix multiplication

case ’transp’

% implementation of the transposed augmented matrix multiplication

otherwise

error(’input transposeFlag has to be ’’transp’’ or ’’notransp’’’)

end end

Python: You can use the function scipy.sparse.linalg.lsqr to implement lsqr. You need to define a transpose and non-transpose as in Matlab. You can do this as follows:

def M_f(f):

% implementation of the augmented matrix multiplication

return M_f

def MT_b(b):

% implementation of the transposed augmented matrix multiplication

return MT_b

The implementation of lsqr also requires function scipy.sparse.linalg.LinearOperator. You can define the linear operator as follow:

A = scipy.sparse.linalg.LinearOperator((M,N),matvec = M_f, rmatvec = MT_b)

where M and N are the size of the M-by-N matrix of the linear system. We need to concate- nate the vectorized image g with a zero-vector which has the same size as g as the input b of lsqr, you can do this as follow:

import numpy as np

size = g.size

b = np.vstack([np.reshape(g,(size,1)),np.zeros((size,1))])

lsqrOutput = scripy.sparse.linalg.lsqr(M,b)

2. Choose a regularisation parameter α

a.) Matlab: You can use the function fzero to find the zero of the discrepancy function.

Python: You can use function scipy.optimize.root or scipy.optimize.brentq to find the zero of the discrepancy function.

b.) Matlab: You can use the loglog function to display the L-Curve.

Python: The function matplotlib.pyplot.loglog can be used for plotting the L-Curve.

3. Using a regularisation term based on the spatial derivative

a.) Matlab: See matlab functions spdiags for constructing the gradient operator using an explicit sparse matrix. Alternatively, you can use the function diff which is the most appropriate choice for the latter, but also multiplying by frequency in the Fourier domain is possible.

Python: You can construct the gradient operator via an explicit sparse matrix by using the scipy.sparse.spdiags function. Also you can use numpy.diff.

b.) Matlab and Python: You can check the correctness of constructing transpose operator D⊤ by simply examining the identity < Du, v >=< u, D⊤v > for random u and v.

c.) Matlab and Python: You can use the methods suggested in Question 2 to choose an optimal value for α.

4. Construct an anisotropic derivative filter

• Matlab: Again you can use the function spdiags to construct the filter matrix γ.

Python: You can use scipy.sparse.spdiags to construct γ. 5. Iterative deblurring

• Matlab and Python: In each iteration you can compute your minimiser with a solver from Exercise 1 (pcg, gmres, lsqr).

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