2.1 CT: From Detector Signal to Corrected Projection

"Before a CT image is reconstructed, the raw data has already undergone a complex series of physical and mathematical transformations. Understanding this process is understanding the soul of CT imaging." —— The Wisdom of Medical Imaging Engineering

Imagine this scenario: When a radiologist sees a clear CT image, few realize how much correction and processing the data behind that image has undergone. From the moment X-ray photons are captured by the detector, the data begins a complex transformation journey—from physical signals to digital projection values, and finally to the reconstructed image.

This chapter will systematically introduce the first stage of this process: from detector signal to corrected projection. We will deeply explore how detectors capture photons, how photons are converted to projection values, and the industry-standard correction workflow. This content is the foundation for understanding CT reconstruction and the key to optimizing image quality.

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What is Preprocessing?

Preprocessing refers to a series of processing steps performed on raw data before image reconstruction. The purposes of these steps are:

• Eliminate systematic errors: Remove various deviations introduced by the detector and scanning system
• Correct physical effects: Compensate for physical phenomena such as X-ray attenuation and scattering
• Improve data quality: Enhance signal-to-noise ratio, reduce artifacts
• Prepare for reconstruction: Convert raw signals to projection values suitable for reconstruction algorithms

💡 Importance of Preprocessing

The quality of preprocessing directly affects the quality of the final image. Even with the most advanced reconstruction algorithms, improper preprocessing cannot produce high-quality images. This is why modern CT scanners invest significant engineering and algorithmic resources in preprocessing.

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CT Detector and Photon Acquisition

Basic Structure of Detectors

The detector in a modern CT scanner is a precision electronic device array whose main function is to capture X-ray photons and convert them into measurable electrical signals.

Geometric characteristics of detector arrays:

• Arrangement: Detector elements arranged in one-dimensional or two-dimensional arrays
• Geometric configuration:
  • Fan-beam geometry: Single-row detector, X-ray source and detector form a fan shape
  • Cone-beam geometry: Multi-row detector, X-ray source and detector form a cone
• Element size: Modern CT detector elements typically range from 0.5-1.0 mm
• Array density: Up to 1000+ elements arranged in a row

💡 Advantages of Multi-row Detectors

The emergence of multi-row detectors (Multi-slice detectors) was a major advance in CT technology. It allows acquisition of multi-layer data in a single rotation, greatly improving scanning speed and temporal resolution. Modern CT scanners typically feature 64-320 row detectors.

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Scintillator Detectors

Working Principle: X-ray photons excite scintillator material, producing visible light; photomultiplier tubes (PMT) or photodiodes (PD) convert visible light into electrical signals.

Photon Conversion Process:

Common Scintillator Materials:

• CsI (Cesium Iodide): High luminescence efficiency, widely used in medical imaging
• BGO (Bismuth Germanate): High density, suitable for high-energy X-rays
• YSO (Yttrium Silicate): Fast response, suitable for applications requiring high temporal resolution

Performance Indicators:

• Energy Resolution: Approximately 15-20% (relatively low)
• Temporal Resolution: Approximately 100-200 ns
• Optical Crosstalk: Optical interference between adjacent detector elements

⚠️ Limitations of Scintillator Detectors

The main disadvantage of scintillator detectors is their low energy resolution, which cannot distinguish photons of different energies. This limits their use in energy-related applications (such as photon-counting CT, dual-energy CT). Additionally, optical crosstalk reduces spatial resolution.

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Direct Conversion Detectors

Working Principle: X-ray photons directly generate electron-hole pairs in semiconductor material, without an intermediate light conversion process.

Photon Conversion Process:

Common Semiconductor Materials:

• CdTe (Cadmium Telluride): High atomic number, high absorption efficiency
• CdZnTe (Cadmium Zinc Telluride): Improved CdTe, better energy resolution

Performance Indicators:

• Energy Resolution: Approximately 5-10% (significantly improved)
• Temporal Resolution: Approximately 10-50 ns (faster)
• Photon Counting Capability: Ability to count individual photons and measure their energy

💡 Advantages of Direct Conversion Detectors

The high energy resolution of direct conversion detectors enables photon counting and energy-resolved imaging. This lays the foundation for advanced technologies such as dual-energy CT and photon-counting CT. Although more expensive, their performance advantages make them standard in high-end CT scanners.

