A journey through the art and science of CT reconstruction

Welcome to this month's edition of our newsletter! Believe it or not, we've reached issue #6. Today, we embark on a thrilling expedition into the very heart of X-ray Computed Tomography (CT) – the fascinating world of image reconstruction. Have you ever marvelled at a crisp, detailed cross-sectional image of a complex industrial component or a delicate biological specimen and wondered, "How on Earth did they create that from a series of simple X-ray projections?" The answer lies in a beautiful interplay of mathematics, physics, and computational ingenuity. So, follow me and let's delve into the magic that transforms shadows into substance.

The grandfather of it all: Johann Radon and his transform

Our story (Figure 1) begins not in a high-tech imaging lab, but in the realm of pure mathematics. In 1917, the brilliant Austrian mathematician Johann Radon, likely without any idea of the medical and industrial imaging revolution he was about to spark, introduced a mathematical concept known as the Radon transform.

Figure 1 ：A brief history of X-ray milestones

 

Imagine you have a 2D object with varying densities. The Radon transform, in essence, describes the process of taking line integrals through this object. In the context of CT, think of an X-ray beam passing through a sample. The attenuation of that beam – the amount of X-ray energy absorbed or scattered – is a line integral of the material's properties along the beam's path. By rotating the X-ray source and detector around the object and collecting these attenuation profiles from multiple angles, we generate a dataset called a sinogram (Figure 2). This sinogram is, in fact, the Radon transform of the object's internal structure. Each point on the sinogram represents the total attenuation along a specific line through the object.

Figure 2：An example of the Maths behind the Radon Transform


The genius of Radon's work was not just in defining this transform (Figure 3), but in proving that it was invertible. This meant that, theoretically, if you had all the possible line integrals (a complete sinogram), you could perfectly reconstruct the original 2D object (Figure 4). It's like having the sum of all the numbers in each row and column of a Sudoku puzzle and then being able to deduce the individual numbers in each cell. For decades, this groundbreaking work remained a mathematical curiosity, a solution waiting for a problem. That problem, as it turned out, was the challenge of seeing inside the human body and industrial parts without cutting them open.

Figure 3：The Radon transform is often called a sinogram because the Radon transform of a Dirac delta function is a distribution supported on the graph of a sine wave

Figure 4：Forward Projection

The workhorse of reconstruction: Filtered Back Projection (FBP)

Fast forward to the 1970s, and the advent of the first CT scanners (developed by Sir Godfrey Hounsfield and Allan Cormack). The theoretical foundation of the Radon transform now had a practical application. But how to perform this "inverse Radon transform" in reality? The answer came in the form of an elegant and computationally efficient algorithm: Filtered Back Projection (FBP).

As the name suggests, FBP involves two main steps:

Back Projection (Figure 5): This is the intuitive part of the process. Imagine taking each projection from the sinogram and "smearing" it back across the image plane from the angle it was acquired. It's like a painter with a very broad brush, applying strokes of varying intensity (representing the attenuation) from different directions. While this process starts to build up the image, it suffers from a significant drawback: blurring. The smearing effect creates a hazy, out-of-focus image because the back projection of a point source isn't a point, but rather a star-like artefact.

Figure 5：Back projection

Filtering (Figure 6): To counteract this blurring, a crucial "filtering" step is applied to the projection data before back projection. This is a mathematical trick that sharpens the data by applying a high-pass filter. Think of it as enhancing the edges and details in each projection before they are smeared back. This filtering process effectively cancels out the blurring introduced by the back projection, resulting in a much sharper and more accurate image.

Figure 6 ： Filtering

From parallel lines to cones: the Feldkamp-Davis-Kress (FDK) algorithm

The initial FBP algorithms were designed for a simple parallel beam geometry (Figure 7), where all X-ray beams are parallel to each other. However, modern CT scanners often use a cone beam geometry, where the X-rays emanate from a point source and form a cone. This allows for faster data acquisition, as a larger volume can be scanned at once.

