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 R^{n} ,
the k dimensional Hausdorff measure H^{k}
(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 H^{k}(A)
for
k < n to
define the (* Hausdorff) dimension* of A as

sup{ j : H^{j}(A) = infty
} or
inf { k : H^{k}(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->0}L(A _{
^ }B(x,r))/ L(B(0,1))r^{n}
of a Lebesgue measurable subset
A of R^{n}
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 R ^{2}*.
After proving many of these, we will turn to
the relatively recent application from image processing called

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) =
*
J*_{U\K} |Du|^{2}dL
+ *
J*_{U} (u-g)^{2}dL
+
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.

2. Exercise 14 was not stated correctly in class. See the correct statement below.

3. Exercise 13 (now corrected) should say Hausdorff measure rather than Lebesgue in the conclusion.

4. In class, I followed Frank Jones' proof of the Theorem on P.539 about the primitive F of an L1 function f .

Haidee (Zheng Meng) made the nice suggestion for the proof of the formula for V_F(a,b). After verifying that F is BV and proving the < or = , (page540 midpage),

one can use the Theorem on p.461 to say that F' = f a.e. and then apply the corollary on p.538 with f = F to get the opposite inequality. This argument replaces p.541 and 1/2 of p.540.

1. Show that if E is a Lebesgue measurable subset of R^{n}
, then the union of E with some set Z having

Lebesgue measure L(Z) = 0 is the intersection of a decreasing
sequence

U_{1} , U_{2} , U_{3} , ...
of open sets. Also E is the union of set Y having L(Y) = 0
and the union of an increasing sequence

K_{1} , K_{2} , K_{3} , ...
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} [H^{1}
(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 R^{n} 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 R^{n} , |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 K_{a} 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 R^{2}, let p_{e}
(z) = z - (z . e)e (so that p_{e} is the projection
onto the line perpendicular to e ). For the “lifted middle-thirds”
set K constructed in class, verify that H^{1}[p_{e}
(K)] = 0 for every such e except (0,1) or (0,-1).

7. Show that Steiner symmetrization S_{1} preserves Lebesgue measure,
i.e. L(S_{1}(E)) = L(E) for E in R^{n}.

(Here S_{1}(E) = { (t,x_{2},x_{3}
,...,x_{ n} ) : (x_{1},...,x_{n})
in E , |t| is at most H^{ 1} { s : (s,x_{2},x_{
3} ,...,x_{ n} ) in E } } . )

8. Show that Steiner symmetrization S_{1} doesn't increase diameter,
i.e. diam(S_{1}(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.

Homeworks 10,11,12 pdf file Homeworks 10,11,12 ps file

Homeworks 13,14,15 pdf file Homeworks 13,14,15 ps file

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*.

too far behind and, if you are, to try to catch up over the Rice break.

For exercises 20-22 , suppose that E is a subset of R^{ N}
and recall that

dim E = inf {t : H^{t}(E) = 0 }.

20. Show that H^{k+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 R^{n} 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.

28. Show that if U is a smoothly bounded domain, u

I

every point of the boundary of U .

29. Show that if f = f(x,t) is a C

d/dt J

30. Suppose that F is a family of closed squares in R

Lebesgue measure. Suppose also that, for all a in A , inf {diam(S) : S in F and a in S } = 0 .

Show that some countable disjoint subfamily of F covers Lebesgue almost all of S .

31. Suppose that G is a (possibly uncountable) family of open rectangles in R

and, for each rectangle I in G , A

Prove that the union, over all I in G , of the A

32. Suppose that f : [a,b] -> R is absolutely continuous, f(a) < -1 and

f(b) > 1. Show that J

33. Let U be the open square (-1,1) x (-1,1) ,

g(x,y) = -3 for -1<x<0, -1 < y < 1 ,

g(x,y) = +3 for 0<x<1, -1 < y < 1 ,

and suppose that (u,K) is a minimizer for the Mumford-Shah functional

H

(1) Show that H

(2) Show that H

Hint: Consider the set of points y such that u(x,y) >-1 for all x in (-1,0)

as well as the set of points y such that u(x,y) < 1 for all x in (0,1) .

(3) Give some ideas about how one would try to find the minimizer (u,K) .

No Final Exam but all problems due May 8 !

There will be no class Thursday, April 25. So April 23 will be the last class.

I will be out of town April 24-April 30. After that I will be available for any questions

about homework, the course, etc.

http://math.rice.edu/~hardt/426S02

This page is maintained by Robert Hardt (
email
)

Last edited 4/22/02.