Pixel-based uncertainty metrics are commonly used in image processing, machine learning, and computer vision to quantify the uncertainty in pixel-level predictions. These metrics are especially relevant in tasks like semantic segmentation, medical image analysis, and generative modeling. Below are some key uncertainty metrics used for pixel-based analysis:

Entropy-Based Uncertainty

Entropy measures the uncertainty of a probability distribution at each pixel. It is particularly useful for classification tasks where the model outputs a probability distribution over multiple classes.

$$ H(x)=-\sum_{i=1}^C p_i(x) \log p_i(x) $$

where:

• $p_i(x)$ is the probability of class $i$ at pixel $x$,
• $C$ is the number of classes.

Higher entropy values indicate more uncertain predictions. Applications: Semantic segmentation, medical imaging (e.g., tumor detection).

Variance-Based Uncertainty (Monte Carlo Dropout)

For models using Monte Carlo Dropout (MC-Dropout), multiple stochastic forward passes are used to compute uncertainty. The variance of the predicted probabilities across runs is used as an uncertainty measure:

$$ \operatorname{Var}(x)=\frac{1}{N} \sum_{i=1}^N\left(p_i(x)-\bar{p}(x)\right)^2 $$

where:

• $p_i(x)$ is the probability from the $i$-th stochastic pass,
• $\bar{p}(x)$ is the mean prediction across all passes,
• $N$ is the number of stochastic passes.

Applications: Bayesian deep learning, image segmentation under uncertainty.

Mutual Information (Epistemic Uncertainty)

Mutual information measures epistemic uncertainty (model uncertainty) by evaluating the disagreement between different stochastic predictions:

$$ I (x)=H(\bar{p}(x))-\frac{1}{N} \sum_{i=1}^N H\left(p_i(x)\right) $$

where: