Image Math

What if you could do math on an image?

I was inspired by 3Blue1Brown's video where he took the logarithm of a Droste art piece to convert it into an Escher-style artwork.

Every point on an image is corresponded with a complex number in the complex plane. Applying a function on every point on an initial image gives a new complex number, cooresponding to its position in the new image.

I was able to recreate Escher's effect and more!

Jump forward to:

1. Applying your own function
2. Creating your own Escher-style images

Some cool effects I got:

Inv-Function	3B1B's image	My Droste Image	Rings
Original Click any image for higher quality
Escher Effect
Rotate $f(z)=iz$
Zoom to corner $f(z)=\frac{z+2+2i}{3}$
Kaleidoscope $f(z)=z^6$
Kaleidoscope (Try any polynomial :D) $f(z) = z^3 + 3z^2$
Logarithm $f(z) = e^z$
Inverse $f(z) = 1/z$
Weierstrass Elliptical function

Applying your own functions:

1. Create the inverse of your function. e.g. if $f(x) = e^x$, the inverse function is $g(x) = log(x)$
2. Create the function in using numpy.

   def f(z):
       return np.log(x) # i.e. log(x)
   

3. Add the @image_function decorator.

   @image_function
   def my_image_function(z):
       return np.log(x)
   

4. Done! Use your function like this:

   my_image_function("input.png", "output.png", zoom = 1.0, output_scale = 1.0)
   

   • zoom: Change this to zoom-in/out the output.
   • output_scale: Output resolution w.r.t input image. Smaller number makes rendering much faster, but loses quality.

Creating an Escher-style Image:

1. Start with a Droste image.
2. Find the singularity of the image, (where all the recursive images meet), generally the center of the image. Find the position w.r.t the center of the image [(0,0) = center; (1,1) = top right corner]
3. Create the escher function:

   scale = 16 #Size of smaller image wrt the larger one.
   center = (0,0) #Center point, (0,0) by default
   subject = (-0.8,-0.8) #Subject of drawing, cannot be center
   esch = escher_function(scale, center,subject)```
   

4. Create the final image:

   input_filename = "droste.png" # Input filename
   output_filename = "escher.png" # Output
   zoom = 1.8 # Used to zoom-in/out the output image.
   res = 1.0 # Set to a low number (e.g. 0.2) for faster rendering and testing, but lower output quality. 1.0 is the maximum quality.
   
   esch(input_filename, output_filename, zoom, res)