image-processing computer-vision image-segmentation entropy source-separation

What does entropy mean in this context?

I'm reading an image segmentation paper in which the problem is approached using the paradigm "signal separation", the idea that a signal (in this case, an image) is composed of several signals (objects in the image) as well as noise, and the task is to separate out the signals (segment the image).

The output of the algorithm is a matrix, $S \in R^{MxT}$ which represents a segmentation of an image into M components. T is the total number of pixels in the image, $s_{ij}$ is the value of the source component (/signal/object) i at pixel j

In the paper I'm reading, the authors wish to select a component m for $m \in [1,M]$ which matches certain smoothness and entropy criteria. But I'm failing to understand what entropy is in this case.

Entropy is defined as the following:

$H(s_m) = - \sum_{n=1}^{256} p_n (s_m) \cdot log_2 (p_n (s_m)), m= 1,..., M$

and they say that '' ${p_n(s_m)}_{n=1}^{256}$ are probabilities associated with the bins of the histogram of ''

The target component is a tumor and the paper reads: "the tumor related component with "almost" constant values is expected to have the lowest value of entropy."

But what does low entropy mean in this context? What does each bin represent? What does a vector with low entropy look like?

link to paper

Solution

They are talking about Shannon's entropy. One way to view entropy is to relate it to the amount of uncertainty about an event associated with a given probability distribution. Entropy can serve as a measure of 'disorder'. As the level of disorder rises, the entropy rises and events become less predictable.

Back to the definition of entropy in the paper:

H(s_m) is the entropy of the random variable s_m. Here is the probability that outcome s_m happens. m are all the possible outcomes. The probability density p_n is calculated using the gray level histogram, that is the reason why the sum runs from 1 to 256. The bins represent possible states.

So what does this mean? In image processing entropy might be used to classify textures, a certain texture might have a certain entropy as certain patterns repeat themselves in approximately certain ways. In the context of the paper low entropy (H(s_m) means low disorder, low variance within the component m. A component with low entropy is more homogenous than a component with high entropy, which they use in combination with the smoothness criterion to classify the components.

Another way of looking at entropy is to view it as the measure of information content. A vector with relatively 'low' entropy is a vector with relatively low information content. It might be [0 1 0 1 1 1 0]. A vector with relatively 'high' entropy is a vector with relatively high information content. It might be [0 242 124 222 149 13].

It's a fascinating and complex subject which really can't be summarised in one post.