Roughness Index Distance

---
name: roughness-index-distance
description: Evaluates surface irregularities and topological consistency between predicted and ground-truth 3D medical segmentation masks. It quantifies local surface roughness, relative roughness differences, and average surface distance to detect spikes, holes, and smoothing artifacts. Use when the user has predictions and gold and needs to compute Roughness Index (RI).
metadata:
  skill_kind: metric
  source_arxiv: 2103.12350
  bibtex_key: rathour2021roughness
  confidence: high
---

# roughness-index-distance

> Roughness Index and Roughness Distance for Benchmarking Medical Segmentation — Rathour et al. (2021) (arXiv:2103.12350, 2021)

## What this evaluates

Evaluates surface irregularities and topological consistency between predicted and ground-truth 3D medical segmentation masks. It quantifies local surface roughness, relative roughness differences, and average surface distance to detect spikes, holes, and smoothing artifacts.

## Datasets

- (no dataset; pure metric skill)

## Metrics

- `Roughness Index (RI)` **(primary)** — range: [0, ∞)
  - RI = (1/M) * Σ_{w=1 to M} [ (1/N) * Σ_{i=1 to N} |ζ_i - Mean(ζ_window)| ], where ζ is the L2 distance from a surface point to the center of gravity, and M is the number of fixed-size surface windows.
- `Roughness Ratio (RR)` — range: [0, ∞)
  - RR = |RI_P - RI_G| / RI_G, measuring the relative roughness difference between predicted and ground-truth segmentations.
- `Average Roughness Distance (ARD)` — range: [0, ∞)
  - ARD = Mean(|ζ̂_m|), where ζ̂_m is the element-wise difference between the predicted and ground-truth distance matrices from the center of gravity.

## Input / output format

**Input**: Two 3D binary segmentation masks: predicted (P) and ground-truth (G), represented as voxel grids.

**Output**: Scalar values for Roughness Index (RI), Roughness Ratio (RR), and Average Roughness Distance (ARD).

## Scoring recipe

```python
def compute_roughness_metrics(P, G):
    # 1. Compute center of gravity C0 for the surface
    C0 = mean([v for v in P if v in surface])
    # 2. Compute distance matrices from C0 for P and G
    zeta_P = array([L2(v, C0) for v in P_surface])
    zeta_G = array([L2(v, C0) for v in G_surface])
    # 3. Compute RI for P and G using fixed windows
    RI_P = mean([mean([abs(d - mean(window_d)) for d in window_d]) for window in windows])
    RI_G = mean([mean([abs(d - mean(window_d)) for d in window_d]) for window in windows])
    # 4. Compute ARD
    zeta_hat = zeta_P - zeta_G
    ARD = mean(abs(zeta_hat))
    # 5. Compute RR
    RR = abs(RI_P - RI_G) / RI_G
    return RI_P, RI_G, RR, ARD
```

## Common pitfalls

- Using a fixed laser-plane height coordinate instead of the center-of-gravity distance, which breaks the closed-contour assumption for medical masks.
- Failing to normalize RI by the number of windows M, causing the metric to scale with surface resolution or windowing strategy.
- Division by zero in RR when the ground-truth segmentation is perfectly smooth (RI_G = 0).

## Evidence (verbatim from paper)

> The Roughness Index (RI) in 3D can be calculated by dividing the segmentation surface S into small surface element ∂S^w of a fixed window size w, and then calculating the average deviation of ζ from the mean ζ_Mean for all surface voxels in the surface element ∂S^w as illustrated in Fig:[9(b)]... We also propose Average Roughness Distance (ARD) which as the name suggest is the average surface/roughness distance between two objects as shown in Eq:[23]... ARD = Mean(|ζ̂_m|)

## Citation

```bibtex
@misc{rathour2021roughness,
  title={Roughness Index and Roughness Distance for Benchmarking Medical Segmentation},
  author={Rathour et al. (2021)},
  year={2021},
  note={arXiv:2103.12350}
}
```

- arXiv: 2103.12350