Estimating Complexity of Images: Entropy & World Perception

[LeoYard]

Let's consider an image of a natural scene ;we 've got some individuals (pixels) that may differ one from another (different colors).
the entropy E of the system is E=sigma(-p.logp), where the sum is computed over the colors and p is the probability of occurence of a given color.
the issue here is that E simply does not depend on the location of the pixels within the image and thus does not depend on the "shapes" or "object" that one can perceive in the image (tress, etc): E only depends on the histogram of the pixels but not on the geometry of the image...

can anyone suggest a way to estimate the complexity of images of the world?

 

[marcusl]

Let's consider an image of a natural scene ;we 've got some individuals (pixels) that may differ one from another (different colors).
the entropy E of the system is E=sigma(-p.logp), where the sum is computed over the colors and p is the probability of occurence of a given color.
the issue here is that E simply does not depend on the location of the pixels within the image and thus does not depend on the "shapes" or "object" that one can perceive in the image (tress, etc): E only depends on the histogram of the pixels but not on the geometry of the image...

That's because you have defined a "color entropy" that, by definition, is geometry independent. That doesn't mean that your color entropy completely defines a scene.

 

[Chris Hillman]

Entropy Measures

Let's consider an image of a natural scene ;we 've got some individuals (pixels) that may differ one from another (different colors).

can anyone suggest a way to estimate the complexity of images of the world?

You wrote down the Shannon entropy; computing this would require you to know the "probability of c" for each color c, whatever that mean. Fortunately there are numerous other entropy measures you can consider which might be more appropriate, depending on what application you have in mind.

Can you say more about how you would use your "entropy"?

For example, if you are planning to compress colorized image files, then the nature of the compression is probably more important than how pixels of a given color occur in relation to one another. In this case, there are immediately applicable notions from coding theory which suggest using some "statistical estimator" for an appropriate Shannon entropy (but the probabilities would most likely not have the interpretation "probability of color c"). If so, don't be afraid of "biased estimators"; these are typically more accurate, which is almost certainly more important for you.

Perhaps you are thinking of images taken by an astronomical instrument? As in, seeking a "relative visual complexity" of two equal-sized areas of the night sky? There has been some work on that kind of thing.

Or perhaps you seek a measure which takes account of the geometric relationship between where pixels of various colors occur in a given image? As in--- ignoring the word "natural" in "natural images"--- trying to seek a novel measure comparing the complexity of a digital image of the Mona Lisa with a digital image of a Warhol print? If so, it might be helpful to note that there are some tricky issues associated with the question, "what is color?", and there are a number of mathematical theories of color (as in paintings) which treat colors as points in certain manifolds, for example. But what about texture? I am thinking of the subtle surface gradations in paintings like http://upload.wikimedia.org/wikipedia/en/7/7b/Mark_rothko_1957_no_20.JPG by Mark Rothko. (His fans are likely to insist that despite initial impressions, a "monochromatic" painting by Rothko is in fact no less complex, or at least no less interesting, than a painting by Kandinsky.)