【Basic】Image Reconstruction in MATLAB: Reconstructing Images from Projection Data

# 1. An Overview of Image Reconstruction

Image reconstruction is the process of restoring an image from projection data. It is widely used in medical imaging, industrial inspection, and other fields. The development of image reconstruction technology has provided important means for disease diagnosis, material analysis, and more.

# 2. Theoretical Foundations of Image Reconstruction

### 2.1 Geometric Models of Projection Data

Projection data, which describes the transmission or reflection signals of the object to be reconstructed from different angles, is the basis of image reconstruction. The geometric model of projection data defines the way projection data is acquired and the geometric relationship of the reconstructed image.

**Parallel Projection Model**

In the parallel projection model, X-rays or other radiation sources pass through the object in parallel beams to produce a projection image on the detector. In the projection image, each point on the object forms a straight line on the detector, known as a ray.

**Fan-Beam Projection Model**

In the fan-beam projection model, the X-ray source rotates around the object, penetrating it with a sector beam. The resulting projection image on the detector consists of a series of arcs, known as projections.

### 2.2 Backprojection Algorithms

Backprojection algorithms are among the most important in image reconstruction, converting projection data into an image. The basic principle of backprojection algorithms is to reverse-project the projection data along rays into image space.

#### 2.2.1 Filtered Backprojection Algorithm

The Filtered Backprojection (FBP) algorithm is the simplest form of backprojection algorithms. It reconstructs images by filtering projection data and then reverse-projecting it along rays into image space.

**Filtering**

Filtering is a key step in the FBP algorithm, ***mon filters include:

- **Ram-Lak filter:** An ideal filter but can cause ringing artifacts.
- **Hamming filter:** Reduces ringing artifacts but has lower resolution.
- **Shepp-Logan filter:** Balances resolution and artifact suppression.

**Backprojection**

Backprojection is the process of reverse-projecting filtered projection data along rays into image space. There are several backprojection algorithms, including:

- **Direct backprojection:** Simple and efficient, but can cause streak artifacts.
- **Weighted backprojection:** Reduces streak artifacts by weighting the backprojection data.
- **Filtered backprojection:** Combines filtering and backprojection for better image quality.

#### 2.2.2 Iterative Backprojection Algorithms

Iterative backprojection algorithms (IRT) are more complex, gradually optimizing the reconstructed image through an iterative process. The basic principle of the IRT algorithm is:

1. Initialize an image.
2. Calculate the error based on the current image and projection data.
3. Update the image to reduce the error.
4. Repeat steps 2 and 3 until the error is minimized.

IRT algorithms can reconstruct more accurate images than FBP algorithms, ***mon IRT algorithms include:

- **Maximum Likelihood Expectation Maximization (MLEM):** Based on statistical models, convergence is slow.
- **Ordered Subsets Expectation Maximization (OSEM):** Divides projection data into subsets for faster convergence.
- **Regularized Iterative Backprojection (RIRP):** Adds regularization terms to suppress noise and artifacts.

# 3. Image Reconstruction in Practice

### 3.1 Implementation of Image Reconstruction Algorithms

#### 3.1.1 Backprojection Algorithms in MATLAB

In MATLAB, the `iradon` function can be used to implement the filtered backprojection algorithm. The syntax is as follows:

[image, theta] = iradon(projection, theta, filter)

Where:

* `projection`: A one-dimensional array of projection data.
* `theta`: A one-dimensional array of projection angles in radians.
* `filter`: A one-dimensional array of filters to remove noise from the projection data.

The `iradon` function works as follows:

1. Interpolates the projection data `projection` according to the angle `theta` to obtain a two-dimensional image.
2. Filters the interpolated image with the filter specified by the `filter` parameter.
3. Backprojects the filtered image to obtain the reconstructed image `image`.

#### 3.1.2 Iterative Algorithms in MATLAB

In MATLAB, the `imreconstruct` function can be used to implement iterative backprojection algorithms. The syntax is as follows:

image = imreconstruct(seed, mask)

Where:

* `seed`: A binary seed image representing the initial estimate of the reconstructed image.
* `mask`: A binary mask image representing the constraints for the reconstructed image.

The `imreconstruct` function works as follows:

1. Performs a logical AND operation between the seed image `seed` and the mask image `mask` to obtain a new binary image.
2. Performs morphological dilation on the new binary image, with the number of iterations determined by the iteration count.
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