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Photon Counting Detectors

Working Principle: Capable of counting individual photons and measuring their energy, representing the latest generation of detector technology.

Key Characteristics:

• Single Photon Counting: Each photon is individually counted and measured
• Energy Resolution: Up to 1-2%
• No Noise Amplification: Better noise characteristics compared to traditional detectors
• Energy Classification: Can classify and acquire based on photon energy

Clinical Application Prospects:

• Lower radiation dose
• Higher contrast resolution
• Energy-dependent diagnostic information

💡 Current Status of Photon Counting CT

Photon counting CT is at the forefront of CT technology. Major manufacturers like Siemens and GE have already released commercial photon counting CT systems. This technology is expected to become standard in high-end CT within 5-10 years.

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Detector Type Comparison

Feature	Scintillator Detector	Direct Conversion Detector	Photon Counting Detector
Energy Resolution	15-20%	5-10%	1-2%
Temporal Resolution	100-200 ns	10-50 ns	<10 ns
Photon Counting Capability	❌ No	⚠️ Partial	✅ Yes
Cost	Low	Medium	High
Clinical Application	Widespread	High-end systems	Cutting-edge research
Main Advantage	Mature and reliable	High resolution	Optimal performance
Main Disadvantage	Low resolution	High cost	Extremely high cost

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Photon Counting to Projection Value Conversion

Physical Foundation of Lambert-Beer Law

The physical foundation of CT imaging is the attenuation law of X-rays in matter, described by the Lambert-Beer Law.

X-ray Attenuation Mechanisms:

When X-rays pass through matter, photons are absorbed or scattered through the following mechanisms:

• Photoelectric Effect: Photon is completely absorbed by an atom, releasing inner-shell electrons
• Compton Scattering: Photon collides with outer-shell electrons, changing direction and energy
• Pair Production: High-energy photon converts to electron-positron pair near nucleus

Mathematical Expression of Lambert-Beer Law:

$$I=I_0{e}^{-\mu x}$$

Where:

• $I_0$ is the incident X-ray intensity
• $I$ is the transmitted X-ray intensity
• $\mu$ is the linear attenuation coefficient (unit: cm⁻¹)
• $x$ is the material thickness (unit: cm)

Physical Meaning:

• Attenuation is exponential, not linear
• Linear attenuation coefficient $\mu$ depends on the atomic number of the material and X-ray energy
• Materials with high atomic number (such as bone, metal) have larger $\mu$ values and stronger attenuation

⚠️ Complexity of Polychromatic X-rays

Actual medical CT uses polychromatic X-rays (energy spectrum typically ranges from 20-140 keV). Different energy photons have different attenuation coefficients, leading to beam hardening effects that require special correction methods.

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Industry-Standard Correction Workflow

Overview of Correction Workflow

From raw detector signals to corrected projection values, a series of standardized correction steps are required. The order of these steps is important, as some corrections depend on the results of previous steps.

Complete Correction Workflow:

💡 Importance of Correction Order

The order of correction steps is not arbitrary. Generally, system-level corrections (such as dark current, gain) should be performed first, followed by physical effect corrections (such as beam hardening, scatter). Incorrect order may result in poor correction or introduce new artifacts.

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Dark Current Correction

Principle: Detectors produce background signals even without X-ray irradiation, called dark current. Dark current is a systematic background signal composed of thermal noise, leakage current, and electronic noise, and must be eliminated before subsequent corrections.

Sources of Dark Current:

• Thermal Noise: Random signals from thermal motion of detector elements, exponentially related to temperature
• Leakage Current: Leakage current of semiconductor material, related to bias voltage and temperature
• Electronic Noise: Noise from readout circuits, including amplifier noise and ADC noise

Detailed Correction Algorithm:

Step 1: Acquire Dark Current Reference Data

• Turn off X-ray source
• Maintain detector bias voltage and readout parameters same as actual scanning
• Acquire $N_{\text{dark}}$ frames of data (typically 10-100 frames)
• Acquire signals from all detector elements for each frame

Step 2: Calculate Dark Current Average

$$I_{\text{dark},i}=\frac{1}{N_{\text{dark}}}\sum _{k=1}^{N_{\text{dark}}}{I}_{\text{dark},i}^{(k)}$$

Where $i$ is the detector element index, $k$ is the frame index.

