Radon Transform: Unveiling Hidden Patterns in Visual Data

By Fouad Sabry

What is Radon Transform

In mathematics, the Radon transform is the integral transform which takes a function f defined on the plane to a function Rf defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line. The transform was introduced in 1917 by Johann Radon, who also provided a formula for the inverse transform. Radon further included formulas for the transform in three dimensions, in which the integral is taken over planes. It was later generalized to higher-dimensional Euclidean spaces and more broadly in the context of integral geometry. The complex analogue of the Radon transform is known as the Penrose transform. The Radon transform is widely applicable to tomography, the creation of an image from the projection data associated with cross-sectional scans of an object.

How you will benefit

(I) Insights, and validations about the following topics:

Chapter 1: Radon transform

Chapter 2: Fourier transform

Chapter 3: Bessel function

Chapter 4: Convolution theorem

Chapter 5: Discrete Fourier transform

Chapter 6: Fourier series

Chapter 7: Integration by parts

Chapter 8: Fractional Fourier transform

Chapter 9: Mellin transform

Chapter 10: Poisson kernel

(II) Answering the public top questions about radon transform.

(III) Real world examples for the usage of radon transform in many fields.

Who this book is for

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of Radon Transform.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Apr 28, 2024

Book preview

Chapter 1: Radon transform

For every function f defined on the plane, the Radon transform maps it onto a function Rf defined on the (two-dimensional) space of lines in the plane, where the value of Rf at any given line is the line integral of f down that line. Johann Radon first described the transform in 1917 and also gave a formula for its inverse. The integral is evaluated over planes, as seen in Radon's three-dimensional transform formulas (integrating over lines is known as the X-ray transform). It was eventually extended outside the realm of integral geometry and to higher-dimensional Euclidean spaces. The Penrose transform is the sophisticated version of the Radon transform. In tomography, where an image is reconstructed from projection data associated with cross-sectional scans of an object, the Radon transform is commonly used.

If a function f represents an unknown density, When a tomographic scan is completed, the projection data is represented by the Radon transform.

Since the Radon transform may be inverted, the original density can be reconstructed from the projection data, hence providing the mathematical basis for tomographic reconstruction, Similarly to the iterative reconstruction technique.

Since the Radon transform of a non-centroidal point source is a sinusoid, the resulting data is often referred to as a sinogram. As a result, the Radon transform of a collection of small objects looks like a bunch of smeared-out sine waves of varying amplitudes and phases in a diagram.

The Radon transform has applications in reflection seismology, computed axial tomography (CAT), electron microscopy of macromolecular assemblies including viruses and protein complexes, and the solution of hyperbolic partial differential equations.

Let {\displaystyle f({\textbf {x}})=f(x,y)} be a function that satisfies the three regularity conditions:

{\displaystyle f({\textbf {x}})} is continuous; the double integral {\displaystyle \iint {\dfrac {\vert f({\textbf {x}})\vert }{\sqrt {x^{2}+y^{2}}}}\,dx\,dy} , covering the entire ground, converges; for any arbitrary point (x,y) on the plane it holds that

{\displaystyle \lim _{r\to \infty }\int _{0}^{2\pi }f(x+r\cos \varphi ,y+r\sin \varphi )\,d\varphi =0.}

Changing to a Radon matrix, Rf , is a function defined on the space of straight lines {\displaystyle L\subset \mathbb {R} ^{2}} by the line integral along each such line as:

{\displaystyle Rf(L)=\int _{L}f(\mathbf {x} )\vert d\mathbf {x} \vert .}

Concretely, the parametrization of any straight line L with respect to arc length z can always be written:

{\displaystyle (x(z),y(z))={\Big (}(z\sin \alpha +s\cos \alpha ),(-z\cos \alpha +s\sin \alpha ){\Big )}\,}

where s is the distance of L from the origin and \alpha is the angle the normal vector to L makes with the X -axis.

It follows that the quantities {\displaystyle (\alpha ,s)} can be considered as coordinates on the space of all lines in \mathbb {R} ^{2} , In these axes, the Radon transform is defined as:

{\displaystyle {\begin{aligned}Rf(\alpha ,s)&=\int _{-\infty }^{\infty }f(x(z),y(z))\,dz\\&=\int _{-\infty }^{\infty }f{\big (}(z\sin \alpha +s\cos \alpha ),(-z\cos \alpha +s\sin \alpha ){\big )}\,dz.\end{aligned}}}

More generally, in the n -dimensional Euclidean space \mathbb {R} ^{n} , the Radon transform of a function f satisfying the regularity conditions is a function Rf on the space \Sigma _{n} of all hyperplanes in \mathbb {R} ^{n} .

