RankSEG: A Consistent Framework

for Segmentation

 

Ben Dai

The Chinese University of Hong Kong

 

 

Segmentation

Input

output

Input: \(\mathbf{X} \in \mathbb{R}^d\)

Outcome: \(\mathbf{Y} \in \{0,1\}^d\)

Segmentation function:

  • \( \pmb{\delta}: \mathbb{R}^d \to \{0,1\}^d\)
  • \( \pmb{\delta}(\mathbf{X}) = ( \delta_1(\mathbf{X}), \cdots, \delta_d(\mathbf{X}) )^\intercal \)

Predicted segmentation set:

  • \( I(\pmb{\delta}(\mathbf{X})) = \{j: \delta_j(\mathbf{X}) = 1 \}\)

Segmentation

Input

output

Input: \(\mathbf{X} \in \mathbb{R}^d\)

Outcome: \(\mathbf{Y} \in \{0,1\}^d\)

Segmentation function:

  • \( \pmb{\delta}: \mathbb{R}^d \to \{0,1\}^d\)
  • \( \pmb{\delta}(\mathbf{X}) = ( \delta_1(\mathbf{X}), \cdots, \delta_d(\mathbf{X}) )^\intercal \)

Predicted segmentation set:

  • \( I(\pmb{\delta}(\mathbf{X})) = \{j: \delta_j(\mathbf{X}) = 1 \}\)

Segmentation

Input

output

$$ Y_j | \mathbf{X}=\mathbf{x} \sim \text{Bern}\big(p_j(\mathbf{x})\big)$$

$$ p_j(\mathbf{x}) :=  \mathbb{P}(Y_j = 1 | \mathbf{X} = \mathbf{x})$$

Probabilistic model:

The Dice and IoU metrics are introduced and widely used in practice:

Evaluation

IoU

The Dice and IoU metrics are introduced and widely used in practice:

Evaluation

Goal: learn segmentation function \( \pmb{\delta} \) maximizing Dice / IoU

Dice

  • Given training data \( \{\mathbf{x}_i, \mathbf{y}_i\} _{i=1, \cdots, n}\), most existing methods characterize segmentation as a classification problem:

Existing frameworks

  • Given training data \( \{\mathbf{x}_i, \mathbf{y}_i\} _{i=1, \cdots, n}\), most existing methods characterize segmentation as a classification problem:

Existing frameworks

Loss functions

Can we directly maximize the Dice/IoU scores?

Direct Dice/IoU Optimization

Evaluation really matters!

Classification

  • Data: \( \mathbf{X} \in \mathbb{R}^d \to Y \in \{0,1\} \)
  • Decision function: \( \delta(\mathbf{X}): \mathbb{R}^d \to \{0,1\} \)
  • Evaluation:

$$ Acc( \delta) = \mathbb{E}\big( \mathbf{1}( Y = \delta(\mathbf{X}) )\big) $$

  • Optimal Rule:

$$ \delta^* = \argmax_{\delta} \ Acc(\delta) \ \ \to \  \delta^*(\mathbf{x}) = \mathbf{1}( p(\mathbf{x}) \geq 0.5 ) $$

Can we directly maximize the Dice/IoU score?

Classification

  • Data: \( \mathbf{X} \in \mathbb{R}^d \to Y \in \{0,1\} \)
  • Decision function: \( \delta(\mathbf{X}): \mathbb{R}^d \to \{0,1\} \)
  • Evaluation:

$$ Acc( \delta) = \mathbb{E}\big( \mathbf{1}( Y = \delta(\mathbf{X}) )\big) $$

  • Optimal Rule:

$$ \delta^* = \argmax_{\delta} \ F1(\delta) \ \ \to \  \delta^*(\mathbf{x}) = \mathbf{1}( p(\mathbf{x}) \geq p_0 ) $$

$$F1(\delta)$$

where \(p_0 = F1^* / 2 \leq 0.5 \), it is also truncation, but not at 0.5!

Direct Dice/IoU Optimization

Evaluation really matters!

$$ \pmb{\delta}^* = \text{argmax}_{\pmb{\delta}} \ \text{Dice}_\gamma ( \pmb{\delta})$$

Optimal segmentation rule

Direct Dice/IoU Optimization

What form would the Bayes segmentation rule take?

