To evaluate the accuracy of the predicted depth maps against the “ground truth” (the actual measured distance), the InfiniDepth paper utilizes a standard set of error and accuracy metrics common in monocular depth estimation research.

These metrics compare the predicted depth $d$ with the ground-truth depth $d^*$ for $N$ total pixels.

1. Error Metrics (Lower is Better)

These metrics quantify the average distance or deviation between the prediction and the truth.

• Absolute Relative Error (Abs Rel): Calculates the average percentage error relative to the ground truth depth.

  $$\text{Abs Rel} = \frac{1}{N} \sum \frac{|d - d^*|}{d^*}$$

• Squared Relative Error (Sq Rel): Similar to Abs Rel but squares the error term to penalize larger outliers more heavily.

  $$\text{Sq Rel} = \frac{1}{N} \sum \frac{\|d - d^*\|^2}{d^*}$$

• Root Mean Squared Error (RMSE):

  Measures the average magnitude of the error in the same units as the depth (e.g., meters). It is very sensitive to large errors.

  $$\text{RMSE} = \sqrt{\frac{1}{N} \sum (d - d^*)^2}$$

• RMSE log:

  Calculates the RMSE in log space, which helps account for the fact that depth errors naturally increase as objects get farther away.

  $$\text{RMSE log} = \sqrt{\frac{1}{N} \sum (\log d - \log d^*)^2}$$

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2. Accuracy Metrics (Higher is Better)

These metrics, often denoted as $\delta < 1.25^n$, measure the percentage of pixels where the ratio of the predicted depth to the ground truth (or vice versa) is within a certain threshold.

The threshold is defined as:

$$\max \left( \frac{d}{d^*}, \frac{d^*}{d} \right) = \delta < 1.25^n$$

The paper typically reports three levels of strictness:

• $\delta_1$: Threshold is $1.25$

• $\delta_2$: Threshold is $1.25^2 = 1.5625$

• $\delta_3$: Threshold is $1.25^3 \approx 1.95$

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3. Log10 Error

This is the mean absolute error calculated using base-10 logarithms, providing a different perspective on the scale-invariant accuracy of the model.

$$\text{Log10} = \frac{1}{N} \sum |\log_{10} d - \log_{10} d^*|$$

4. Why these matter for InfiniDepth

Because InfiniDepth focuses on fine-grained geometry, the authors look closely at how these metrics behave at high resolutions. Standard metrics can sometimes “hide” errors in thin structures (like a power line) because those structures only occupy a few pixels. By evaluating on their new Synth4K dataset, they demonstrate that their model maintains high $\delta_1$ accuracy even when rendering at 4K, where traditional models often see a significant performance drop.

Since you’ve worked with Structure from Motion (SfM), you’ll know that high RMSE can lead to “warped” 3D reconstructions, while poor $\delta_1$ accuracy usually results in “noisy” or “floating” artifacts in the point cloud.