To handle this cone beam data, an extension of the FBP algorithm was developed in 1984 by Feldkamp, Davis, and Kress, known as the FDK algorithm (https://opg.optica.org/josaa/abstract.cfm?uri=josaa%E2%80%901%E2%80%906%E2%80%90612). The FDK algorithm is an approximation that essentially treats the cone beam data as a collection of tilted 2D fan beams, applies a weighting factor to account for the cone's geometry, and then uses a modified back projection to reconstruct the 3D volume. For small cone angles, the FDK algorithm provides excellent results and remains a widely used and computationally efficient method for cone beam CT.

Figure 7： an example of the FBP algorithm

Pros and Cons of FBP and FDK:

Pros:

  • Speed: 🏎️ FBP and FDK are incredibly fast, which is crucial for clinicalapplications and high-throughput industrial inspection.

  • Simplicity: The underlyingmathematics is well-understood and relatively straightforward toimplement.

Cons:

  • Noise Amplification: The high-passfilter used in FBP can amplify noise present in the raw data, leading tograiny images, especially in low-dose scans.

  • Artefacts: FBP is susceptible tovarious artefacts, such as streak artefacts from dense objects(like metal implants) and beam hardening artefacts.

  • Cone Beam Artefacts (for FDK):As the cone angle increases, the approximations made by the FDK algorithmbecome less accurate, leading to artefacts, particularly at the topand bottom of the reconstructed volume.

The rise of the iterative approach

While FBP was the undisputed king of CT reconstruction for decades, its limitations, particularly with the push for lower radiation doses and higher image quality, paved the way for a new paradigm: iterative reconstruction.

Instead of the direct, one-shot approach of FBP, iterative reconstruction is more like a conversation. It starts with an initial guess of the 3D volume and then refines this guess through a series of iterations. Here's a simplified breakdown of the process:

  1. Initial Guess: The algorithm startswith a rough estimate of the image. This could be a blank image or animage reconstructed with a fast but less accurate method like FBP.

  2. Forward Projection: The algorithmthen simulates the CT scan by performing a "forward projection"on this initial guess. This creates a virtual sinogram.

  3. Comparison: This virtual sinogramis then compared to the actual measured sinogram from the scanner. Thedifferences between the two represent the errors in the current imageestimate.

  4. Correction and Update: Thealgorithm uses these error values to update and improve the image. Thiscorrection step is where the different iterative algorithms diverge intheir specific mathematical approaches.

  5. Repeat: The process of forwardprojection, comparison, and correction is repeated multiple times (iterated)until the virtual sinogram closely matches the measured sinogram, and ahigh-quality image is achieved.

The Edge-Preserving magic: Total Variation (TV) regularization

A key challenge in reconstruction is removing noise without destroying important details. How do you tell the algorithm, "Smooth out this grainy area, but don't you dare blur this critical edge"? This is where a powerful technique called Total Variation (TV) regularisation comes into play.

Imagine an image as a landscape of intensity values. Noisy areas are like bumpy, rugged terrain, while smooth regions are like flat plains. The "Total Variation" is a measure of the total "bumpiness" in the image. TV regularisation works by adding a constraint to the iterative process: it tries to find a solution that not only matches the measured data but also has the minimum possible total variation.

The result is almost magical. The algorithm preferentially smooths out the noisy, "bumpy" regions (where pixel values are randomly fluctuating) while preserving the sharp "cliffs" that represent genuine edges in the object (Figure 8). It's a key reason why iterative reconstruction can produce such clean-looking images without the "blurry" side effects of simpler noise reduction filters. This technique is a cornerstone of many modern advanced reconstruction packages.

Figure 8 - An example of TV denoising algorithm.

A parade of iterative algorithms:

The world of iterative reconstruction is rich and diverse, with various algorithms developed over the years, often incorporating clever techniques like Total Variation.

  • Algebraic Reconstruction Technique (ART): This is one of the earliest iterative methods. It works on aray-by-ray basis, correcting the image pixels that contribute to each rayin the projection data. While conceptually simple, it can becomputationally intensive.

  • Statistical Iterative Reconstruction (SIR): These algorithms take a more sophisticated approach byincorporating statistical models of the noise in the data. This allowsthem to differentiate between genuine signal and noise more effectivelythan FBP, leading to significant noise reduction. Many commercialiterative reconstruction techniques, such as GE's ASiR (AdaptiveStatistical Iterative Reconstruction) and Siemens' SAFIRE (SinogramAffirmed Iterative Reconstruction), are based on these principles.