Bayes segmentation rule

Theorem 1 (Dai and Li, 2023). A segmentation rule \(\pmb{\delta}^*\) is a global maximizer of \(\text{Dice}_\gamma(\pmb{\delta})\) if and only if it satisfies that

\( \tau^*(\mathbf{x}) \) is called optimal segmentation volume, defined as

$$ \tau^* = \arg\max_{\tau \in \{0,1,\cdots,d\}} \Big( \sum_{j \in J_\tau(\mathbf{x})}  \mathbb{E} \big( \frac{2p_j(\mathbf{x})}{\tau + \Gamma_{-j}(\mathbf{x}) + \gamma + 1 } \big) + \gamma \mathbb{E} \big( \frac{1}{\tau + \Gamma + \gamma} \big) \Big)  $$

The Dice measure is separable w.r.t. \(j\)

Bayes segmentation rule

Bayes segmentation rule

Bayes segmentation rule

Bayes segmentation rule

Theorem 1 (Dai and Li, 2023). A segmentation rule \(\pmb{\delta}^*\) is a global maximizer of \(\text{Dice}_\gamma(\pmb{\delta})\) if and only if it satisfies that

\( \tau^*(\mathbf{x}) \) is called optimal segmentation volume, defined as

$$ \tau^* = \arg\max_{\tau \in \{0,1,\cdots,d\}} \Big( \sum_{j \in J_\tau(\mathbf{x})}  \mathbb{E} \big( \frac{2p_j(\mathbf{x})}{\tau + \Gamma_{-j}(\mathbf{x}) + \gamma + 1 } \big) + \gamma \mathbb{E} \big( \frac{1}{\tau + \Gamma + \gamma} \big) \Big)  $$

Theorem 1 (Dai and Li, 2023). A segmentation rule \(\pmb{\delta}^*\) is a global maximizer of \(\text{Dice}_\gamma(\pmb{\delta})\) if and only if it satisfies that

\( \tau^*(\mathbf{x}) \) is called optimal segmentation volume, defined as

Obs: both the Bayes segmentation rule \(\pmb{\delta}^*(\mathbf{x})\) and the optimal volume function \(\tau^*(\mathbf{x})\) are achievable when the conditional probability \(\mathbf{p}(\mathbf{x}) = ( p_1(\mathbf{x}), \cdots, p_d(\mathbf{x}) )^\intercal\) is well-estimated

Bayes segmentation rule

$$ \tau^* = \arg\max_{\tau \in \{0,1,\cdots,d\}} \Big( \sum_{j \in J_\tau(\mathbf{x})}  \mathbb{E} \big( \frac{2p_j(\mathbf{x})}{\tau + \Gamma_{-j}(\mathbf{x}) + \gamma + 1 } \big) + \gamma \mathbb{E} \big( \frac{1}{\tau + \Gamma + \gamma} \big) \Big)  $$

Theorem 1 (Dai and Li, 2023). A segmentation rule \(\pmb{\delta}^*\) is a global maximizer of \(\text{Dice}_\gamma(\pmb{\delta})\) if and only if it satisfies that

\( \tau^*(\mathbf{x}) \) is called optimal segmentation volume, defined as

where \(J_\tau(\mathbf{x})\) is the index set of the \(\tau\)-largest probabilities, \(\Gamma(\mathbf{x}) = \sum_{j=1}^d {B}_{j}(\mathbf{x})\), and \( {\Gamma}_{- j}(\mathbf{x}) = \sum_{j' \neq j} {B}_{j'}(\mathbf{x})\) are Poisson-binomial random variables.

RankDice inspired by Thm 1 (plug-in rule)

  1. Ranking the conditional probability \(p_j(\mathbf{x})\)

Plug-in rule

Theorem 1 (Dai and Li, 2023+). A segmentation rule \(\pmb{\delta}^*\) is a global maximizer of \(\text{Dice}_\gamma(\pmb{\delta})\) if and only if it satisfies that

\( \tau^*(\mathbf{x}) \) is called optimal segmentation volume, defined as

RankDice inspired by Thm 1

  1. Ranking the conditional probability \(p_j(\mathbf{x})\)

  2. searching for the optimal volume of the segmented features \(\tau(\mathbf{x})\)

Plug-in rule

RankSEG

RankSEG

RankSEG

O( d log(d) )

RankSEG

O( d^2 )

RankSEG

$$ \tau^* = \arg\max_{\tau \in \{0,1,\cdots,d\}} \Big( \sum_{j \in J_\tau(\mathbf{x})} \mathbb{E} \big( \frac{2p_j(\mathbf{x})}{\tau + \Gamma_{-j}(\mathbf{x}) + \gamma + 1 } \big) + \gamma \mathbb{E} \big( \frac{1}{\tau + \Gamma + \gamma} \big) \Big)  $$

Blind approximation (BA; Dai and Li 2023). In high-D segmentation, the difference in distributions between \(\Gamma(\mathbf{x})\) and \(\Gamma_{-j}(\mathbf{x})\) is negligible.

$$ \approx $$

$$\to 0 (d \to \infty)$$

Lemma 5 in Dai and Li (2023)

RankSEG: BA Algo

Blind approximation (BA; Dai and Li 2023). In high-D segmentation, the difference in distributions between \(\Gamma(\mathbf{x})\) and \(\Gamma_{-j}(\mathbf{x})\) is negligible.

