Why does first calling the function radon and then iradon on the result produce extra rows and columns?

[Stephen23]

P = phantom(64);
size(P)

ans = 1×2

64 64

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theta = 0:179;
R = radon(P,theta);
size(R)

ans = 1×2

95 180

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Q = iradon(R,theta);
size(Q)

ans = 1×2

66 66

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Answer 1

[Stephen23]

"Why does first calling the function radon and then iradon on the result produce extra rows and columns?"

The functions radon and iradon are not exact inverses in the sense of “64×64 in -> 64×64 out”. The size change comes from geometry: how long the projections are in radon, and how iradon chooses the reconstruction size N from that projection length. Lets consider your example:

P = phantom(64);
size(P)

ans = 1×2

64 64

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theta = 0:179;
R = radon(P,theta);
size(R)

ans = 1×2

95 180

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Q0 = iradon(R,theta)

Q0 = 66×66

0.0028 0.0019 -0.0069 0.0079 -0.0021 -0.0029 0.0030 0.0033 -0.0045 -0.0010 0.0007 -0.0043 0.0025 0.0022 -0.0052 0.0033 0.0020 -0.0030 0.0010 -0.0015 -0.0006 0.0010 -0.0072 0.0077 -0.0015 -0.0032 0.0078 -0.0042 0.0030 0.0038 -0.0001 0.0061 0.0006 -0.0073 0.0069 -0.0030 -0.0080 0.0122 -0.0010 -0.0052 0.0058 -0.0051 -0.0052 0.0085 -0.0056 -0.0061 0.0072 -0.0059 -0.0023 0.0068 0.0006 -0.0057 -0.0020 0.0019 -0.0013 0.0006 0.0028 -0.0027 0.0009 0.0015 -0.0021 0.0004 0.0013 -0.0052 -0.0002 0.0035 -0.0019 -0.0050 0.0091 -0.0066 0.0028 0.0067 -0.0081 0.0071 0.0014 -0.0096 0.0087 0.0048 -0.0066 0.0002 0.0081 -0.0097 0.0063 0.0044 -0.0068 0.0127 -0.0084 0.0070 0.0097 -0.0026 -0.0027 -0.0039 0.0016 0.0027 -0.0073 0.0042 0.0040 -0.0109 0.0027 -0.0025 -0.0044 0.0059 -0.0065 -0.0004 0.0043 -0.0056 -0.0001 0.0006 0.0035 -0.0050 0.0046 -0.0040 0.0052 0.0039 -0.0133 0.0091 -0.0202 -0.0070 0.0277 0.1329 0.0007 -0.0002 -0.0019 0.0025 0.0029 -0.0035 0.0019 0.0051 -0.0055 0.0037 -0.0069 -0.0026 -0.0011 0.0020 -0.0049 -0.0015 0.0048 -0.0095 -0.0005 -0.0001 -0.0084 -0.0046 -0.0057 -0.0067 0.0040 0.0535 0.1792 0.2167 0.3318 0.6804 0.0011 0.0044 -0.0041 -0.0052 0.0078 -0.0037 -0.0023 0.0075 0.0016 -0.0006 0.0040 -0.0012 -0.0045 0.0089 -0.0076 -0.0054 0.0130 -0.0028 -0.0036 0.0121 0.0006 0.0147 0.0166 0.0308 0.1485 0.2977 0.6682 0.8036 0.7994 0.8656 -0.0068 0.0007 0.0053 -0.0057 -0.0034 0.0133 -0.0079 -0.0006 0.0046 -0.0035 0.0015 0.0112 -0.0026 0.0099 0.0092 -0.0080 0.0073 0.0030 -0.0012 0.0093 -0.0093 -0.0164 0.0431 0.2867 0.7029 0.8029 0.8210 0.8437 0.7391 0.4639 0.0043 -0.0062 0.0052 0.0079 -0.0080 0.0019 0.0076 -0.0070 0.0039 -0.0006 -0.0175 0.0047 -0.0062 -0.0051 -0.0066 0.0007 0.0032 -0.0126 -0.0074 -0.0145 0.0504 0.1617 0.3069 0.6705 0.8433 0.7580 0.4691 0.3506 0.3280 0.2121 0.0053 -0.0044 -0.0072 0.0057 -0.0044 -0.0084 0.0032 -0.0010 -0.0035 0.0061 -0.0055 0.0006 -0.0074 0.0106 -0.0061 -0.0024 0.0088 -0.0129 0.0280 0.0364 0.2742 0.7028 0.8291 0.7358 0.4429 0.3144 0.2439 0.1840 0.2208 0.1943 -0.0001 0.0042 -0.0062 -0.0052 0.0075 -0.0026 -0.0038 0.0002 -0.0010 0.0052 0.0053 0.0116 -0.0016 0.0072 -0.0005 -0.0104 0.0287 -0.0149 0.0205 0.2983 0.7099 0.9159 0.7638 0.4485 0.2350 0.1960 0.1931 0.2018 0.2009 0.1917 -0.0056 0.0041 0.0075 -0.0025 0.0021 0.0091 0.0040 -0.0004 0.0017 0.0003 -0.0163 -0.0045 -0.0137 -0.0072 0.0103 0.0022 -0.0193 0.0556 0.3085 0.6863 0.9535 0.7769 0.4189 0.2152 0.1888 0.2129 0.1946 0.2120 0.1801 0.2214 0.0028 -0.0015 0.0002 0.0085 -0.0009 -0.0056 0.0031 -0.0040 -0.0057 0.0098 -0.0066 0.0075 0.0153 0.0151 -0.0307 0.0065 0.0308 0.1190 0.6877 0.9176 0.7892 0.4243 0.2402 0.2138 0.2049 0.1985 0.1948 0.2025 0.2130 0.1924 0.0031 0.0033 -0.0079 -0.0056 0.0020 -0.0097 -0.0104 0.0022 -0.0065 0.0016 -0.0003 0.0029 -0.0159 -0.0039 0.0259 -0.0109 0.0377 0.3352 0.8328 0.7623 0.4307 0.2176 0.1995 0.1775 0.2134 0.1835 0.2155 0.1874 0.1964 0.1981 -0.0081 -0.0018 0.0071 -0.0066 0.0045 0.0109 0.0024 0.0120 0.0062 0.0014 -0.0095 -0.0080 0.0131 -0.0006 -0.0300 0.0516 0.2793 0.6823 0.7274 0.4023 0.2595 0.1819 0.2247 0.1938 0.2037 0.2015 0.1934 0.1996 0.2048 0.2078 0.0036 -0.0049 0.0136 0.0090 -0.0098 0.0047 -0.0053 -0.0046 -0.0025 0.0046 0.0043 -0.0023 0.0132 -0.0109 0.0229 0.1612 0.6769 0.7365 0.4501 0.2282 0.2002 0.2022 0.1996 0.1944 0.1998 0.2083 0.2062 0.2057 0.1959 0.2151

