So I suppose that the $L^1$ norm is something like $\int_0^1 |f|\,dx$, and the $L^k$ norm is something like $\left(\int_0^1 |f|^k\,dx\right)^{1/k}$, implying that $L^\infty$ norm is $\max_{x=0}^1 f(x)$, but what is the $L^0$ norm?

keenan

Well, suppose that for vectors in R^n you define the L0 norm as the number of nonzero elements. Is this really a norm? Which properties hold and which do not?

Now you can think about defining an L0 norm for functions. Rather than counting the number of elements, you could say that L0 is the total length over which the function is nonzero. Is this really a norm? Why or why not? Which properties still hold?

bqdx

I was curious why the integral has to be over the interval (0,1).
Why not use (-INFINITY,+INFINITY) as the interval?

keenan

Because it's easier to draw pictures of functions on the unit interval. :-)

But yes, you can define an L^p norm on lots of domains, not just [0,1].

omegaiota

How is L^p norm used in computer graphics? Is there a particular preference for larger p or smaller p?

keenan

Yeah, I'd guess about 90% of graphics algorithms use either L^2 or L1. In a nutshell:

L2 is really easy to minimize/optimize (often amounting to linear equations)

L1 tends to produce solutions that are "sparse" (often requiring more sophisticated optimization)

The idea of using L1 for graphics/signal processing problems follows the fairly recent trend of "compressed sensing." There's a pretty nice article here about compressed sensing and the "one pixel camera."

So I suppose that the $L^1$ norm is something like $\int_0^1 |f|\,dx$, and the $L^k$ norm is something like $\left(\int_0^1 |f|^k\,dx\right)^{1/k}$, implying that $L^\infty$ norm is $\max_{x=0}^1 f(x)$, but what is the $L^0$ norm?

Well, suppose that for vectors in R^n you define the L0 norm as the number of nonzero elements. Is this really a norm? Which properties hold and which do not?

Now you can think about defining an L0 norm for functions. Rather than counting the number of elements, you could say that L0 is the total length over which the function is nonzero. Is this really a norm? Why or why not? Which properties still hold?

I was curious why the integral has to be over the interval (0,1). Why not use (-INFINITY,+INFINITY) as the interval?

Because it's easier to draw pictures of functions on the unit interval. :-)

But yes, you can define an L^p norm on lots of domains, not just [0,1].

How is L^p norm used in computer graphics? Is there a particular preference for larger p or smaller p?

Yeah, I'd guess about 90% of graphics algorithms use either L^2 or L1. In a nutshell:

The idea of using L1 for graphics/signal processing problems follows the fairly recent trend of "compressed sensing." There's a pretty nice article here about compressed sensing and the "one pixel camera."