  • Model-Based Iterative Reconstruction (MBIR): This is the current state-of-the-art. MBIR algorithmsincorporate not only statistical models of the noise but also detailedphysical models of the CT scanner itself, including the geometry of theX-ray source and detectors, and the physics of how X-rays interact withmatter. This comprehensive modelling allows MBIR to produce images withexceptional detail and clarity, even at very low radiation doses. Philips'IMR (Iterative Model Reconstruction) and Canon's FIRST (Forward projectedmodel-based Iterative Reconstruction SoluTion) are examples of thisadvanced approach.

Pros and Cons of Iterative Reconstruction:

Pros:

Superior Image Quality: ✨ Iterative methods excel at reducing noise and artefacts, leading to clearer and more detailed images.

Dose Reduction: Their ability to handle noisy data allows for significant reductions in radiation dose without compromising diagnostic or inspection quality.

Flexibility: They can be adapted to handle incomplete or sparse data, which is beneficial in certain applications.

Cons:

Computational Cost: ⏳ The iterative nature of these algorithms makes them significantly more computationally demanding and time-consuming than FBP. However, with the rapid advancements in GPU technology, this is becoming less of a bottleneck.

"Plastic" Appearance: Some earlier or aggressively applied iterative reconstruction algorithms could produce images with a "waxy" or "plastic-like" texture, which took some getting used to for radiologists and analysts. Newer algorithms have largely overcome this issue.

Geometries and their challenges

The choice of reconstruction algorithm is often intertwined with the geometry of the CT scanner.

  • Parallel and fan beam: For 2Dimaging with parallel or fan beams, both FBP and iterative methods can beused effectively. The choice often comes down to a trade-off between speed(FBP) and image quality/dose (iterative).

  • Cone beam: For 3D cone beam CT, theFDK algorithm is a fast and effective solution for smaller cone angles.However, for larger cone angles, iterative reconstruction methods areincreasingly favoured as they can more accurately model the cone beamgeometry and produce artefact-free images.

  • Laminography (discussed in a previousissue, please go and find it :-)): This is a special case where alimited range of projection angles is used to reconstruct a series ofslices through an object. This is particularly useful for imaging flatobjects, like printed circuit boards, where a full 360-degree rotation isnot feasible. Because of the limited data, FBP can produce significantartefacts. Iterative reconstruction techniques, with their ability tohandle incomplete data (a concept known as compressed sensing, where TVregularisation plays a starring role!), are exceptionally well-suited forlaminography, producing images with significantly improved clarity andreduced out-of-plane blur.

Commercial Vendors: the turnkey solution

Commercial manufacturers like ZEISS, Nikon, TESCAN, North Star Imaging (NSI), Waygate, Diondo, Bruker, RX Solutions, Comet Yxlon, and Rigaku (and many more, I cannot name you all) sell integrated systems where the hardware and software are a complete, validated package. Their goal is to provide reliable, fast, and accurate results with a user-friendly interface.

The workhorse (FDK): The default reconstruction method for virtually all commercial systems is a highly optimised version of Filtered Back Projection (FBP/FDK). It's very fast, computationally simple, and produces predictable, high-quality results for a wide range of standard applications, which is essential for industrial throughput and quality control.

The premium feature (proprietary IR): These companies invest heavily in developing their own proprietary Iterative Reconstruction (IR) algorithms, which are sold as advanced, often optional, modules. Below are some approaches from companies I know very well since we own their systems:

ZEISS: With its strong background in metrology, ZEISS focuses on accuracy. Their IR offerings (like OptiRecon and DeepRecon, part of ART Toolbox, https://www.zeiss.com/microscopy/en/products/software/advanced-reconstruction-toolbox.html) are designed to reduce noise and artefacts, which enables more precise dimensional measurements from CT data, even with faster or lower-power scans.