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size(Q0)

ans = 1×2

66 66

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So the key numbers are:

• original image: 64×64
• projection length: len = size(R,1) = 95
• reconstructed image: N×N with N = 66

Inside iradon

The value of N is defined when the inputs are parsed:

[p,theta,filter,d,interp,N] = images.internal.iradon.parseInputs(args{:});

N is the output image size. When you call

Q = iradon(R,theta)

without specifying N, parseInputs picks a default value of N based on size(p,1) (the number of rows in R). So iradon does notknow that your original image was 64×64; it only sees a sinogram of length 95 and chooses an N that fits that. Later inside iradon:

len = size(p,1);

ctrIdx = ceil(len/2); % index of the center of the projections

% Zero pad the projections to size 1+2*ceil(N/sqrt(2)) if this

% quantity is greater than the length of the projections

imgDiag = 2*ceil(N/sqrt(2))+1; % largest distance through image.

if size(p,1) < imgDiag

rz = imgDiag - size(p,1); % how many rows of zeros

p = [zeros(ceil(rz/2),size(p,2)); p; zeros(floor(rz/2),size(p,2))];

ctrIdx = ctrIdx+ceil(rz/2);

end

Interpretation:

• len = size(p,1) is the length of each projection (95 in your case).
• imgDiag = 2*ceil(N/sqrt(2)) + 1 is the number of samples needed along a projection to cover the longest line through an N×N image (the image “diagonal” in pixel units).
• If the projections are too short for that N, they get zero-padded to length imgDiag.

