MATLAB Implementation of Radon Transform Principle with Code Analysis

Radon transform is a widely used technique in image processing and medical imaging, primarily employed for image reconstruction from projection data. The core concept involves measuring line integrals (projections) of an object at different angles and utilizing these projections to reconstruct the original image. Basic Principles of Radon Transform The Radon transform maps a two-dimensional function (such as an image) to its integrals (projections) along a set of lines. Mathematically, for a given function f(x,y), its projection at a specific angle θ and distance ρ can be expressed as the line integral along that path. The result of Radon transform is typically a two-dimensional matrix where rows represent different angles and columns represent projection values at various distances. MATLAB Implementation Approach MATLAB provides built-in functions `radon` and `iradon` for computing the Radon transform and its inverse, respectively. These functions implement efficient algorithms for projection calculation and image reconstruction. Radon Transform Computation The `radon` function accepts an input image and a set of angle parameters, returning projection data. Users can specify angle increments (e.g., 1° or 5°) or directly provide an angle vector. Code implementation: The function internally calculates line integrals using interpolation methods, generating a sinogram - a visual representation of projection data where each column corresponds to projections at a specific angle. For medical CT images, this produces characteristic sinusoidal patterns in the sinogram matrix. Inverse Radon Transform (Image Reconstruction) The `iradon` function employs the Filtered Back Projection (FBP) algorithm to reconstruct the original image from projection data. Users can optimize reconstruction quality by selecting different filter types (e.g., Ram-Lak, Shepp-Logan) that control frequency response during the back-projection process. Algorithm details: FBP first applies a ramp filter to the projections in frequency domain, then smears these filtered projections back across the image plane along their original paths. Reconstruction quality depends on the number of projection angles and noise levels - denser angular sampling produces images closer to the original. Application Examples Medical Imaging: CT scanners use the inverse Radon transform process to reconstruct cross-sectional images from X-ray projection data. Industrial Testing: Non-destructive evaluation employs Radon transform to analyze internal material structures. Seismology: Used for interpreting acoustic reflection data to map subsurface geological structures. By implementing Radon transform in MATLAB, researchers can gain intuitive understanding of its mathematical principles and explore practical applications in image reconstruction through hands-on code experimentation and parameter adjustment.