Hello everyone and welcome back to Computer Vision Lecture Series.

This is Lecture 11, Part 2 and this is also our last recording for the Computer Vision

Lecture Series this semester.

So let's just get on with it.

In the last recording we saw how we can create a dense set of 3D points from multiple scans

or from different depth images.

And we almost saw how we could easily register all the point clouds in such a way that we

can get a final nice consistent surface reconstruction from the point clouds or from different multiple

scans.

So the first problem of surface reconstruction was solved that is to have a nicely registered

point cloud.

The second problem we focus now is to convert this point cloud from the registered point

clouds to a surface.

Basically to create a surface from this point clouds.

An example is shown here where you have on the left hand side a very well registered

point cloud of an animal and on the right hand side a surface reconstruction from those

points.

Now the simplest way to create those surfaces is to just choose randomly 3 points, 3 neighboring

points and interpolate a surface or fit a surface or a plane between them and create

a surface.

However, this can be very tricky because mainly these registered points are not always very

well registered and the scans are not very good.

Usually there could be a lot of outliers like noise or incorrectly registered points and

therefore ideally we should not take this approach because if we take this approach

we get a surface reconstruction as shown on the right hand side which is not the best

because there is a lot of discontinuities, there are a lot of mismatches in the surfaces

especially in the space regions or where there are lesser density of point clouds or meshes.

It's not easy to reconstruct the surfaces from the sparse set of points that we have

there and therefore this method is trivial and not preferred.

The second approach is to use implicit functions.

Now what are implicit functions?

An implicit function can be any function which we define such that when you fit it across

some set of points or model some set of points all the points lying inside the function will

be considered greater than 0 inside as well as those points which are lying outside this

function value are considered less than 0.

So for example in this case we fit an implicit function along the boundary points here based

on the color codes and we extract the zero set so all those points which are lying along

the points of the function where its value is 0.

We extract this function and essentially what we have done here is out of all these points

lying on this 2D grid we have created a surface which runs along a zero set of the implicit

function and at the end we get a kind of a representation of the surface in this 2D grid.

So extracting the zero surfaces of implicit function has been very well studied and known

in graphics specifically and also in visualization because in such areas you begin by defining

sparse set of points and then you interpolate the surfaces between them and therefore the

usefulness of implicit functions is quite more rampant in such areas of application.

So let's look into a bit deeper into what implicit functions actually mean.

For that we start with a problem on 2D surfaces to describe how to describe 2D curves basically.

So we define polygons from isolines.

Now what are isolines?

Let's say we have a function which is a 2D function x and y and the third value of the