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I hope this question is appropriate for this site.

I came across this passage in The Three Body Problem, a novel by Liu Cixin:

The professor had put up two pictures: One was the famous Song Dynasty painting Along the River During the Qingming Festival, full of fine, rich details; the other was a photograph of the sky on a sunny day, the deep blue expanse broken only by a wisp of a cloud ... The photograph's information content - its entropy- exceeded the painting's by one or two orders of magnitude

Representative pictures:

Here is the painting Blue sky Is this true? How does one explain this counterintuitive phenomenon?

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  • $\begingroup$ Is there more context in the book? $\endgroup$
    – endolith
    Jun 9, 2017 at 19:13
  • $\begingroup$ @endolith no, unfortunately not. $\endgroup$
    – RSS
    Jun 9, 2017 at 19:22
  • $\begingroup$ I wish entropy was the only measure of the content. But no. RGB Images are created for humans to look at, both paintings and photographs. So look at it yourself. Which one do you think is more informative and rich? Your choice is correct, regardless the computer measures we invent. $\endgroup$
    – Tolga Birdal
    Jun 9, 2017 at 19:32
  • $\begingroup$ @TolgaBirdal Fair enough, but I would still be interested to understand why the computers are getting it wrong in this case. $\endgroup$
    – RSS
    Jun 9, 2017 at 19:41
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    $\begingroup$ Great SciFi novel by the way $\endgroup$
    – Laurent Duval
    Apr 28, 2020 at 14:08

3 Answers 3

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It depends how you define the term "information" or "entropy".

The conventional definition of entropy of an image is to think the image as a two-dimensional matrix of pixels and $$H = - \sum_k p_k \log_2(p_k),$$ where $p_k$ is the probability, which is calculated from histogram, associated with gray level $k$.

This kind of entropy is correct if we ignore the correlation between pixels. For example the two images have the same entropy by this definition.

identical entropy image 1 identical entropy image 2

It is not true if the correlation between pixels is considered. For example if the color first pixel in top-left has probability $p_k$, the next pixel surely has the same color and its color does not have the same probability $p_k$.

We human being, with you as an example, use this kind of correlation to perceive the images. This kind of correlation are called "details" and, therefore, we/you expect that rich-detail images should have more information/entropy than simple ones. This is the reason why you have found it counter-intuitive.

PS:

I tried calculating the entropy of the two images you have posted, but they are not different "by one or two orders of magnitude" !!!!

"Along the River During the Qingming Festival" entropy about 7

"The sky" entropy about 6

They must not be the same files of the professor.

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  • $\begingroup$ Thanks, I think this is the answer I was looking for. Of course the images I uploaded were meant to be representative only, I have no idea of what the fictional professor actually showed to the class :D $\endgroup$
    – RSS
    Jun 9, 2017 at 20:06
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Based on your provided images, the painting should involve more information.

However, the sky scene might include perceptually invisible components that a mathematical algorithm would treat as information so that the entropy of the image would increase...

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  • $\begingroup$ You've raised a good distinction i.e. did the professor compare a photograph with the actual painting (let's call this the weaker hypothesis) or would even a scan of the painting contain less info (stronger hypothesis). So, as per your explanation, only the weaker hypothesis is true? $\endgroup$
    – RSS
    Jun 9, 2017 at 19:39
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Both contain same information i.e both have 1 bit of information . Consider on board level there is 2 two images one of painting and other photograph . So probability of one image is 1/2 = 0.5 . As you don't know which one is the image before seeing them.

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