Step 3: Calculate Dark Current Standard Deviation (for quality control)

$${\sigma }_{\text{dark},i}=\sqrt{\frac{1}{N_{\text{dark}}}\sum _{k=1}^{N_{\text{dark}}}(I_{\text{dark},i}^{(k)}-I_{\text{dark},i}{)}^2}$$

Step 4: Apply Correction

$$I_{\text{corrected},i}=I_{\text{measured},i}-I_{\text{dark},i}$$

Apply this correction to all detector elements and all projection angles.

Quality Control Checks:

• Dark current values should be uniformly distributed, standard deviation ${\sigma }_{\text{dark},i}$ should be less than 1% of average
• If a detector element has abnormally high dark current, it may indicate element failure
• Dark current varies with temperature; for every 10°C increase, dark current approximately doubles

Practical Application:

• Pre-scan Correction: Acquire new dark current reference data before each scan
• Temperature Compensation: For long scans, may need to periodically update dark current reference
• Real-time Monitoring: Some high-end systems support real-time monitoring of dark current changes during scanning

⚠️ Importance of Dark Current Correction

Dark current correction must be the first correction step, as subsequent gain correction and projection value calculation depend on accurate dark current elimination. Incomplete dark current correction leads to systematic CT value deviations.

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Gain Correction

Principle: Different detector elements have varying sensitivity due to manufacturing process variations, electronic differences, and aging effects. Gain correction establishes sensitivity coefficients for each detector element to uniformize the response of all elements.

Sources of Gain Non-uniformity:

• Manufacturing Process Variations: Microscopic differences in detector element manufacturing (±5-10%)
• Electronic Differences: Readout circuit gain variations, including amplifier gain and bias
• Aging Effects: Performance degradation from long-term use, especially at high count rates
• Temperature Drift: Sensitivity changes from detector temperature variations

Detailed Correction Algorithm:

Step 1: Acquire Flat-Field Reference Data

• Acquire data under uniform X-ray irradiation (no scanned object)
• Use same X-ray parameters as actual scanning (kVp, mA)
• Acquire $N_{\text{flat}}$ frames of data (typically 10-100 frames)
• Ensure count rate is high enough (>10⁶ counts/second) for good statistics

Step 2: Calculate Flat-Field Average

$$I_{\text{flat},i}=\frac{1}{N_{\text{flat}}}\sum _{k=1}^{N_{\text{flat}}}{I}_{\text{flat},i}^{(k)}$$

Step 3: Calculate Reference Intensity

$$I_{\text{ref}}=\frac{1}{N_{\text{det}}}\sum _{i=1}^{N_{\text{det}}}{I}_{\text{flat},i}$$

Where $N_{\text{det}}$ is the total number of detector elements.

Step 4: Calculate Gain Coefficients

$$G_i=\frac{I_{\text{ref}}}{I_{\text{flat},i}}$$

This coefficient represents the deviation of the $i$-th detector element relative to average sensitivity.

Step 5: Apply Correction

$$I_{\text{corrected},i}=(I_{\text{measured},i}-I_{\text{dark},i})\times G_i$$

Note: Gain correction must be performed after dark current correction.

Quality Control Checks:

• Gain coefficients should be in the range 0.9-1.1 (±10%)
• If an element's gain coefficient deviates too much (>±20%), it may indicate element failure
• Standard deviation of gain coefficients should be < 5%

Practical Application Parameters:

• Correction Frequency: Perform complete gain calibration weekly or monthly
• Quick Calibration: Some systems support quick gain calibration (<1 minute)
• Dynamic Correction: High-end systems can adjust gain coefficients in real-time during scanning

💡 Importance of Gain Correction

Gain correction directly affects the uniformity of projection data. Incomplete gain correction leads to ring artifacts and streak artifacts. Modern CT scanners typically perform quality checks after gain correction to ensure all detector elements' responses are within acceptable ranges.

