This course is a continuation of Math 425 which treated Lebesgue Integration Theory and Lebesgue Measure. Math 426 will focus around Hausdorff measure. Whereas Lebesgue (outer) measure L provides a notion of the n dimensional size for any subset A of Rn , the k dimensional Hausdorff measure Hk (A) gives, for every nonnegative number k , a precise notion of k dimensional size . While H n (A) = L (A), one may use the other numbers Hk(A) for k < n to define the ( Hausdorff) dimension of A as
sup{ j : Hj(A) = infty } or inf { k : Hk(A) = 0 }.
Then countable sets have dimension 0 , smooth curves have dimension 1, and smooth surfaces have dimension 2. A more striking example is the Cantor ternary set which has dimension k = log2/ log3 and in fact finite positive Hausdorff measure in this dimension. In general, one expects a self-similar set , consisting of m essentially disjoint congruent pieces each similar to the whole set but scaled down by a factor 1/n , to have dimension log m / log n . Sets of non-integer dimension are examples of fractals. The Lebesgue integral employing Hausdorff measure is an important and useful tool, but not all properties of Lebesgue measure have analogues with Hausdorff measure. For example, Lebesgue’s beautiful theorem that the n dimensional density function
lim r->0L(A ^ B(x,r))/ L(B(0,1))rn of a Lebesgue measurable subset A of Rn coincides almost everywhere with the characteristic function of A is no longer true for the k dimensional density of a compact subset of finite k dimensional measure when k < n . Nevertheless, there are some general density properties that are true for H k which we will discuss. For k being an integer there are even some results about tangency, the extent to which the set locally looks near some points at small scales like a k dimensional plane. These are important both historically and for applications in the special case of one dimensional sets in the plane R2. After proving many of these, we will turn to the relatively recent application from image processing called image segmentation.
Here the problem concerns improving the appearance of a noisy image which
is given as a (grayscale intensity) function
g(x) defined on a planar region
U
or digitally on a discrete grid (pixels) in the plane.
One has two competing goals in improving the
image: 1)
to improve the sharpness of the edges (as in a profile curve)
and 2) to improve the regularity
in smooth regions (as in a facial area).
It is mathematically difficult to describe or
characterize these processes and hence to find some kind of algorithmic
procedure to effect these.
There have been many models proposed and used for image segmentation
and related problems. We will discuss some of the variational theory
and mathematics associated with the Mumford-Shah model where one considers
minimizers of an energy
E(u,K) =
JU\K |Du|2dL
+
JU (u-g)2dL
+
H 1 (K)
corresponding to an image
u with edge
set K .
Although there is now a considerable literature on this
model, there are still many open questions. We will try to highlight
some aspects that fit very well with our study of 1 dimensional subsets
of R 2
.
Background for the course is a knowledge of Lebesgue integration as in Math 425.
TThurs 9:25-10:40 in Herman Brown 423
Robert Hardt Office: Herman Brown 430; Office hours: 1-2 MWF (and others by appt.),
Email: hardt@rice.edu, Telephone: ext 3280
Frank Jones, Lebesgue Integration on Euclidean Space
First Course, Jones and Bartlett, 1993.
(See Frank for a good deal if you don't already have
a copy.)
Other references:
K.J. Falconer, The Geometry of Fractal Sets, (paperback) Cambridge University Press, 1985.
J-M. Morel and S. Solimini, Variational Methods in Image Segmentation, Birkhauser, 1995.
1. Show that if E is a Lebesgue measurable subset of Rn
, then the union of E with some set Z having
Lebesgue measure L(Z) = 0 is the intersection of a decreasing
sequence
U1 , U2 , U3 , ...
of open sets. Also E is the union of set Y having L(Y) = 0
and the union of an increasing sequence
K1 , K2 , K3 , ...
of compact sets.
2. For the “lifted middle-thirds” set K constructed in class, verify
that lower and upper densities,
liminf r -> 0 r -1 [H1
(K intersect B(0,r))] and limsup r -> 0 r -1
[H 1 (K intersect B(0,r))] ,
are different (where 0 = (0,0) is the lower left corner
of the set).
3. Show that if A is a convex open subset of Rn and f
is continuously differentiable on A , then
Lip f = sup { |grad f(x)| : x is in A } . ( Note
that this formula is correct for f being real-valued.
For f being vector-valued one should replace |grad f(x)|
by sup { e . D f(x) : e in Rn , |e|=1 }. )
(#1-3 due Thursday, January 24)
4. Let S = [0,1]2 (the unit square). Using only the definition
of Hausdorff measure (and the fact that
the Lebesgue measure of S is 1 ), show that 0 < H
2 (S) < 2 .
5. Suppose that 0 < a < 1 and that Ka is the Cantor
set obtained by recursively omitting from intervals I
the middle interval of length a [diam I ]. (So the usual
Cantor set is K 1/3 .) Find the (self-similarity) dimension
of K a .
6. For each unit vector e in the plane R2, let pe (z) = z - (z . e)e (so that pe is the projection onto the line perpendicular to e ). For the “lifted middle-thirds” set K constructed in class, verify that H1[pe (K)] = 0 for every such e except (0,1) or (0,-1).
(#4-6 due Thursday, January 31)7. Show that Steiner symmetrization S1 preserves Lebesgue measure,
i.e. L(S1(E)) = L(E) for E in Rn.
(Here S1(E) = { (t,x2,x3
,...,x n ) : (x1,...,xn)
in E , |t| is at most H 1 { s : (s,x2,x
3 ,...,x n ) in E } } . )
8. Show that Steiner symmetrization S1 doesn't increase diameter, i.e. diam(S1(E)) is at most diam(E) for E in R n .
9. For any outer measure m on X , show that the family M = { m measurable
subsets of X } is a
sigma algebra and the triple ( X, M, m restricted to M
) is a measure space.
16. Exercise number 1 on Page 512 of Jones.
17. Exercise number 16 on Page 530 of Jones.
18. Exercise number 17 on Page 532 of Jones.
19. Exercise number 19 on Page 536 of Jones.
For exercises 20-22 , suppose that E is a subset of R N
and recall that
dim E = inf {t : Ht(E) = 0 }.
20. Show that Hk+1( E x [0,1] ) = 0 if and only if H k(E) = 0 .
21. Show that dim (E x E) is at most 2k .
22. Show that dim { x-y : x and y belong to E } is at most 2k.
23. Show that every sequence in a precompact metric space contains a Cauchy
subsequence.
(Recall that a metric space is precompact (or totally bounded) if it admits,
for each positive d, a finite cover by open balls of radius d.)
24. Show that a metric space E is compact if and only if E is precompact
and complete.
25. Show that if K is a compact subset of Rn with H
n-1( K ) = 0 , then the complement of K is connected.
26. Show that for W equalling the open unit square (0,1) x (0,1)
in R 2, the Sobolev space H(W) is separable, that is,
has a countable dense subset. Hint: First approximate by smooth functions,
then by
certain continuous piece-wise affine functions.
27. Show that H(W) has a countable Hilbert basis.
http://math.rice.edu/~hardt/426S02
This page is maintained by Robert Hardt (
email
)
Last edited 4/22/02.