To put it simply,:

{\displaystyle Rf(\xi )=\int _{\xi }f(\mathbf {x} )\,d\sigma (\mathbf {x} ),\quad \forall \xi \in \Sigma _{n}}

when the integral is done in terms of the measure of natural hypersurfaces, d\sigma (generalizing the {\displaystyle \vert d\mathbf {x} \vert } term from the 2 -dimensional case).

Observe that any element of \Sigma _{n} is characterized as the solution locus of an equation \mathbf {x} \cdot \alpha =s , where {\displaystyle \alpha \in S^{n-1}} is a unit vector and {\displaystyle s\in \mathbb {R} } .

Thus the n -dimensional Radon transform may be rewritten as a function on {\displaystyle S^{n-1}\times \mathbb {R} } via:

{\displaystyle Rf(\alpha ,s)=\int _{\mathbf {x} \cdot \alpha =s}f(\mathbf {x} )\,d\sigma (\mathbf {x} ).}

It is also possible to generalize the Radon transform still further by integrating instead over k -dimensional affine subspaces of \mathbb {R} ^{n} .

One common application of this framework is the X-ray transform, and by integrating along curved lines.

There is a close relationship between the Radon transform and the Fourier transform. The Fourier transform of a single variable is defined as:

{\displaystyle {\hat {f}}(\omega )=\int _{-\infty }^{\infty }f(x)e^{-2\pi ix\omega }\,dx.}

For a function of a 2 -vector \mathbf {x} =(x,y) , The Fourier Transform in One Dimension:

{\displaystyle {\hat {f}}(\mathbf {w} )=\iint _{\mathbb {R} ^{2}}f(\mathbf {x} )e^{-2\pi i\mathbf {x} \cdot \mathbf {w} }\,dx\,dy.}

For convenience, denote {\displaystyle {\mathcal {R}}_{\alpha }[f](s)={\mathcal {R}}[f](\alpha ,s)} .

As a corollary, the Fourier slice theorem states:

{\displaystyle {\widehat {{\mathcal {R}}_{\alpha }[f]}}(\sigma )={\hat {f}}(\sigma \mathbf {n} (\alpha ))}

where \mathbf {n} (\alpha )=(\cos \alpha ,\sin \alpha ).

Thus the two-dimensional Fourier transform of the initial function along a line at the inclination angle \alpha is the one variable Fourier transform of the Radon transform (acquired at angle \alpha ) of that function.

The Radon transform and its inverse can be calculated by using this information.

This finding is n-dimensionally generalizable:

{\displaystyle {\hat {f}}(r\alpha )=\int _{\mathbb {R} }{\mathcal {R}}f(\alpha ,s)e^{-2\pi isr}\,ds.}

An adjoint to the Radon transform is the dual Radon transform.

Beginning with a function g on the space \Sigma _{n} , the dual Radon transform is the function {\displaystyle {\mathcal {R}}^{*}g} on Rn defined by:

{\displaystyle {\mathcal {R}}^{*}g(\mathbf {x} )=\int _{\mathbf {x} \in \xi }g(\xi )\,d\mu (\xi ).}

The integral here is taken over the set of all hyperplanes incident with the point {\displaystyle {\textbf {x}}\in \mathbb {R} ^{n}} , and the measure d\mu is the unique probability measure on the set {\displaystyle \{\xi |\mathbf {x} \in \xi \}} invariant under rotations about the point \mathbf {x} .

In particular, the dual transform of the Radon transform in dimension two is given by:

{\displaystyle {\mathcal {R}}^{*}g(\mathbf {x} )={\frac {1}{2\pi }}\int _{\alpha =0}^{2\pi }g(\alpha ,\mathbf {n} (\alpha )\cdot \mathbf {x} )\,d\alpha .}

Since the dual transform smears or projects a function specified on each line in the plane back over the line to create an image, it is often referred to as back-projection in the field of image processing.

Let \Delta denote the Laplacian on \mathbb {R} ^{n} defined by:

{\displaystyle \Delta ={\frac {\partial ^{2}}{\partial x_{1}^{2}}}+\cdots +{\frac {\partial ^{2}}{\partial x_{n}^{2}}}}

This is a second-order differential operator that is rotationally invariant by definition.

On \Sigma _{n} , the radial second derivative Lf(\alpha ,s)\equiv {\frac {\partial ^{2}}{\partial s^{2}}}f(\alpha ,s) is also rotationally invariant.

For these two differential operators, the Radon transform and its dual are coupled operators in the sense that:

{\displaystyle {\mathcal {R}}(\Delta f)=L({\mathcal {R}}f),\quad {\mathcal {R}}^{*}(Lg)=\Delta ({\mathcal {R}}^{*}g).}

The translational representation of Lax and Philips is based on the interweaving property, which is useful for analyzing solutions to the wave equation in higher-than-three-dimensional space. This is used as a dimensional splitting technique to simplify multidimensional issues into one-dimensional ones.