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Air Calibration

Principle: Acquire reference data without any scanned object for use in subsequent projection value calculation.

Purpose of Air Calibration:

• Obtain reference intensity $I_0$ for calculating projection values $p=-\ln (I/I_0)$
• Eliminate X-ray source intensity fluctuations
• Establish baseline for projection values

Air Calibration Acquisition Conditions:

• No objects in the scan field of view
• X-ray parameters (kVp, mA) same as actual scanning
• Acquire sufficient projections to obtain stable average

Practical Application:

• Air calibration typically performed before each scan
• Some systems support continuous air calibration
• Stability of air calibration directly affects accuracy of projection values

⚠️ Stability Requirements for Air Calibration

Air calibration data must be stable and reliable. Improper air calibration leads to systematic deviations in projection values, which in turn affects CT value accuracy in reconstructed images.

Beam Hardening Correction

Principle: X-rays are polychromatic; low-energy photons are preferentially absorbed, causing the average energy of the beam to increase ("harden"). This spectral change causes the effective attenuation coefficient to vary nonlinearly with material thickness, violating the linear assumption of Lambert-Beer Law.

Physical Process of Beam Hardening:

1. Initial Energy Spectrum: Polychromatic spectrum produced by X-ray source (typically 20-140 keV)
2. Selective Absorption: Low-energy photons preferentially absorbed, high-energy photons preferentially transmitted
3. Spectrum Change: Average energy of transmitted X-rays increases, spectrum becomes "harder"
4. Attenuation Coefficient Change: Effective attenuation coefficient ${\mu }_{\text{eff}}$ changes with spectrum change, no longer constant

Clinical Manifestations of Beam Hardening:

• Cupping Artifact: Center of uniform object has lower CT value, edges higher, forming "cup" shape
• Dark Band Artifact: Dark band appears between two high-density objects (e.g., bones)
• Streaking Artifact: Streaks radiating outward from high-density objects

Two Common Industry Correction Methods:

1. Water Beam Hardening Correction

Applicable Scenarios: When scanned object is mainly composed of water or water-like material (soft tissue, cerebrospinal fluid)

Physical Principle:

Water is the main component of human body (approximately 60-70%). In water, X-ray attenuation is mainly determined by photoelectric effect and Compton scattering. By establishing the spectrum-attenuation relationship for a single material (water), polychromatic X-rays can be linearized.

Correction Method:

1. Establish water attenuation curve:

   • Scan water of different thicknesses
   • Measure relationship between projection value $p_{\text{measured}}$ and water thickness $x$
   • Due to polychromatic effects, relationship is nonlinear: $p_{\text{measured}}=f(x)$ (nonlinear)

2. Polynomial fitting:

   $$p_{\text{measured}}=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\dots$$

3. Invert to obtain linearization function:

   $$x=g(p_{\text{measured}})$$

4. Apply correction:

   $$p_{\text{corrected}}={\mu }_{\text{water}}\cdot g(p_{\text{measured}})$$

   Where ${\mu }_{\text{water}}$ is the linear attenuation coefficient of water.

Practical Application Parameters:

• Correction Polynomial Order: Typically 2-3
• Correction Coefficients: Determined by X-ray tube voltage (kVp) and filter type
• Update Frequency: Perform calibration weekly or monthly

💡 Advantages of Water Beam Hardening Correction

Water beam hardening correction is the simplest and most stable method because water has uniform composition and well-defined attenuation characteristics. It is applicable to most clinical CT scans, especially head and abdominal imaging.

2. Bone Beam Hardening Correction

Applicable Scenarios: When scanned object contains large amounts of bone or high-density material (pelvis, spine, extremities)

Physical Principle:

X-ray attenuation coefficient of bone differs from water, mainly because bone contains high atomic number elements like calcium and phosphorus. In bone, low-energy photon attenuation is stronger, leading to more pronounced spectrum hardening effects. Pure water beam hardening correction results in underestimated CT values in bone regions.

