Entropy primarily based Uncertainty Prediction | by François Porcher | Sep, 2023

This text explores how Entropy will be employed as a software for uncertainty estimation in picture segmentation duties. We’ll stroll by way of what Entropy is, and methods to implement it with Python.

Picture by Michael Dziedzic on Unsplash

Whereas working at Cambridge College as a Analysis Scientist in Neuroimaging and AI, I confronted the problem of performing picture segmentation on intricate mind datasets utilizing the newest Deep Studying strategies, particularly the nnU-Net. Throughout this endeavor, I noticed a big hole: the overlooking of uncertainty estimation. But, uncertainty is essential for dependable decision-making.

Earlier than delving into the specifics, be at liberty to take a look at my Github repository which comprises all of the code snippets mentioned on this article.

On the planet of pc imaginative and prescient and machine studying, picture segmentation is a central drawback. Whether or not it’s in medical imaging, self-driving vehicles, or robotics, correct segmentation are important for efficient decision-making. Nonetheless, one usually neglected side is the measure of uncertainty related to these segmentations.

Why ought to we care about uncertainty in picture segmentation?

In lots of real-world functions, an incorrect segmentation may lead to dire penalties. For instance, if a self-driving automobile misidentifies an object or a medical imaging system incorrectly labels a tumor, the implications may very well be catastrophic. Uncertainty estimation provides us a measure of how ‘positive’ the mannequin is about its prediction, permitting for better-informed choices.

We will additionally use Entropy as a measure of uncertainty to enhance the educational of our neural networks. This space is is aware of as ‘lively studying’. This concept will likely be explored in additional articles however the principle thought is to determine the zones on which the fashions are essentially the most unsure to deal with them. For instance we may have a CNN performing medical picture segmentation on the mind, however performing very poorly on topics with tumours. Then we may focus our efforts to amass extra labels of this kind.

Entropy is an idea borrowed from thermodynamics and data concept, which quantifies the quantity of uncertainty or randomness in a system. Within the context of machine studying, entropy can be utilized to measure the uncertainty of mannequin predictions.

Mathematically, for a discrete random variable X with chance mass perform P(x), the entropy H(X) is outlined as:

Or within the continous case:

The upper the entropy, the larger the uncertainty, and vice versa.

A traditional instance to totally grasp the idea:

State of affairs 1: A biased coin

Picture by Jizhidexiaohailang on Unsplash

Think about a biased coin, which lands on head with a chance p=0.9, and tail with a chance 1-p=0.1.

Its entropy is

State of affairs 2: Balanced coin

Now let’s think about a balanced coin which lands on head and tail with chance p=0.5

Its entropy is:

The entropy is bigger, which is coherent with what we stated earlier than: extra uncertainty = extra entropy.

Really it’s attention-grabbing to notice that p=0.5 corresponds to the utmost entropy:

Entropy visualisation, Picture by creator

Intuitively, do not forget that a uniform distribution is the case with maximal entropy. If each consequence is equally possible, then this corresponds to the maximal uncertainty.

To hyperlink this to picture segmentation, contemplate that in deep studying, the ultimate softmax layer often offers the category possibilities for every pixel. One can simply compute the entropy for every pixel primarily based on these softmax outputs.

However How does it work?

When a mannequin is assured a few specific pixel belonging to a selected class, the softmax layer exhibits excessive chance (~1) for that class, and really small possibilities (~0) for the opposite courses.

Softmax layer, assured case, Picture by creator

Conversely, when the mannequin is unsure, the softmax output is extra evenly unfold throughout a number of courses.

Softmax layer, unsure case, Picture by creator

The chances are rather more diffuse, near the uniform case in case you keep in mind, as a result of the mannequin can not determine which class is related to the pixel.

In case you have made it till now, nice! It is best to have an awesome instinct of how entropy works.

Let’s illustrate this with a hands-on instance utilizing medical imaging, particularly T1 Mind scans of fetuses. All codes and pictures for this case examine can be found in my Github repository.

1. Computing Entropy with Python

As we stated earlier than, we’re working with the softmax output tensor, given by our Neural Community. This method is model-free, it solely makes use of the possibilities of every class.

Let’s make clear one thing essential in regards to the dimensions of the tensors we’re working with.

If you’re working with 2D Photos, the form of your softmax layer needs to be:

Which means that for every pixel (or voxel), we’ve got a vector of dimension Courses, which provides us the possibilities of a pixel to belong to every of the courses we’ve got.

Subsequently the entropy needs to be pc alongside the primary dimension:

def compute_entropy_4D(tensor):
Compute the entropy on a 4D tensor with form (number_of_classes, 256, 256, 256).

tensor (np.ndarray): 4D tensor of form (number_of_classes, 256, 256, 256)

np.ndarray: 3D tensor of form (256, 256, 256) with entropy values for every pixel.

# First, normalize the tensor alongside the category axis in order that it represents possibilities
sum_tensor = np.sum(tensor, axis=0, keepdims=True)
tensor_normalized = tensor / sum_tensor

# Calculate entropy
entropy_elements = -tensor_normalized * np.log2(tensor_normalized + 1e-12) # Added a small worth to keep away from log(0)
entropy = np.sum(entropy_elements, axis=0)

entropy = np.transpose(entropy, (2,1,0))

total_entropy = np.sum(entropy)

return entropy, total_entropy

2. Visualizing Entropy-based Uncertainty

Now let’s visualize the uncertainties by utilizing a heatmap, on every slice of our picture segmentation.

T1 scan (left), Segmentation (center), Entropy (Proper), Picture by creator

Let’s have a look at an different instance:

T1 scan (left), Segmentation (center), Entropy (Proper), Picture by creator

The outcomes look nice! Certainly we will see that that is coherent as a result of the zones of excessive entropy are on the contour of the shapes. That is regular as a result of the mannequin does probably not doubt the factors on the center of every zone, however its fairly the delimitation or contour that’s tough to identify.

This uncertainty can be utilized in loads of other ways:

  1. As medical specialists work an increasing number of with AI as a software, being conscious of the uncertainty of the mannequin is essential. This imply that medical specialists may spend extra occasions on the zone the place extra fine-grained consideration is required.

2. Within the context of Energetic Studying or Semi-Supervised Studying, we will leverage Entropy primarily based Uncertainty to deal with the examples with maximal uncertainty, and enhance the effectivity of studying (extra about this in coming articles).

  • Entropy is an especially highly effective idea to measure the randomness or uncertainty of a system.
  • It’s potential to leverage Entropy in Picture Segmentation. This method is mannequin free and solely makes use of the softmax output tensor.
  • Uncertainty estimation is neglected, however it’s essential. Good Information Scientists know methods to make good fashions. Nice Information Scientists know the place their mannequin fail and use this to enhance studying.

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