In your case, no extra padding is added here, because len already matches imgDiag for N = 66:

len = 95;

imgDiag = 2*ceil(66/sqrt(2)) + 1 = 2*47 + 1 = 95;

size(p,1) < imgDiag % false, so no padding

So the “extra 2 pixels” in each direction do not come from this padding step – they come from how N was chosen earlier.

Why N=66 and not 64?

The geometric requirement is:

2*ceil(N/sqrt(2)) + 1 <= len

so that the projections are long enough to cover the diagonal of an N×N image.

Given len = 95, the largest N that satisfies this inequality is N = 66:

• For N = 66:imgDiag = 2*ceil(66/sqrt(2)) + 1 = 2*47 + 1 = 95 <= 95 % just right
• For N = 67:imgDiag = 2*ceil(67/sqrt(2)) + 1 = 2*48 + 1 = 97 > 95 % too large

So, from the sinogram length 95 alone, the “maximal” consistent reconstruction size is 66×66. Since you didn’t tell iradonotherwise, parseInputs chooses that size. Notice that N = 64 would also satisfy the inequality:

2*ceil(64/sqrt(2)) + 1 = 2*46 + 1 = 93 <= 95

but iradon doesn’t know you started from 64×64; it just sees that it can go up to 66×66 with the available projection length, so it does. At this point you have:

• radon: created projections long enough to cover your 64×64 image (and a bit more), resulting in 95 samples per projection.
• iradon: looked at those 95 samples and chose the largest N consistent with them, which is N = 66.

That’s why a 64×64 image becomes 66×66 after radon → iradon with default parameters.

What about radon giving 95 rows?

For completeness: radon chooses the number of samples per projection to cover the full image at all angles (roughly a diagonal-based formula plus some discrete/rounding margin). For a 64×64 image, that works out to 95 samples, which is exactly what iradon then interprets as sufficient for a 66×66 reconstruction.

So the chain is:

64×64 image

↓ (radon chooses length 95 to cover geometry)

95×180 sinogram

↓ (iradon chooses largest N such that diagonal fits in length 95)

66×66 reconstruction

How to get back to 64×64

If you want iradon to give you a 64×64 image, you must tell it explicitly:

Q1 = iradon(R, theta, 64)

Q1 = 64×64

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size(Q1)

ans = 1×2

64 64

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% or, in syntax with named arguments:
Q2 = iradon(R, theta, 'linear', 'Ram-Lak', 1.0, 64)

Q2 = 64×64

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size(Q2)

ans = 1×2

64 64

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isequal(Q1,Q2)

ans = logical
   1

Now N is fixed at 64, and the output will be 64×64.

Summary

The extra rows and columns are not a bug; they are a result of iradon’s default choice of reconstruction size N, which is derived from the projection length size(R,1), not from the original image size. For your len = 95, the largest consistent square is 66×66, so that’s what you get unless you override N.

[dpb]

If you want iradon to give you a 64×64 image, you must tell it explicitly:"

Q1 = iradon(R, theta, 64)

That syntax surely seems at odds with what the doc says...

I = iradon(R,theta,interp,filter,frequencyScaling,outputSize)

specifies additional parameters to use in the inverse Radon transform. You can specify any combination of the last four arguments. To omit an argument, specify the value as an empty array ([]). iradon uses default values for arguments that you omit.

[Jöran]

Thank you for your excellent answers - regrettably I overlooked using the optional input "outputSize."

Answer 2

[dpb]

The doc for iradon has the optional input 'OutputSize' which states that if not specified the reconstruction size is calculated from

outputSize = 2*floor(size(R,1)/(2*sqrt(2)))

P = phantom(64);
theta = 0:179;
R = radon(P,theta);
size(R)

ans = 1×2

95 180

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outputSize = 2*floor(size(R,1)/(2*sqrt(2)))

outputSize = 66

Syntax says--

I = iradon(R,theta,interp,filter,frequencyScaling,outputSize)

specifies additional parameters to use in the inverse Radon transform. You can specify any combination of the last four arguments. To omit an argument, specify the value as an empty array ([]). iradon uses default values for arguments that you omit.

