Exam 2023 Q16

Exam 2023 Q16

by Lorie Xu -
Number of replies: 5

Hello,

I was doing question 16 and was confused by the fact that the third option, "The visual hull is included in the convex hull" was graded as false in the corrections. I had understood the convex hull as the smallest convex shape defined by the cameras rays that contains the visual hull. Using that definition, the third statement would be true. I was wondering what I wasn't understanding?

Thank you very much.

In reply to Lorie Xu

Re: Exam 2023 Q16

by Zhantao Deng -
Hi,

Good question.

The intersection of the back-projected silhouette cones from the cameras defines the visual hull. The smallest convex set containing the object itself defines the convex hull.

For example, imagine four cameras surround horizontally around a vertical cylinder, with 90-degree angular spacing between neighboring cameras.

The convex hull of the cylinder is itself. However, the visual hull reconstructed from these four views would be a square/rectangular prism enclosing the cylinder. Therefore, in this case, the visual hull contains the convex hull rather than being contained in it.

Best,
In reply to Zhantao Deng

Re: Exam 2023 Q16

by Lorie Xu -
Thanks for your reply!

I'm just still a bit confused about the defition of the convex hull. I intitially thought edges of the convex polygon that makes the convex hull had to be formed by the cameras views. As you said since the convex hull of the cylinder is itself, regardless of the cameras views, it is just the smallest polygon (in the cylinder's case, I suppose a polygon with infinitely-many edges) that contains the object, regardless of the cameras?
In reply to Lorie Xu

Re: Exam 2023 Q16

by Zhantao Deng -
Hi,

Yes, exactly. The convex hull of a shape or point set is defined only by the shape or point set itself. You can think of it as an intrinsic geometric property of the object. External factors such as the cameras or lighting do not affect this definition.

So in the cylinder example, the convex hull is the cylinder itself, regardless of the camera views.
In reply to Lorie Xu

Exam 2023 Q16

by Daniel Alex Alves Ataíde -
Hi Zhantao,

I get the feeling slide 20 from the ‘Shape from Contours’ lecture says that indeed ‘visual hull is included in the convex hull’. Taking your cylinder example, the convex hull would be the rectangular prism enclosing it, not the visual hull (like we can see with the tea pot, where the convex hull completely encloses it). Van you verify this ?

Thank you very much.
In reply to Daniel Alex Alves Ataíde

Re: Exam 2023 Q16

by Zhantao Deng -
Hi,

Based on the video recording of the 2025 lecture (link, 28:22 ~ 29:27), Slide 20 should be interpreted as follows:

1. for a teapot, bounding box is the smallest cube or rectangle that contains it.
2. its convex hull is essentially the smallest convex object that contains it.
3. its visual hull is the volume you will get by doing space carving with a number of cameras.
4. The smallest/best visual hull can be obtained by doing space carving with infinite number of views. 
5. The best visual hull you can get is actually between the convex hull and the real thing.

Therefore, regarding Question 16, “its visual hull as seen from N cameras” is not necessarily contained in the convex hull. For example, when N = 1, the visual hull may extend over an infinitely large region, and therefore would not be contained in the convex hull.

Regarding the cylinder example, since the cylinder itself is convex, its convex hull, i.e. the smallest convex object that contains it, should be the cylinder itself rather than the rectangular prism.

Best,