Curvature Flow & Smoothing curves

I am getting addicted to curves. They are the perfect beautiful representation which we can numerically compute. I don't want to express more my feelings towards it which may be boring to you.

Coming to the topic, curvature flow is a kind of method to modify the curve.
Take any planar non intersecting curves , find the curvature at each point and multiply with it the normal there, then move the curve along that direction. This is the concept. It is analogs to the heat exchange. Heat will eventually spread uni-formally , no matter how you wrap it.

So take a curve {X(s),Y(s)} and it's curvature and normal , Say  {K(s)} and {N(s)}.
Then curvature flow vector is defined as {K(s)*N(s) }.

Using this technique we can smooth the curve from noise. Eventually this curve will become circular and will vanish. it is possible to know the exact point where it will vanish.

I just made a quick demo of this with matlab. See the images below

Original curve.
We can see that some edges are not smooth (think of it like created by noise).These spikes(non-smooth) in edges are not influencing our capability to detect the shape.We humans normally pick up a smooth shape from a contour. Now we want to remove some noise (or say smooth it) using curvature flow.

After 10 iteration

See that curve is more smooth now.

Curve after 50 Iterations
Curve is getting circular. Yeah, it will become circular at one point , because circle is the only one shape with a non-zero uniform curvature.

This is type of smoothing is also possible with Gaussian filtering along the curve,but with lesser accuracy. This type of technique can also be extended to 3D, Say you are having an object and you want to add a higher layer of layer/cover to it. Rather than just extending the surface along normal , use curvature flow.Then the surface will be more appealing and natural (I Guess).

Gaussian filter

Gaussian Filter modifies the input data by convolution with a Gaussian distribution. Gaussian filter is often used to smooth out images.In this post i will try to show the frequency response of Gaussian filter.  It is very important to understand the frequency domain behaviorism.When we consider the frequency response of gaussian one diamension filter we can see that , the filter reponse is inversely proportional to the frequency, lower the frequency, its response is high, that is more smoothing happens to lower frequency components.

An unnormalized Gaussian distribution is 

where is the standard deviation. This function is will form a belll shaped curve with center at 0.  This function is non zero every where(high value at center and decreases ). curve is shown below

A normalized gaussian distribution can be found by normalizing the above equation with the total area, which can be found by integrating it over -Infinite to +infinite. 

so the normalized equation is(from wiki) : 

Frequency response of a 1-D Gaussian filter can be found by doing DFT over the 1-D kernel,

We can find it in mathematica simply with following commands

 DftResults =    Fourier[   Table[PDF[NormalDistribution[0,1],x],{x,-3,3,.01}] ];
 ListLinePlot [ Abs[DftResults],PlotRange->All ]

This will plot the power spectrum of Gaussian 1-D filter. It will look like this (X axis-frequency,Y axis magnitude), we can see that at lower frequency level , filer gives better output.. the extreme right shows the nyquist frequency( ignore that for now. ). From this we can see Gaussian filter act as a low pass filter.