Detailed Analysis of CT Reconstruction Concepts and Algorithms

History of CT Image Reconstruction

The proposal of Radon transform and inverse transform lays the mathematical foundation for CT image reconstruction (1917) The
proposal of convolutional back-projection algorithm/filtered back-projection algorithm opened the door to accurate image reconstruction (1971-1974) The
FDK algorithm proposed by Feldkamp et al. opened the door A New Era of 3D Image Reconstruction (1980)
Katsevich solved the problem of axial truncation in accurate reconstruction of cone-beam helical CT images (2002)
Pan et al proposed a back-projection filter algorithm to solve the problem of data transverse truncation (2004)
Zhang et al proposed Introduced an intelligent reconstruction method based on artificial intelligence technology/deep learning technology, and innovated the CT reconstruction algorithm (2019)

Radon transform

Radon transformation reveals the relationship between functions and projections. If the function is f (x, y), the projections at different angles are:

$$p(t,i)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }f(x,y)d(xcos\theta +ysin\theta -t)dxdy$$

The distribution of relative linear attenuation coefficients within an infinitely thin slice is uniquely determined by the set of all its line integrals

projection

The absorption of X-rays by matter can be described by Lambert-Beer's law:
Assume that the emitted X-ray intensity is $N_0$, the incident X-ray intensity is $N_i$
$N_0=N_i{e}^{-\mu d}$
is the superposition form below:

Chord diagram

The chord diagram is a method to describe the projection. The vertical axis of the chord space represents the detector unit, and the horizontal axis represents the projection angle. A unit projection is represented as a sample set on a line parallel to the horizontal axis.
In this way, all the data collected at different scanning angles form a two-dimensional image.

Will project $p(s,\theta )$ is shown in$s-In\; the\; \theta$ coordinate system, a point in the space domain is a sine curve in the chord space.
Think of the object as a combination of many points, which is a series of overlapping images of sinusoids in the string space.

matlab code to generate chord diagram

I=phantom(256);
theta=0:179;
P=radon(I,theta);
figure;
imshow(I,[]),title('256*256头模型图像');
figure;
imagesc(P),colormap(gray),colorbar,title('180°平行束投影图像');

Sampling Geometry

image reconstruction

The available algorithms for solving CT images from known chord diagrams are:
direct matrix inversion,
iterative,
Fourier reconstruction,
back-projection, and
filtered back-projection ( Filtered back projection)

direct matrix

Suppose the image size to be reconstructed is: $2^8\ast 2^8=256\ast 256,$
then the size of matrix F is:$2^{16}\ast 2^{16}=2^{32},$
not convenient for practical

Iterative method

Given the initial matrix
Use the initial matrix to form a projection
Compare the projection of the object to be reconstructed with the simulated projection
If the error meets the requirements, stop the iteration

More accurate, but slow, mostly used for image reconstruction of nuclear medicine equipment and low-dose CT

Fourier reconstruction

1. Collect projections from various angles of CT scanning.
2. Calculate 1D FT for each projection.
3. Regularize 2D coordinate FT plane.
4. Calculate the projection of the original
image along a certain direction through 2D inverse FT. A line
\qquad of the 2D Fourier transform plane \qquad
The more projections, the more samples, similar to the K space in MRI reconstruction. The
following figure is more conducive to understanding the one-dimensional and two-dimensional frequency spaces. The following figure

shows the transformation of the three spaces.

Limitations of Fourier reconstruction

1. The points in the 2D frequency domain are not arranged in a matrix. Before performing the inverse Fourier transform, the sample interpolation needs to be converted into a Cartesian coordinate representation. The points in the high-frequency area are relatively sparse, and the interpolation results are affected. The interpolation error in the real domain only affects the small area around the pixel, but the interpolation error
in the frequency domain will affect the quality of the entire image. It also produces obvious shadow artifacts
(F(0, 1) represents the dc component of the image f(x, y) in the horizontal direction and the first harmonic in the vertical direction, and the estimated error will lead to changes in image intensity and vertical A single-period sinusoidal shadow in the vertical direction)
3. It is difficult to achieve target reconstruction. The size of the inverse Fourier transform is inversely proportional to the size of the ROI. For a small ROI, the matrix is ​​too large to handle.
4. The projection of the fault is one-dimensional. The inverse transformation of the object image is two-dimensional. Therefore, all the data must be stored, and the two-dimensional inverse transformation can only be performed after all the data is complete. This requires a large hardware memory and a long waiting time, making it difficult to realize real-time images. Rebuild required.

back projection

Principle: The density value of a certain point in the fault plane can be regarded as the sum (average value) of all ray projections passing through the point in this plane

.
Prone to star artifacts.
Cause: The back-projection method evenly back-smears (back-projects) the projection taken from the limited object space to each pixel in the infinite space touched by the ray, including the point where the original pixel value is 0.
It is easy to form star-shaped artifacts
. Improvement method: the filter function processes the image (including pre-processing and post-processing).
Pre-processing: filter before back-projection. Theoretical basis: In the linear system, the filter operator can exchange the order.
Post-processing: use The 2D filter function processes the image obtained by the back projection method to improve the image quality without changing the reconstruction

