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From idea to observe with the Otsu thresholding algorithm
Let me begin with a really technical idea:
An Picture will probably be seen, handled, analyzed, and processed as a 2D sign.
And a few correct definitions:
- A sign is a amount that modifications over house or time and can be utilized to transmit a type of data.
- An picture is nothing however a amount of sunshine that hits an optical system, that’s the digital camera or the canvas the place you might be portray it.
On this sense, a picture is nothing however a 2D sign, an electromagnetic sign that carries some data that’s retrieved by a bodily system.
In order we have now established that a picture is certainly a sign, we are able to consider making use of a sign processing method to an picture processing activity. We are able to thus cease discussing philosophy and begin with the nerd half.
Talking of philosophy. Let’s take this picture:
There may be the thinker within the picture doing his job: considering. After which there may be this very white background, that we actually don’t care about. Can we do away with it? Can we get one thing like that?
If I’m asking you, it signifies that we are able to. 😅
Each one who is aware of a bit little bit of photoshop can do this, however how will you do this mechanically and with Python? Once more, sure.
Let me present you the way 🚀
So let’s take an easy case.
Yep. A small sq. inside a much bigger sq.. That is an very simple case. What we need to do is to set all of the values within the smaller sq. to 1 and every thing that’s exterior to 0.
We are able to extract the 2 values with this code:
After which do one thing like:
This converts the picture from two values to 1 and 0.
That is very simple, proper? Let’s make it a bit bit tougher.
Now we’ll do the little sq. inside the larger sq. however each the squares have some noise.
What I imply is that we don’t have solely 2 values however we are able to theoretically have all of the values between 0 and 255 which is the entire vary of values within the encoding.
How can we take care of this?
Nicely, the very first thing we need to do is to flatten the picture (2D sign) and alter it right into a 1D one.
The picture was a 50×50 picture and we have now a “raveled” 50×50=2500 lengthy 1D sign.
Now if we examine the distribution of our 1D Sign we acquired one thing like this:
As we are able to see, we have now two regular distributions. That is precisely the place the Otsu algorithm performs greatest. The underlying concept is that the background and the topic of the picture have two completely different natures and two completely different domains. For instance, on this case, the primary gaussian bell is the one associated to the background (let’s say from 0 to 50), whereas the second Gaussian bell is the one of many smaller sq. (from 150 to 250).
So let’s say that we resolve that we set every thing that’s bigger than 100 as 1 and every thing that’s smaller as 0:
And the result’s the next masks between the background and the topic:
That is it. That is the entire concept of the Otsu algorithm:
- Import/Learn the picture as a 2D sign
- Flatten the picture right into a 1D vector
- Select a threshold
- Set every thing that’s beneath that threshold as 0 and every thing that’s above as 1
Very simple proper?
However how can we select the correct threshold? What’s the greatest one? Let’s speak about math.
Let’s formalize this idea a bit bit.
Now we have a area of pixels in a picture. The complete area goes from 0 to 255 (white to black) however it doesn’t need to be that extensive (it may be from 20 to 200 for instance).
Now, a number of factors can have the identical pixel depth in fact (we are able to have two black pixels in the identical picture). Let’s say that we have now 3 pixels with an depth of 255 in a picture that has 100 pixels. Now the chance of getting depth 255 in that picture is 3/100.
Generally, we are able to say that the chance of getting pixel i in a picture is:
Now let’s say that the pixel on which we’re doing the break up is pixel ok (in our earlier instance ok was 100). This classifies the information factors. All of the factors earlier than ok belong to class 0 and all of the factors after ok belong to class 1.
Which means the chance of choosing some extent from class 0 is the next:
Whereas the chance of choosing some extent from class 1 is the next:
As we are able to see, each possibilities are clearly depending on ok.
Now, one other factor that we are able to compute is the variance for every class:
The place:
And
The sigma worth is the variance of every class aka how a lot the category is unfold across the imply values which might be mu_0 and mu_1.
Now theoretically the concept is to seek out the worth that creates that little valley that we noticed on this image earlier:
However the method that we use is barely completely different and extra rigorous. Through the use of the identical concept of the Linear discriminant analysis (LDA). Within the (Fisher) LDA we need to discover a hyperplane that splits the 2 distributions in a manner that the variance between lessons is as large as attainable (in order that the 2 means are the furthest away from one another) and the variance inside the lessons is as small as attainable (in order that we don’t have an excessive amount of overlap between the 2 lessons knowledge factors).
On this case, we don’t have any hyperplane and the brink that we set (our ok) just isn’t even a line, however it’s extra of a chance worth that we use to discriminate knowledge factors and classify them.
It may be confirmed (full proof right here within the authentic paper) that the greatest break up between the background and topic (is given the belief that the area of the background is completely different from the area of the topic) is obtained by minimizing this amount:
Which means we are able to attempt all completely different ks and simply choose the one with the bottom ok.
The speculation would possibly look complicated and obscure, however the implementation is extraordinarily simple and it’s manufactured from three blocks:
2.1 Importing the libraries
The very first thing that we need to do is to import 4 primary libraries that we’ll want.
2.2 The brink operate
As soon as you discover the proper threshold, that is methods to apply it to your picture:
2.3 The Otsu criterion
The operate that can compute this amount:
Is the next:
2.4 Finest threshold computation
This different operate simply runs all around the attainable ks and finds the most effective one in response to the criterion above:
2.5 The entire course of
So the picture we’re utilizing is the next one:
If we save that picture in a path and we apply the Otsu algorithm we get:
And if we examine im (the unique picture) and im_otsu (the one after the algorithm) we get:
As we are able to see, the black a part of the fitting higher a part of the image is misinterpreted as the topic as a result of it has the identical tone as among the topics. Individuals are not good and neither are the algorithms 🙃
Thanks for being right here with me all through the entire path of this Otsu algorithm tutorial.
On this transient article, we noticed:
- That an Picture will be handled as a 2D sign and may then be analyzed utilizing Sign Processing method
- The belief of the Otsu algorithm, that’s the background and the topic of a picture have two, steady, non-overlapping, distinguished domains
- The best way to discover the greatest discrimination between the background and topic of a picture given the Otsu algorithm. How we are able to interpret the Otsu algorithm as a Fisher Linear Discriminant.
- The best way to implement the Otsu algorithm utilizing Python
- The best way to apply this algorithm in an actual picture
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