$$ \approx $$

$$\to 0 (d \to \infty)$$

Lemma 5 in Dai and Li (2023)

$$ \tau^* = \arg\max_{\tau \in \{0,1,\cdots,d\}} \Big( \sum_{j \in J_\tau(\mathbf{x})} \mathbb{E} \big( \frac{2p_j(\mathbf{x})}{\tau + \Gamma_{-j}(\mathbf{x}) + \gamma + 1 } \big) + \gamma \mathbb{E} \big( \frac{1}{\tau + \Gamma + \gamma} \big) \Big)  $$

RankSEG: BA Algo

GPU via CUDA

O( d log(d) ) 😄

(approx 100 times slower than T (at 0.5) 🥲)

RankSEG: BA Algo

RankSEG: RMA

Chao, M. T., & Strawderman, W. E. (1972). JASA

Wooff, David A. (1985) JRSS-b

RankSEG: RMA

Theorem 2 (Wang and Dai, 2025). (Reciprocal moment approximation to RankSEG). Let \(\Gamma\) be a Poisson-binomial r.v., then for any \(\tau \geq 1\),

$$ (\mathbb{E}\Gamma + \tau)^{-1} \leq \mathbb{E}(\Gamma + \tau)^{-1} \leq \left(\frac{d+1}{d}\mathbb{E}\Gamma + \tau - 1\right)^{-1}. $$

Zixun Wang (CUHK)

$$ \approx $$

$$ \approx $$

$$\to 0 (d \to \infty)$$

Theorem 2 in Wang and Dai (2025)

RankSEG: RMA

$$ \approx $$

$$ \approx $$

$$\to 0 (d \to \infty)$$

Theorem 2 in Wang and Dai (2025)

RankSEG: RMA

  • Three segmentation benchmark: VOC, CityScapes, ADE20K

RankSEG: Experiments

Source: Visual Object Classes Challenge 2012 (VOC2012)

  • Three segmentation benchmark: VOC, CityScapes, ADE20K

RankSEG: Experiments

Source: Visual Object Classes Challenge 2012 (VOC2012)

Source: The Cityscapes Dataset: Semantic Understanding of Urban Street Scenes

  • Three segmentation benchmark: VOC, CityScapes, ADE20K

RankSEG: Experiments

Source: Visual Object Classes Challenge 2012 (VOC2012)

Source: The Cityscapes Dataset: Semantic Understanding of Urban Street Scenes

Zhou, Bolei, et al. "Semantic understanding of scenes through the ade20k dataset." IJCV

  • Three segmentation benchmark: VOC, CityScapes, ADE20K

RankSEG: Experiments

  • Three segmentation benchmarks: VOC, CityScapes, ADE20
  • Commonly used models: DeepLab-V3+, PSPNet, SegFormer
  • The proposed framework  VS.  the existing frameworks
    • Based on the SAME trained neural networks
    • Open-Source python module and codes for replication
    • All trained neural networks available for download

RankSEG: Experiments

RankSEG: Experiments

RankSEG: Experiments

RankSEG: Experiments

The optimal threshold is NOT 0.5, and it is adaptive over different images/inputs

RankSEG: Experiments

The optimal threshold is NOT fixed, and it is adaptive over different images/inputs

RankSEG: Experiments

RankSEG: Experiments

RankSEG: Experiments

More experimental results in Dai and Li (2023) and Wang and Dai (2025)

RankSEG: Experiments

Fisher consistency or Classification-Calibration

(Lin, 2004, Zhang, 2004, Bartlett et al 2006)

Classification

Segmentation

RankSEG: Theory

RankSEG: Theory

RankSEG: Theory

RankSEG: Theory

RankSEG: Theory

  • mRankDice: extension and challenge
  • RankIoU
  • Simulation
  • Probability calibration
  • ....

More results

  • To our best knowledge, the proposed ranking-based segmentation framework RankSEG, is the first consistent segmentation framework with respect to the Dice/IoU metric.

  • BA and RMA algorithms with GPU parallel execution and are developed to implement the proposed framework in large-scale and high-dimensional segmentation.

  • We establish a theoretical foundation of segmentation with respect to the Dice metric, such as the Bayes rule, Dice-calibration, and a convergence rate of the excess risk for the proposed RankDice framework, and indicate inconsistent results for the existing methods.

  • Our experiments suggest that the improvement of RankDice over the existing frameworks is significant.

Contribution

Thank you!

If you like RankSEG  please star 🌟 our Github repository, thank you for your support!

  • Dai, B., & Li, C. (2023). RankSEG: A Consistent Ranking-based Framework for Segmentation. Journal of Machine Learning Research.
  • Wang, Z., & Dai, B. (2025). RankSEG-RMA: An Efficient Segmentation Algorithm via Reciprocal Moment Approximation. NeurIPS.