Filtered Back Projection (FBP)

It mainly introduces filtering operation and convolution operation, and uses a correction function to filter P ( ω , θ ) P(\omega, \theta) before backprojection

$P(\omega ,\theta )$ means corresponding to$The\; Fourier\; transform\; of\; the\; unit\; projection\; of\; the\; \theta$ angle; the integral in the inner layer is$P(\omega ,\theta )|\omega |$ inverse Fourier transform, denoted as$g(t,\theta )$ , in the spatial domain, it represents the unit projection by a frequency domain response as$The\; function\; of\; |\omega |$ performs filtering operation, so it is called filtered back projection.

If the filter is properly designed, when the filtered back-projection signals are superimposed, the positive and negative values ​​of "radiation" just cancel each other out, and the original target can be accurately reflected.
$g(t,i)=g(xcos\theta ,ysin\theta )={\int }_{-\infty }^{\infty }P(\omega ,i)|\omega |e^{j2say(xcos\theta +ysin\theta )}d\omega$
filter波function：
$z\left(t\right)={\int }_{-\infty }^{\infty }|\omega |e^{j2\pi \omega t}d\omega$
above is an ideal filter, due to the infinite frequency band, it is necessary to use the window function to obtain different filters, which

can be realized with matlab

I=phantom(256);
theta=0:1:179;
P=radon(I,theta);
rec=iradon(P,theta,'linear','None');
rec_RL=iradon(P,theta,'Ram-Lak');
rec_SL=iradon(P,theta,'linear','Shepp-Logan');
figure;
subplot(2,2,1);imshow(I,[]),title('原始图像');
subplot(2,2,2);imshow(rec,[]),title('直接反投影图像');
subplot(2,2,3);imshow(rec_RL,[]),title('RL滤波反投影图像');
subplot(2,2,4);imshow(rec_SL,[]),title('SL滤波反投影图像');

Shepp-logan filter: smoothes the image and loses some high-frequency information.
Hamming filter: reduces high-frequency noise
Bone filter and soft tissue filter: Different filter functions can be selected according to diagnostic requirements
1) Smoothing is used to observe soft tissues
2) Sharpness is used to observe high-resolution images
FBP Complement 0 Operation
The original filtering operation includes a non-periodic convolution operation, which is a periodic convolution after being changed to the frequency domain. Direct calculation will produce interference artifacts, the so-called warp-around effect. Therefore, it is necessary to add 0 to each projection before Fourier transform and filtering operations to avoid artifacts.

Summarize

From left to right: back projection method, filtered back projection method, Fourier transform

back projection

1. Back projection method (sum method): It is to use the projection value to approximate the two-dimensional distribution of the absorption coefficient. Its basic principle is to evenly distribute the measured projection values ​​to each point according to its original path, after the projection values ​​in each direction are back-projected, they are superimposed on the image to infer the original image.
2. The image of the object reconstructed by the back-projection method of the square object is not a square, but becomes a "star" object. The absorption coefficient value is the largest at the center, and the value is lower the farther away from the center, resulting in sharp edges of the image.
3. The back projection method will cause the edge of the image to be unclear. If within a uniform tissue density, there are parts with extremely non-uniform absorption coefficients, image artifacts will appear in the back-projection image.

filtered back projection

1. Filtered back-projection reconstruction method: the original density function is obtained by first correcting and then back-projecting. The basic method of reconstructing an image by filtered back projection is: after obtaining a projection function (one-dimensional function) at a certain projection angle, filter it to obtain a corrected projection function. Then the modified projection function is back-projected to obtain the required density function.
2. When the filtered back projection method realizes image reconstruction, it only needs to perform one-dimensional Fourier transform. Because the time-consuming two-dimensional Fourier transform is avoided, the filtered back projection method significantly shortens the time of image reconstruction.

Fourier

1. Fourier transform reconstruction method: For each measured projection data, one-dimensional Fourier transform is performed first. According to the central slice theorem, the result of this transformation can be regarded as a straight line passing the origin at the same angle in the two-dimensional frequency domain on the value. The one-dimensional transformation function obtained under different projection angles can form a complete two-dimensional Fourier transformation function in the frequency domain, and the two-dimensional transformation function is inversely transformed to obtain the required density function in the space domain.
2. When the Fourier transform method is used to reconstruct an image, the one-dimensional Fourier transform of the projection function is expressed in the form of polar coordinates in the frequency domain. When the data in the form of polar coordinates is converted into data in the form of rectangular coordinates through interpolation operations , the calculation workload is relatively large. In addition, in the frequency domain data in the form of polar coordinates, the data of the part with higher frequency farther from the origin is relatively sparse. When the data at these positions are converted to rectangular coordinates, they need to be interpolated, which will introduce a certain degree of error. That is, in the reconstructed image, high-frequency components may have obvious distortion.