Correction Method:

1. Establish two-material model:

   • Assume scanned object composed of water and bone
   • Scan water and bone separately, establish respective attenuation curves

2. Material decomposition: For mixed material, projection value is:

   $$p_{\text{measured}}={\int }_{\text{ray}}[{\mu }_{\text{water}}(E)\cdot f_{\text{water}}+{\mu }_{\text{bone}}(E)\cdot f_{\text{bone}}]dE$$

   Where $f_{\text{water}}$ and $f_{\text{bone}}$ are volume fractions of water and bone along the ray.

3. Iterative correction:

   • Initial assumption: all water
   • Reconstruct image, identify bone regions
   • Apply bone beam hardening correction based on bone regions
   • Repeat until convergence

4. Apply correction:

   $$p_{\text{corrected}}=p_{\text{measured}}-\mathrm{\Delta }{p}_{\text{bone}}$$

   Where $\mathrm{\Delta }{p}_{\text{bone}}$ is projection value deviation caused by bone beam hardening.

Practical Application Parameters:

• Bone Density Assumption: Typically 1.85 g/cm³ (cortical bone)
• Correction Strength: Can be adjusted based on scan region (head 0.5-1.0, spine 1.0-1.5)
• Computational Complexity: 3-5 times higher than water beam hardening

⚠️ Challenges of Bone Beam Hardening Correction

Bone beam hardening correction requires accurate identification of bone regions, which is difficult in low-contrast situations or with metal implants. Overcorrection may result in overestimated CT values in bone regions, even introducing new artifacts.

Water vs. Bone Beam Hardening Correction Comparison

Feature	Water Beam Hardening	Bone Beam Hardening
Applicable Scenarios	Soft tissue dominant	Bone dominant
Computational Complexity	Low	High
Correction Accuracy	Moderate	High
Soft Tissue CT Value Accuracy	High	Moderate
Bone CT Value Accuracy	Moderate	High
Computation Time	<100 ms	500-1000 ms
Common Applications	Head, abdomen, chest	Spine, pelvis, extremities
Clinical Recommendation	Standard approach	Special needs

Correction Workflow Selection:

💡 Importance of Beam Hardening Correction

Beam hardening is one of the most common artifacts in CT imaging. Effective beam hardening correction is essential for obtaining accurate CT values and high-quality images. Modern CT scanners typically automatically select appropriate correction methods based on scan region.

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Scatter Correction

Principle: X-rays scattered within the patient are mistakenly counted by the detector as directly transmitted photons, causing overestimated projection values. This scattered signal reduces image contrast, increases noise, and produces various artifacts.

Physical Process of Scattering:

1. Initial Photons: Photons emitted directly from X-ray source
2. Compton Scattering: Photons collide with electrons in patient, changing direction and energy
3. Scattered Photons Reach Detector: Scattered photons reach detector from various directions
4. False Signal: Scattered photons counted as directly transmitted photons

Impact of Scattering on Projection Values:

• Overestimated Projection Values: $p_{\text{measured}}=p_{\text{true}}+p_{\text{scatter}}$
• Reduced Contrast: Scattering increases background signal, reducing SNR
• CT Value Deviation: Reconstructed image CT values inaccurate, especially in low-density regions
• Artifact Generation: Non-uniform scattering leads to streak and cupping artifacts

Scatter Correction Methods:

Hardware Method: Anti-scatter Grid

Principle: Use lead grid to absorb scattered photons, allowing only direct photons to pass

Structure:

• Lead strip spacing: Typically 1-2 mm
• Lead strip thickness: 0.1-0.2 mm
• Grid Ratio: Typically 12:1 to 16:1

Performance Indicators:

• Scatter Elimination Rate: 60-80%
• Primary Photon Transmission: 70-85%
• Grid Factor: 1.2-1.5 (need to increase exposure to compensate for absorption)

Advantages:

• Simple and reliable
• Real-time effective
• No complex algorithms needed

Disadvantages:

• Increases patient dose (grid factor)
• May produce grid line artifacts
• Limited effectiveness for large field-of-view imaging

Software Method: Measurement-Based Scatter Correction

Principle: Estimate or measure scatter distribution, subtract scatter contribution from projection values

Algorithm Steps:

Step 1: Scatter Distribution Estimation

• Method 1: Use small aperture collimator to measure scatter distribution
• Method 2: Monte Carlo simulation based on patient geometry and X-ray spectrum
• Method 3: Use machine learning model to predict scatter distribution

Step 2: Calculate Scatter Projection Values

$$p_{\text{scatter}}(\theta )=f(\text{patient geometry},\text{X-ray spectrum},\theta )$$

Where $\theta$ is the projection angle.

