Implement functions to determine ITSP bounds and compute quantification ambiguity range (QAR)

[AjeyPaiK]

@JorenB and I spent some time to understand errors that occur when a segmentation model makes predictions. By using the dice metric, we can come up with upper and lower bounds expected for the errors when a deep learning model generates segmentations. These bounds can be used to derive something called the "quantification ambiguity range" that represents the range of ambiguity we can expect when the model estimates ITSP. It's much like the perceptual ambiguity in humans.

[AjeyPaiK]

Sensitivity of segmentation errors on volume and dice score

Note: \mathcal isn't supported on github markdown.

Today, I looked at how significantly do the volume and the dice score effect the segmentation errros.
For Eq 5 and Eq 6 from my lab journal, we can write the partial derivative of errors with respect to volume and dice score as follows:

Sensitivity of Segmentation Errors to Volume:

$$ \frac{\partial E_{\text{over}}}{\partial V} = \frac{2}{d} - 2 $$

And

$$ \frac{\partial E_{\text{under}}}{\partial V} = \frac{2 (1 - d)}{2 - d}. $$

These partial derivatives show that sensitivity to $V$ does not depend on $V$ itself but instead depends on $d$. As $d$ increases, these derivatives approach zero, meaning that for high dice scores, segmentation error becomes almost independent of $V$.

Sensitivity of Segmentation Errors to Dice Score:

$$ \frac{\partial \mathcal{E}_{\text{over}}}{\partial d} = -\frac{2V}{d^2} $$

And

$$ \frac{\partial \mathcal{E}_{\text{under}}}{\partial d} = -\frac{2V}{(2 - d)^2}. $$

These partial derivatives also indicate a larger sensitivity to $d$ due to the inverse square proportionality. Consequently, we can also conclude that QAR is more sensitive to the dice score compared to the volumes.

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Below is an output from a simulation that illustrates the points made above. The simulation is now part of itsper.

[AjeyPaiK]

Along the same lines, I concluded that the dependence of QAR on the proportions of tumor and stroma volumes behaves similarly as the perceptual ambiguity seen in humans conducting the same task. In that, the QAR is maximum when the tumor volume and stroma volume are equal. Whereas, QAR decreases when the volumes are unequal. Humans also have most ambiguity when the tumor and stroma compartments are equal in a given image.

[AjeyPaiK]

Implemented better ways to simulate tumor and stroma compartments. Now it closely mimics real situations.