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Project 8 Radon transform
Please do the following Matlab project, and the three problems at the end (Probs. 5.435.45) which should not be done with Matlab.
Read in an image and compute the Radon transform and its inverse using the commands
radon and iradon respectively, display the results, and measure the MSE. When taking
the forward transform, be sure to use enough angle resolution to give reasonable results
(a minimum of one projection every 1 degree).
When calculating the inverse transform on a computer there will be an error when
compared to the original image. Use four different combinations of filters and
interpolations in the inverse Radon command and indicate which one gives the best
result for your input image determined by the lowest mean squared error (MSE) between
the original image and the reconstructed image.
The MSE is a scalar value that is the sum of ((original image – reconstructed
image).^2)/(number of pixels in an image).
Turn in:
• A description of what you did
• Your image
• Its Radon transform
• A table with the MSE values four different combinations of rconstruction
parameters.
• Your code
For example:
Parameter
Ram-Lak/linear
Hamming/linear
Cosine/spline
Cosine/nearest
MSE
Below are partial descriptions of the Radon commands in Matlab.
R = radon(I,THETA) returns the Radon transform of the intensity image I
for the angle THETA degrees. If THETA is a scalar, the result R is a
column vector containing the Radon transform for THETA degrees. If
THETA is a vector, then R is a matrix in which each column is the Radon
transform for one of the angles in THETA. If you omit THETA, it
defaults to 0:179.
I = iradon(R,THETA,INTERPOLATION,FILTER,FREQUENCY_SCALING,OUTPUT_SIZE)
specifies parameters to use in the inverse Radon transform. You can
specify any combination of the last four arguments. iradon uses default
values for any of these arguments that you omit.
INTERPOLATION specifies the type of interpolation to use in the
backprojection. The default is linear interpolation. Available methods
are:
‘nearest’ – nearest neighbor interpolation
‘linear’ – linear interpolation (default)
‘spline’ – spline interpolation
‘pchip’ – shape-preserving piecewise cubic interpolation
‘v5cubic’ – the cubic interpolation from MATLAB 5, which does not
extrapolate and uses ‘spline’ if X is not equally spaced.
FILTER specifies the filter to use for frequency domain filtering.
FILTER is a string or a character vector that specifies any of the
following standard filters:
‘Ram-Lak’ The cropped Ram-Lak or ramp filter (default). The
frequency response of this filter is |f|. Because this
filter is sensitive to noise in the projections, one of
the filters listed below may be preferable.
‘Shepp-Logan’ The Shepp-Logan filter multiplies the Ram-Lak filter by
a sinc function.
‘Cosine’ The cosine filter multiplies the Ram-Lak filter by a
cosine function.
‘Hamming’ The Hamming filter multiplies the Ram-Lak filter by a
Hamming window.
‘Hann’
The Hann filter multiplies the Ram-Lak filter by a
Hann window.
‘none’
No filtering is performed.
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