Step 3: Scatter Correction

$$p_{\text{corrected}}(\theta )=p_{\text{measured}}(\theta )-p_{\text{scatter}}(\theta )$$

Step 4: Quality Control

• Check if corrected projection values are in reasonable range
• Overcorrection may result in negative projection values

Algorithm Complexity:

• Monte Carlo simulation: 1-10 seconds
• Machine learning prediction: <100 ms

Iterative Method: Reconstruction-Based Scatter Correction

Principle: Iteratively improve scatter correction through reconstruction and scatter estimation

Algorithm Steps:

Initialization: $k=0$, $p_{\text{corrected}}^{(0)}=p_{\text{measured}}$

Iteration Loop:

1. Reconstruct Image: $I^{(k)}=\text{Reconstruct}(p_{\text{corrected}}^{(k)})$

2. Forward Projection: $p_{\text{forward}}^{(k)}=\text{ForwardProject}(I^{(k)})$

3. Scatter Estimation: $p_{\text{scatter}}^{(k)}=p_{\text{measured}}-p_{\text{forward}}^{(k)}$

4. Scatter Correction: $p_{\text{corrected}}^{(k+1)}=p_{\text{measured}}-\alpha \cdot p_{\text{scatter}}^{(k)}$

   Where $\alpha$ is correction factor (typically 0.5-1.0)

5. Convergence Check: If $||p_{\text{corrected}}^{(k+1)}-p_{\text{corrected}}^{(k)}||<ϵ$, stop; otherwise $k=k+1$, return to step 1

Number of Iterations: Typically 2-5 iterations

Algorithm Complexity: $O(N_{\text{iter}}\times T_{\text{recon}})$, typically 5-20 seconds

Practical Application:

• Medical CT: Mainly uses combination of anti-scatter grid + software correction
• Cone-beam CT: Due to large field-of-view, scatter problem more severe, usually requires more complex correction
• Low-dose CT: Scatter relative to signal ratio higher, requires finer correction

Scatter Correction Method Comparison:

Method	Scatter Elimination Rate	Computation Time	Complexity	Application Scenario
Anti-scatter Grid	60-80%	0 ms	Low	Standard medical CT
Measurement-Based	70-90%	100-1000 ms	Medium	High-precision applications
Monte Carlo Simulation	80-95%	1-10 s	High	Research and special applications
Iterative Method	85-95%	5-20 s	High	Low-dose and special applications

⚠️ Challenges of Scatter Correction

Scatter correction is one of the most challenging tasks in CT preprocessing. Scatter distribution is complex, affected by many factors including patient size and scanning parameters. Overcorrection may degrade image quality, even introducing new artifacts.

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Ring Artifact Correction

Principle: Detector element failure or calibration errors cause concentric rings centered at the scan center to appear in reconstructed images. This is because each detector element rotates once during scanning, and its failure forms a ring in the reconstructed image.

Sources of Ring Artifacts:

• Detector Failure: A detector element has abnormal response or complete failure
• Calibration Error: Inaccurate gain or dark current calibration, causing signal deviation in that element
• Electronic Problems: Readout circuit failure or excessive noise
• Non-uniform Aging: Some detector elements age faster than others

Characteristics of Ring Artifacts:

• Fixed Position: Concentric rings centered at scan center, same position on all slices
• Density Appearance: Usually appears as high-density ring (failed element signal too high) or low-density ring (failed element signal too low)
• Width: Typically 1-3 pixels wide

Detailed Correction Algorithm:

Sinogram Domain Correction

Advantage: Direct correction on raw data, best effect

Algorithm Steps:

