# Ideal Observer CT Resolution Calculator

> Interactive calculator for the minimum detectable feature size and spatial-frequency limit of a CT scanner, derived from the projection-domain ideal observer (Rose criterion, SKE/BKE). The result is reconstruction-independent — it is a property of photon statistics and scan geometry. Supports both the at-isocentre closed-form and an off-axis fan-beam extension that maps d_min(P) across the phantom.

Authors: Falk L. Wiegmann & Nancy L. Ford, University of British Columbia, 2026.

## What this is for

Given a CT scanner's photon flux and geometry, this calculator answers: *what is the smallest disc of a given contrast that the scanner can theoretically detect?* The model is non-prewhitening, signal-known-exactly (SKE), background-known-exactly (BKE), and assumes Poisson photon statistics. The result is an **upper bound** on detectability — every real reconstruction (FDK, SIRT, neural) performs worse.

The page exposes two computations:

1. **At isocentre (closed-form)** — a single d_min value assuming the feature is at the rotation centre and projections cover ≥ 180°.
2. **Off-axis / partial-arc map** — a 2D heatmap (and matching 3D surface) of d_min(P) across the phantom for arbitrary point-source distance D and projection arc.

## Quick formulas (for agents who just want to compute)

Shared inputs:
- `N0` — photons/pixel in the unattenuated beam
- `mu_bg` — background linear attenuation coefficient (mm⁻¹)
- `L` — background path length through phantom diameter (mm)
- `N_theta` — total number of projection angles
- `delta_a_obj` — object-plane pixel pitch at isocentre (mm)
- `delta_mu` — linear attenuation contrast of the feature (mm⁻¹); for HU contrast: `delta_mu = (delta_HU / 1000) * mu_water`
- `d_prime_threshold` — Rose detectability threshold (classical 3; Hsieh et al. 2022 argue 5)

### At isocentre (full-arc, parallel-beam idealisation)

```
N_bg = N0 * exp(-mu_bg * L)
d_min = (3 * d_prime_threshold^2 * delta_a_obj / (2 * N_theta * N_bg * delta_mu^2))^(1/3)   # mm
resolution = 1 / (2 * d_min)                                                                # lp/mm
```

### Off-axis (fan-beam, arbitrary arc)

Extra inputs:
- `R` — phantom radius (mm); typically L/2
- `D` — source-to-isocentre distance (SOD, mm)
- arc covers angles [phi_start, phi_end]

For a feature at P = (x, y), the source at projection angle theta sits at S(theta) = (D*cos theta, D*sin theta). The fan-beam chord through the phantom is

```
d_hat = (P - S(theta)) / |P - S(theta)|
chord(theta, P) = 2 * sqrt( (S . d_hat)^2 - (D^2 - R^2) )
```

(reduces to 2*sqrt(R^2 - r^2 * sin^2(theta - alpha)) as D -> infinity). The local object-plane pixel pitch is

```
delta_a_obj(P, theta) = delta_a_obj * |P - S(theta)| / D
```

Substituting into the d'^2 sum and solving:

```
d_min(P) = ( 3 * d_prime_threshold^2 * delta_a_obj
             / ( 2 * delta_mu^2 * D
                 * sum_theta [ N0 * exp(-mu_bg * chord(theta, P)) / |P - S(theta)| ] ) )^(1/3)
```

where the sum runs over all N_theta projection angles in the scan arc.

## Resources

- [llms-full.txt](llms-full.txt) — full derivation including Step 7 (fan-beam off-axis extension).
- [formulas.json](formulas.json) — typed JSON: inputs, outputs, both formulas, worked example.
- [index.html](index.html) — interactive UI with TIFF upload for N₀ estimation and a radial-resolution heatmap toggle.

## Primary reference

Barrett HH, Myers KJ. *Foundations of Image Science.* Wiley, 2004.

## Other references

- Hsieh SS, Leng S, Yu L, Huber NR, McCollough CH. "A minimum SNR criterion for computed tomography object detection in the projection domain." *Med Phys* 49(8): 4988–4998, 2022.
- Hanson KM. "Detectability in computed tomographic images." *Med Phys* 6(5): 441–451, 1979.
- Gang GJ, Tward DJ, Lee J, Siewerdsen JH. "Anatomical background and generalized detectability in tomosynthesis and cone-beam CT." *Med Phys* 37(5): 1948–1965, 2010.
- Rose A. *Vision: Human and Electronic.* Plenum Press, 1973.