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C'est la Z

Seam Carving and Dynamic Programming

It's spring break and for me that's always been a good time to explore some new ideas.

Here's one that some of you might like, particularly if you're teaching APCS-A or something similar.

Many APCS-A teachers do a unit on image processing using the picture lab (alternate resource). Image processing is a nice platform to explore two dimensional arrays. You basically use a 2D array of pixels (points) to represent an image. You can just use a 2D array of ints and store 0-255 at each location for a grayscale image and three ints per pixel for red, green, blue. For this you can just use a simple 2D array of some color class - I think there's even one built into Java that you can use.

I like to use the PPM format to save and later reload the images. PPM is a simple text format. It starts with a header, then an optional comment (starting with #), the width and height (4x4 in the example below), then the maximum color number. Here's an example:

# feep.ppm
4 4
 0  0  0    0  0  0    0  0  0   15  0 15
 0  0  0    0 15  7    0  0  0    0  0  0
 0  0  0    0  0  0    0 15  7    0  0  0
15  0 15    0  0  0    0  0  0    0  0  0

You can even cut and paste the above into a file and view it with an image viewer.

You can find the full format specification here (alternate resource).

The picture lab has a bunch of interesting exercise ideas but if you're looking for something "next level" check this out:

<iframe width="560" height="315" src="" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

Seam Carving is no longer a new cutting edge technique but it still blows me away. It's also a really cool project for more advanced students for two reasons: one, it's not a toy assignment - it's from real research and two, it contains a very approachable example of dynamic programming which I'll talk about in a bit.

If you watch the video, there are two pieces that can sound intimidating. One is the "dynamic programming algorithm." The other is when they talk about the "gradient magnitude."

Gradient magnitude. That sounds pretty hard. Actually, it could be worse. When I first starting looking at the basics of image processing it was described as "the derivative of the pixels." The dirivative of the pixels???? I have to know calculus for this?????? Not really. I hate it when they use big words for simple ideas. All they're really talking about is the difference between the color of neighboring pixels. That is, how much the color changes from one pixel to the next. Let's say you have a greyscale image (or have converted an image to greyscale by averaging the red green and blue values), if you have a vertical row of pixels with all the values of, let's say 200 and the row right next to it is also of values 200, then the two lines are of the same color - there is no difference. If one pixel has a value of 200 and its neighbor has a value of 50, it changes quite a lot.

The APCS-A picture lab describes a simple way to calculate this in exercise 9 (linked above). You can also just do a search on "edge detection tutorial" or "sobol edge detection tutorial" or something similar.

For Seam Carving, you have to find a sequence of pixels from one side of the image to the other where the sum off all the differences is the lowest. This is where the dynamic programming comes in.

Dynamic Programming is a technique that, in basic terms takes a problem that decomposes into subproblems and you store the optimal subproblem solution rather than recalculating it.

Probably the easiest example is Fibonacci numbers. You can generate Fibonacci numbers recursively using something like this:

  def fib(n):
      if n<=2:
          return 1
          return fib(n-1)+fib(n-2)

It's very simple but it gets very slow very fast. That's because it keeps recalculating the same subproblems over and over and over.

You can "fix" this by creating a list of previously found Fibonacci numbers and just return them rather than recalculating:

  fibs = [0]*1000 # make room for 1000 fib numbers

  def fib(n):
      if fibs[n] != 0:
          # we already have the answer, just return it
          return fibs[n]
          # otherwise, calculate it, store it, return it
          f = fib(n-1)+fib(n-2)
          return f

and that's a dynamic programming solution for Fibonacci numbers, specifically using a technique known as memoization.

How does this apply to finding the proper seam to remove? The one with the lowest total change?

First, go through the image array and build a second 2D array where each element contains the pixels "gradient magnitude" or change using either the method described in task 9 of the picture lab or elsewhere.

For the first row, the sum of each pixels path so far is just the array element.

For each successive row, the value of any pixel is going to be it's value plus one of the values abov it, either up left, up center, or up right. Specifically, the elements new value will be it's value + the smallest of the above three.

For example, given this array representing the color changes:

<figure class="z_image_center"><img src="/img/seam-mat1.png"/> </figure>

When calculating the second item in the second row, you'll consider the three values above it:

<figure class="z_image_center"><img src="/img/seam-mat2.png"/> </figure>

The smallest is 0 so that 2 remains a 2. If we do this for every element in the second row we get the following:

<figure class="z_image_center"><img src="/img/seam-mat3.png"/> </figure>

The green boxes added a 0 from the line above but the yellow ones added the smallest non zero value from the above row. Note that the edges just considered the two values above them and didn't wrap.

Working the rest of the way through, you get this final array:

<figure class="z_image_center"><img src="/img/seam-mat4.png"/> </figure>

You can now easily identify the best seam to remove by finding the smallest value in the bottom row and working your way up to the top.

All together this is very doable by an advanced APCS-A student and the results are very cool, particularly if you do it in an interactive environment like processing.

So, check out seam carving. I'll leave you with one more link to a longer presentation on the topic:

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