Implicit surfaces and Level Set

Things are more complicated in discretized form. Say a line when discretized is not actually a line, Right(remembering the bresenham line algorithm ) ? Same thing goes to circle or any shapes. So for computing the associated properties we need more techniques. Implicit surface helps us to compute such properties.

Before I explain about implicit surfaces, you tell be what is the gradient of a scalar surface/curve.
Say you define a 3D surface by the equation x*x + y*y + z*z = 9, we can easily see that this surface is nothing but a sphere with radius 3. Right ?
So Q(x,y,z) = x*x + y*y + z*z - 9 = 0.
What is the gradient of Q then ? it represents the normal at any point on this surface , and it is (2x,2y,2z) (not normalized).

Why i said this was to show you how easy it is compute the gradient of a surface with an explicit equation. We can also easily compute other properties related with that surface like tangent,curvature,etc.

But what can you do if you don't have such explicit equations. In practical things will be like this.
In Implicit form we can define a shape implicitly. Our shape must be closed and non-self intersecting. With this agreement we can define the shape with following definitions.

Let P ( { Xi,Yi } ) be our point set which denotes the boundaries of our shape. We can define shape implicitly based on the following conditions.

1. For all points in shape boundaries ∅ (Pi) = 0
2. For all other points outside the shape ∅(Pi) must be > 0 
3. For all other points inside the shape ∅(Pi) must be < 0 . (Conditions 2,3 can be interchanged though) 

Based on the earlier definitions consider the above picture. Pixel's with green boundary is our shape where ∅(Pi) will be 0. pixels having red color will have negative value, and rest of the pixels (blue) will have positive value. This is how we define implicit functions for complicated shapes. In the next post I will show you how we can numerically compute the properties of these shapes from this definitions also will introduce about level sets. It is not a big deal( Actually i had intention to write more about this, but I lost my mood so stopping now )

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).

Checking a point inside Quadratic curve

In the last post I had written about qudratic equation. After that one thought came to my mind is How to check a point inside the curve ?. 

Below shows the qudratic curve equation from our 3 points. t^2(p1-p0'-p0) + p0' * t + p0 = P.  P is a point on the curve corresponding to t.

That is  a*t^2 + b* t + c = p, where a,b,c ,p are vectors. p yields the points on the curve accoding to the 't'.

So how can we check a point is in curve or not ?  we have 'p' and equation of the curve.As you guess this can be done by equation solving.But here a,b,c, are vectors.

So there must be some way to produce scalar values from these vetors , like taking magnitude of vectors in the equation like 

     |a|*t^2 + |b|*t + |c| = |p|.

This equation has some problmes , because it is just looking the magnitude of the vectors , direction is missing.A More wiser solution may be to solve for X,Y (Z in 3D)independently,like

a.x* t^2 + b.x *t + c.x = p.x --------> (1)

a.y* t^2 + b.y *t + c.y = p.y --------> (2)

As we know when we draw curve from p0 to p1 the t will vary from 0 -1 , We can check a point is inside the curve by solving one equation ( example : equation 1 ) and substitute that value of t in other equation ( here in equation2 ) . If you got the correct result as that of the input point (res == p.y) you can say the point is in the curve. Otherwise not.

When you solving the above equation you will get two values of t , so take the value which is inbetween 0 and 1 , and susbstitue it in other equation. 

To solve linear equation in computer you can use gauss elimination method ( you can try the above method in matheamtica easily ).