So, let's try

  I = iradon(R,theta,[],[],[],64);

Error using images.internal.iradon.parseInputs (line 45)
Invalid input parameters.

Error in iradon (line 7)
[p,theta,filter,d,interp,N] = images.internal.iradon.parseInputs(args{:});
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

That seems funky but I don't know enough about the radon transform nor have the toolbox to investigate locally what it thinks is wrong and the error message certainly isn't of much help.

[dpb]

Just taking a flyer,

P = phantom(64);
theta = 0:179;
R = radon(P,theta);
outputSize = 2*floor(size(R,1)/(2*sqrt(2)))

outputSize = 66

I = iradon(R,theta,[],[],64/66,64);

Error using images.internal.iradon.parseInputs (line 45)
Invalid input parameters.

Error in iradon (line 7)
[p,theta,filter,d,interp,N] = images.internal.iradon.parseInputs(args{:});
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

size(I)

Well, that didn't work, either. No idea which is thought to be invalid nor why...

[dpb]

type iradon

function [img,H] = iradon(varargin)

narginchk(2,6);

args = matlab.images.internal.stringToChar(varargin);

[p,theta,filter,d,interp,N] = images.internal.iradon.parseInputs(args{:});

% Determine if single precision computation has to be enabled. Cast the
% inputs p and theta accordingly.
[p, theta, useSingleForComp, isMixedInputs] = images.internal.iradon.postProcessInputs(p, theta);

% Design the filter used to filter the projections
[p,H] = images.internal.iradon.filterProjections(p, filter, d, useSingleForComp, isMixedInputs);

% Define the x & y axes for the reconstructed image so that the origin
% (center) is in the spot which RADON would choose.
center = floor((N + 1)/2);
xleft = -center + 1;
x = (1:N) - 1 + xleft;

ytop = center - 1;
y = (N:-1:1).' - N + ytop;

len = size(p,1);
ctrIdx = ceil(len/2);     % index of the center of the projections

% Zero pad the projections to size 1+2*ceil(N/sqrt(2)) if this
% quantity is greater than the length of the projections
imgDiag = 2*ceil(N/sqrt(2))+1;  % largest distance through image.
if size(p,1) < imgDiag
    rz = imgDiag - size(p,1);  % how many rows of zeros
    p = [zeros(ceil(rz/2),size(p,2)); p; zeros(floor(rz/2),size(p,2))];
    ctrIdx = ctrIdx+ceil(rz/2);
end

interpStr = images.internal.iradon.convertEnumsToInterpModes(interp);

% Backprojection - vectorized in (x,y), looping over theta
if (interp == images.internal.iradon.InterpModes.Linear) || ...
   (interp == images.internal.iradon.InterpModes.Nearest) 
    % Converting to a linear buffer to enable linear indexing in the halide
    % code
    p = reshape(p, numel(p), []);

    % Theta expected to be NumThetax1
    theta = reshape(theta, numel(theta), []);

    % Coordinates are expected to have dimensions NumCoordsx1
    x = reshape(x, numel(x), []);
    y = reshape(y, numel(y), []);

    img = images.internal.builtins.iradon_halide(p, theta, x, y, N, interpStr);
else
    % The builtin-code performs the math in double precision. Use this code
    % path to ensure all computatations are done in single precision.
    %'spline','pchip','cubic','v5cubic'
    interp_method = sprintf('*%s',interpStr); % Add asterisk to assert
    % even-spacing of taxis

    % Generate trigonometric tables
    costheta = cos(theta);
    sintheta = sin(theta);

    % Allocate memory for the image
    img = zeros(N,'like',p);

    for i=1:length(theta)
        proj = p(:,i);
        taxis = (1:size(p,1)) - ctrIdx;
        t = x.*costheta(i) + y.*sintheta(i);
        projContrib = interp1(taxis,proj,t(:),interp_method);
        img = img + reshape(projContrib,N,N);
    end

end

img = img*pi/(2*length(theta));

end

%   Copyright 1993-2022 The MathWorks, Inc.

type images.internal.iradon.parseInputs

Error using type
File 'images.internal.iradon.parseInputs' not found.