## Principle of sub-pixel localization via charge sharing
Charge-sharing occurs when a single interaction’s deposited charge induces measurable signals on two or more neighboring strips. The relative amplitudes (or collected charges) on adjacent strips encode the lateral position within the strip pitch with resolution finer than the strip pitch. For CZT strip detectors, sub-pixel interpolation uses the measured charge ratios and calibrated charge-spread functions (CSFs) to estimate the interaction coordinate between strip centers.
## Requirements on detector physics and geometry
- Sufficient lateral charge spread: The electron cloud transverse size at the anode (after drift) must be comparable to or larger than the strip pitch fraction to produce measurable sharing. Cloud broadening arises from initial charge-cloud size, carrier diffusion during drift, and Coulomb repulsion for large energy deposits.
- Strip pitch and inter-strip gap design: Pitch must be small enough to allow multi-strip hits for the energies of interest but large enough to keep inter-strip capacitance and noise manageable.
- Weighting-field profile: Strip geometry and dielectric/passivation define the weighting potentials; shallow weighting gradients (wider induced footprints) favor sharing but affect depth dependence—design must balance localization and depth sensitivity.
- Low trapping and uniform drift: Nonuniform trapping or depth-dependent attenuation changes charge fractions unpredictably; reliable sub-pixel algorithms assume predictable depth corrections or use simultaneous DOI estimation.
## Signal and readout prerequisites
- Low-noise, matched front-ends per strip: Accurate amplitude ratios require high SNR; channel noise should be small compared with shared-signal fractions.
- Synchronous digitization and timing alignment: All strips contributing to an event must be sampled and time-stamped with matched timing to correctly associate shared signals and to separate simultaneous events.
- High dynamic range and linearity: Front-end must preserve linearity of charge amplitudes across the anticipated energy range so interpolation remains valid.
- Fast shaping or waveform digitization: Waveforms allow pulse-shape separation of overlapping hits and extraction of drift-time/rise-time for DOI correction.
## Calibration: charge-spread function and mapping
- Empirical CSF measurement: Use a finely collimated beam scanned across the strip pitch to measure mean charge fractions vs. true lateral position at representative depths and energies.
- Depth-dependent CSFs: Because charge sharing often depends on interaction depth (diffusion scales with drift distance), calibrate CSFs as a function of DOI (using cathode/anode ratio or drift time) or build a 2D LUT CSF(x,z).
- Energy dependence: For some ranges, initial cloud size and pulse-splitting behavior change with deposited energy; include energy as a calibration axis if needed.
- Statistical characterization: Measure charge-fraction variance vs. position to model localization uncertainty and to weight measurements optimally.
## Basic interpolation algorithms
- Center-of-gravity (CoG) / charge-weighted centroid: For two- or three-strip clusters, compute x_est = (Σ qi * xi) / (Σ qi), where qi are collected charges and xi are strip center coordinates. Simple and low-compute but biased by weighting-field nonlinearity and depth effects.
- Ratio-based interpolation: For two adjacent strips with charges qL and qR, use a calibrated mapping x = f(qR/(qL+qR)), where f is the inverse CSF measured in calibration. This accounts for nonlinear charge-fraction relationships.
- Maximum-likelihood estimation (MLE): Given measured charges and a parametrized CSF plus noise model, estimate the lateral position that maximizes likelihood; provides statistically optimal unbiased estimates when the model is accurate.
- Template matching / lookup tables: Precompute expected charge vectors for discrete positions (and depths); find best-matching template for each measured cluster (fast with indexed LUTs).
## Advanced approaches accounting for depth and nonidealities
- Joint DOI and lateral estimation: Use cathode/anode ratio or drift-time to estimate depth z, then apply the CSF(z) mapping for lateral interpolation; joint MLE can estimate (x,z) simultaneously from multi-strip amplitudes and pulse-shapes.
- Pulse-shape-assisted sharing: Fit digitized waveforms to depth-dependent templates per strip to separate shared charge arrival times or slow tail contributions, improving fraction estimates especially when trapping or ballistic deficits occur.
