Conformal Mapping

In maths we call this type of image distortion CONFORMAL MAPPING.

Mathematicians write equations for conformal mapping, which means they produce equations that can turn your picture into an image you might see reflected in a fun park mirror or even in pond ripples.

Visualizing Functions of a Complex Variable

2D picture of y = sin(x)

Graph of z = sin(x*y)

3D surface Density plot set of contour lines

A function of a complex variable, w = f(z), can be thought in terms of its real components:

We will demonstrate a number of ways to visualize the set of points (x, y, u, v) satisfying this equation.

Explicitly dropping one of the coordinates allows us to render a 3D surface. We can also encode the remaining coordinate by coloring the surface.

o (x, y, u) can be seen by graphing Re(f(x+iy))o (x, y, v) can be seen by graphing Im(f(x+iy))o (x, u, v) can be seen by graphing the real part of the inverse of f.o (y, u, v) can be seen by graphing the imaginary part of the inverse of f.o We can also graph the magnitude or phase of f(x+iy).

The points (x, y, u, v) represent a surface in four dimensions. This surface can be drawn using perspective projection in much the same way 3D surfaces are rendered on screen.

We can encode the magnitude and phase of f(x+iy) onto a 2D color map using the hue and intensity to represent the complex value of the function.

We can represent the complex value of the function as a vector, drawing f(x+iy) as a 2D vector plot.

2D models using complex variables

sin πz (z3 - 1)(z - 1)

ez z2

3D models using complex variables

Re(z2) Im(z2)

|z2| arg(z2)

|z2| |sin πz|

4D models using complex variables 

All of the images above are different views of the complex exponential function. In general, one can see the sinusoidal variation along the imaginary or y-axis and exponential variation along the real or x-axis. The white grid is the (x,y) plane at u = v = 0. In some views, it is seen edge-on as a white line segment.

Using conformal maps

APPLICATION OF BILINEAR TRANSFORMATION IN IMAGE PROCESSING

In all the images above, the lines and picture are the left are mapped under the complex inverse function to produce the image on the right.

 In the color maps, hue indicates phase, saturation and lightness indicate magnitude. Real values are cyan, pure imaginary values are red.