The process of reconstruction produces the image (or function f in the previous section) from its projection data.

Inverse problems like reconstruction.

For the simple situation of two dimensions, the most commonly used analytical formula to recover f from its Radon transform is the Filtered Back-projection Formula or Radon Inversion Formula:

{\displaystyle f(\mathbf {x} )=\int _{0}^{\pi }({\mathcal {R}}f(\cdot ,\theta )*h)(\left\langle \mathbf {x} ,\mathbf {n} _{\theta }\right\rangle )\,d\theta }

where h is such that {\displaystyle {\hat {h}}(k)=|k|} .

The convolution kernel h is referred to as Ramp filter in some literature.

Intuitively, The formula for filtered back-projection, similar to how differentiation works, for which {\textstyle \left({\widehat {{\frac {d}{dx}}f}}\right)\!(k)=ik{\widehat {f}}(k)} , The filter appears to be carrying out a function analogous to a derivative.

Roughly speaking, then, The filter gives everything a more distinct identity.

Radon inversion is quantitatively stated to be ill-posed as follows:

{\displaystyle {\widehat {{\mathcal {R}}^{*}{\mathcal {R}}g}}(k)={\frac {1}{\|\mathbf {k} \|}}{\hat {g}}(\mathbf {k} )}

where {\displaystyle {\mathcal {R}}^{*}} is the previously defined adjoint to the Radon Transform.

Thus for {\displaystyle g(\mathbf {x} )=e^{i\left\langle \mathbf {k} _{0},\mathbf {x} \right\rangle }} , we have:

{\displaystyle {\mathcal {R}}^{*}{\mathcal {R}}g={\frac {1}{\|\mathbf {k_{0}} \|}}e^{i\left\langle \mathbf {k} _{0},\mathbf {x} \right\rangle }}

The complex exponential {\displaystyle e^{i\left\langle \mathbf {k} _{0},\mathbf {x} \right\rangle }} is thus an eigenfunction of {\displaystyle {\mathcal {R}}^{*}{\mathcal {R}}} with eigenvalue {\textstyle {\frac {1}{\|\mathbf {k} _{0}\|}}} .

Thus the singular values of {\mathcal {R}} are {\textstyle {\frac {1}{\sqrt {\|\mathbf {k} \|}}}} .

Since these singular values tend to {\displaystyle 0} , {\displaystyle {\mathcal {R}}^{-1}} is unbounded.

Iterative reconstruction is more computationally intensive than the Filtered Back-projection approach, which limits its use. However, in the presence of discontinuity or noise, the Filtered Back-projection approach may be unfeasible due to the ill-posedness of Radon Inversion. Many researchers are interested in iterative reconstruction methods (such iterative Sparse Asymptotic Minimum Variance) because they have the potential to reduce metal artifacts, noise, and dosage in the reconstructed output.

Radon transform and its counterpart inversion formulas are available and are both explicit and computationally efficient.

The Radon transform in n dimensions can be inverted by the formula:

{\displaystyle c_{n}f=(-\Delta )^{(n-1)/2}R^{*}Rf\,}

where {\displaystyle c_{n}=(4\pi )^{(n-1)/2}{\frac {\Gamma (n/2)}{\Gamma (1/2)}}} , and the power of the Laplacian {\displaystyle (-\Delta )^{(n-1)/2}} is defined as a pseudo-differential operator if necessary by the Fourier transform:

{\displaystyle \left[{\mathcal {F}}(-\Delta )^{(n-1)/2}\varphi \right](\xi )=|2\pi \xi |^{n-1}({\mathcal {F}}\varphi )(\xi ).}

Logic-based computation, the power of the Laplacian is commuted with the dual transform R^{*} to give:

{\displaystyle c_{n}f={\begin{cases}R^{*}{\frac {d^{n-1}}{ds^{n-1}}}Rf&n{\text{ odd}}\\R^{*}{\mathcal {H}}_{s}{\frac {d^{n-1}}{ds^{n-1}}}Rf&n{\text{ even}}\end{cases}}}

where {\displaystyle {\mathcal {H}}_{s}} is the Hilbert transform with respect to the s variable.

Using only two axes, the operator {\displaystyle {\mathcal {H}}_{s}{\frac {d}{ds}}} appears in image processing as a ramp filter.

One can prove directly from the Fourier slice theorem and change of variables for integration that for a compactly supported continuous function {\displaystyle f} of two variables:

{\displaystyle f={\frac {1}{2}}R^{*}H_{s}{\frac {d}{ds}}Rf.}

Thus in an image processing context the original