Nikon Metrology: Focused on non-destructive testing (NDT), Nikon's software provides fast FDK for rapid inspection. Their iterative reconstruction is marketed to enhance the probability of detection for subtle defects (like porosity or cracks) and to reduce scan times without sacrificing image quality. They recently introduced AI reconstruction to their portfolio. (https://industry.nikon.com/en-gb/products/x-ray-ct/ai-reconstruction/)

TESCAN: Often working in materials and life sciences, sometimes with correlative microscopy (CT + SEM), TESCAN's approach is about high-resolution results. Their iterative methods are tailored to reduce noise in high-magnification scans of complex microstructures (https://info.tescan.com/micro-ct/products/dynatom-micro-ct-info?utm_term=tescan%20ct&utm_campaign=Micro+CT+May+2025&utm_source=adwords&utm_medium=ppc&hsa_acc=3637663133&hsa_cam=22523753479&hsa_grp=187602959188&hsa_ad=750583167756&hsa_src=g&hsa_tgt=kwd-2418604497771&hsa_kw=tescan%20ct&hsa_mt=p&hsa_net=adwords&hsa_ver=3&gad_source=1&gad_campaignid=22523753479&gbraid=0AAAAADHzWiCEqQfzmELo7_MMFuEmDmqEm&gclid=CjwKCAjwyb3DBhBlEiwAqZLe5Fz6z3dSpgEKumcGj-ipE67YYL_Agj2xfFTT0ve_o8psbQpDVVpXShoCSmgQAvD_BwE).

In short, commercial vendors provide a "black box" solution: you input raw data, and a validated, high-quality image comes out. The user doesn't need to worry about the parameters of a SIRT algorithm; they select "Metal Artefact Reduction" or "Low Dose Mode."

Open-source toolboxes: the researcher's playground 🔬

Research toolboxes like ASTRA, TIGRE, and CIL are built for a completely different user: the scientist or developer. Their philosophy is to provide a flexible, transparent, and powerful set of building blocks to create and test novel reconstruction methods. You are very familiar with these, so you know their power.

  • ASTRA Toolbox: ASTRA's corestrength is raw speed and flexibility. It provides highly optimised,GPU-accelerated primitives for forward and back projection. It's not afull application but a library you call from Python or MATLAB. Researchersuse ASTRA as an engine to build almost any conceivable reconstructionalgorithm (FDK, SIRT, CGLS, etc.) for any 2D or 3D geometry. Its primarygoal is to be the fastest "Lego brick" for tomographic research.https://astra-toolbox.com/

  • TIGRE (Tomographic IterativeGPU-based Reconstruction): As the name suggests, TIGRE's focus is squarelyon iterative algorithms for GPUs. While it shares goals with ASTRA, itcomes "pre-packaged" with a wider array of ready-to-use advancediterative algorithms (like OS-SART for large datasets and ASD-POCS forTotal Variation regularisation). It's designed for researchers who want toquickly implement and compare different state-of-the-art iterative methodswithout building them from scratch using lower-level primitives. https://github.com/CERN/TIGRE

  • CIL (Core Imaging Library): CIL,from the Rutherford Appleton Lab (STFC), takes a higher-level, moreholistic approach. Given its origin at a national lab with unique,large-scale facilities (like synchrotrons), its philosophy is to create aunified framework for the entire imaging pipeline. CIL is designed to be amodular system that can use powerful backends like ASTRA or TIGRE for theheavy lifting. However, CIL's unique contribution is its extensive libraryof optimisation algorithms and regularisation techniques (like TV, TGV,FGP), allowing researchers to formulate and solve complex reconstructionproblems easily. It's less about the projectors themselves and more aboutsolving the inverse problem robustly and flexibly, especially forchallenging, real-world scientific data. (https://ccpi.ac.uk/cil/ )

In short, these toolboxes offer a "glass box" approach. They provide researchers with full control over every aspect of the reconstruction, enabling the development and validation of the next generation of algorithms that may eventually be integrated into commercial systems.

The future is bright

The journey of CT reconstruction is far from over. With the rise of artificial intelligence and deep learning, we are witnessing the emergence of a new generation of reconstruction techniques. These AI-powered algorithms can learn from vast datasets of existing CT images to produce reconstructions with unprecedented speed and quality, potentially pushing the boundaries of what is possible in terms of resolution and dose reduction even further.

So, the next time you gaze upon a stunning CT image, take a moment to appreciate the incredible journey that raw projection data has undertaken. From the elegant mathematics of Radon to the workhorse efficiency of FBP and the refined intelligence of iterative and AI-driven methods, it is a testament to human ingenuity and our relentless pursuit of seeing the unseen.

Until next time, keep exploring the frontiers of imaging