- Charge-conservation constraints: Enforce total collected charge consistency (sum over cluster corrected for known charge loss/efficiency vs. z) to reduce bias and detect partial-loss or multi-site events.
- Multi-hit discrimination: Use timing and waveform-shape differences to distinguish true charge sharing from closely spaced multiple interactions that mimic sharing; reject or deconvolve multi-site events.
## Noise, discretization, and uncertainty handling
- Noise propagation: Calculate position uncertainty from noise variances on each channel via error propagation in the chosen estimator (e.g., centroid variance formula) and use it to weight or reject low-confidence events.
- Thresholds and charge truncation: Finite thresholds can truncate small shared fractions producing bias; calibration must account for threshold-induced truncation or implement lower thresholds and baseline subtraction.
- Capacitance and cross-talk correction: Inter-strip capacitive coupling alters apparent charge fractions; deconvolve known cross-talk matrix from measured amplitudes before localization.
## Implementation considerations and real-time processing
- FPGA/ASIC-friendly algorithms: Ratio mapping, small LUTs, and simple centroid calculations are implementable in low-latency hardware for real-time localization; MLE and template fits can be accelerated with lookup-assisted coarse-to-fine searches.
- Clustering and channel routing: Efficient cluster-finding (identify contiguous above-threshold strip groups) and channel synchronization are prerequisites for real-time sub-pixel computation.
- Throughput vs. accuracy trade-off: For high-rate systems, use fast approximate estimators online and apply heavier template/MLE processing offline for selected events or calibration.
## Performance limits and achievable resolutions
- Theoretical limits: Best-case lateral resolution scales with cloud size and SNR; for Gaussian-sharing with sigma s and total charge Q and noise σn, centroid precision ~ s / sqrt(Q/σn^2). Practically, sub-pitch resolutions of order 1/10 to 1/20 of pitch are achievable in optimized systems.
- Dominant degrading factors: Low SNR, insufficient cloud spread, severe depth dependence, charge trapping, cross-talk, and improper calibration reduce achievable sub-pixel accuracy.
- Empirical expectations: With small-pitch (100–500 μm) strip geometries, good SNR, and DOI correction, typical reported sub-pixel resolutions are in the tens of microns to few hundred microns depending on energy and detector thickness.
## Validation and characterization methods
- Micro-collimated scans: Map reconstructed position vs. known beam position to quantify bias, linearity, and resolution across pitch and depth.
- Flat-field and imaging tests: Use patterned masks to evaluate reconstructed image fidelity and modulation-transfer function (MTF) for spatial-frequency response.
- Monte Carlo plus detector-response simulation: Model diffusion, weighting fields, noise, and electronics to predict achievable resolution and to design optimal pitch and front-end specs.
## Design trade-offs and optimization guidelines
- Optimize pitch vs. SNR: Smaller pitch improves intrinsic localization granularity but raises capacitance and noise—balance for target energies.
- Control weighting fields and passivation: Engineering the electrode and passivation stack can tailor the induced footprint to a beneficial shape (broader but predictable).
- Encourage predictable sharing: Slightly larger diffusion (via moderate thickness or optimized bias) can increase useful sharing without excessive multi-strip spread.
- Integrate DOI measurement: Always combine lateral interpolation with depth estimation to remove dominant bias sources.
## Summary
Sub-pixel resolution in CZT strip detectors exploits predictable charge sharing between adjacent strips. Success requires detector geometry and weighting fields that produce measurable sharing, low-noise synchronous readout, depth-aware calibrated charge-spread models, and algorithms ranging from simple centroids and ratio mappings to MLE and template fitting. With rigorous calibration, DOI compensation, and careful electronics/layout design, sub-pitch localization down to a small fraction of the strip pitch is achievable and can be realized in both real-time (LUT/ratio) and high-accuracy offline (template/MLE) implementations.
CdZnTe Association (CdZnTe.com)
https://www.cdznte.com/blog/how-can-sub-pixel-resolution-be-achieved-through-charge-sharing-in-czt-strip-detectors.html