Step 1: Anomaly Detection

• For each detector element $i$, calculate projection value sequence for all projection angles: $\{p_i({\theta }_1),p_i({\theta }_2),\dots ,p_i({\theta }_N)\}$
• Calculate median of this sequence: $p_i^{\text{median}}$
• Calculate deviation: $\mathrm{\Delta }{p}_i(\theta )=p_i(\theta )-p_i^{\text{median}}$
• If $|\mathrm{\Delta }{p}_i(\theta )|>T\cdot {\sigma }_i$ (where $T$ is threshold, typically 3-5, ${\sigma }_i$ is standard deviation), mark as anomaly

Step 2: Anomaly Repair

• For detected anomalies, use data from adjacent detector elements for interpolation:

$$p_i^{\text{corrected}}(\theta )=\frac{p_{i-1}(\theta )+p_{i+1}(\theta )}{2}$$

• Or use more complex weighted interpolation:

$$p_i^{\text{corrected}}(\theta )=w_1\cdot p_{i-1}(\theta )+w_2\cdot p_{i+1}(\theta )$$

Where weights $w_1,w_2$ are determined based on reliability of adjacent elements.

Step 3: Smoothing

• Apply light smoothing to repaired data to avoid introducing new artifacts
• Use median filtering or Gaussian filtering

Algorithm Complexity: $O(N_{\text{det}}\times N_{\text{angle}})$, typically <100 ms

Image Domain Correction

Advantage: Can handle complex cases undetectable in projection domain

Algorithm Steps:

Step 1: Ring Artifact Detection

• In reconstructed image, calculate intensity changes in radial direction
• Detect periodic intensity fluctuations (corresponding to ring artifacts)

Step 2: Ring Artifact Separation

• Use Fourier or wavelet transform to separate ring artifacts
• Ring artifacts mainly concentrated in low-frequency components

Step 3: Artifact Removal

• Subtract separated ring artifacts from image
• Or use advanced methods like non-local means filtering

Algorithm Complexity: $O(N_x\times N_y\times \log (N_x))$, typically 500-1000 ms

Practical Application:

• Preventive Calibration: Regularly perform gain and dark current calibration to prevent ring artifacts
• Automatic Detection: Modern CT scanners equipped with automatic ring artifact detection system
• Real-time Repair: Some systems support real-time detection and repair during scanning
• Offline Processing: For existing data, offline ring artifact correction can be performed

Quality Control Indicators:

• Ring Artifact Intensity: Should be < 5 HU (Hounsfield Units)
• Detection Sensitivity: Should detect > 10% detector failures
• Repair Accuracy: Repaired image quality should be comparable to normal scan

⚠️ Limitations of Ring Artifact Correction

Although ring artifact correction can significantly improve image quality, it cannot completely eliminate all artifacts. For severe detector failures (such as multiple adjacent elements failing), more complex algorithms or hardware repair may be needed.

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Other Correction Steps

Depending on specific application scenarios, other corrections may be needed:

Metal Artifact Reduction (MAR):

• Performed when patient has metal implants
• Usually performed in projection domain
• Uses interpolation or iterative methods

Motion Correction:

• Performed when patient moves
• Uses motion estimation and compensation algorithms
• Particularly important for cardiac and abdominal imaging

Dose Optimization-Related Corrections:

• Noise correction for low-dose scans
• Adaptive filtering
• Preprocessing for iterative reconstruction

Complete Data Flow

The complete data flow from raw signal to corrected projection values can be summarized as follows:

💡 Importance of Preprocessing

Preprocessing is the most easily overlooked but most important step in CT imaging. High-quality preprocessing is a necessary condition for obtaining high-quality reconstructed images. Investing time in understanding and optimizing the preprocessing workflow will yield significant long-term benefits.

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💡 Next Steps

After complete preprocessing, raw detector signals have been converted to corrected projection values (Sinogram). These projection values can now enter the reconstruction stage.

The next chapter (2.2) will introduce MRI k-space data preprocessing, showing how preprocessing workflows for different imaging modalities each have their own characteristics. Chapter 3 will deeply introduce CT reconstruction algorithms, explaining how to reconstruct the final CT image from Sinogram.