Singular Sets of Minimizers for the Mumford-Shah Functional

Contents......Page 6
Foreword......Page 9
1 The Mumford-Shah functional and image segmentation......Page 13
2 Definition of the functional......Page 17
3 Minimizing in u with K fixed......Page 20
4 H[sup(n–1)] is not semicontinuous but J has minimizers......Page 23
5 Simple examples, no uniqueness, but few butterflies in Hong-Kong......Page 26
6 The Mumford-Shah conjecture and some known results......Page 42
7 Many definitions of almost- and quasiminimizers......Page 48
8 Reduced minimizers and pairs, coral pairs......Page 62
9 Absolute continuity on lines......Page 74
10 Some removable sets for W[sup(1,p)]......Page 78
11 Composition with other functions......Page 82
12 Poincaré estimates for f∈W[sup(1,p)](Ω)......Page 86
13 Boundary values and restrictions to hypersurfaces......Page 91
14 A simple welding lemma......Page 97
15 Energy-minimizing functions with prescribed boundary values......Page 101
16 Energy minimizers for an increasing union of open sets......Page 106
17 Conformal invariance and the Neumann condition......Page 109
18 Local Ahlfors-regularity: the statements......Page 116
19 A small density result......Page 124
20 Proof of local Ahlfors-regularity in dimension 2: the scheme......Page 125
21 Energy estimates for the Ahlfors-regularity result (n=2)......Page 133
22 Simple estimates on the Poisson kernel......Page 148
23 Carleson measures......Page 154
24 The property of projections (n=2)......Page 159
25 The concentration property (n=2)......Page 166
26 Average number of points on a circle......Page 171
27 Uniform rectifiability when n=2......Page 174
28 The co-area formula......Page 179
29 K contains big pieces of connected sets locally......Page 181
30 Simple arcs in connected sets with finite H[sup(1)]-measure......Page 197
31 Big pieces of connected sets and uniform rectifiability......Page 202
32 Application to quasiminimizers and a temporary conclusion......Page 215
33 Vitali covering lemmas......Page 217
34 Local Hausdorff convergence of sets......Page 223
35 Uniform concentration and lower semicontinuity of Hausdorff measure......Page 227
36 A little more on the existence of minimizers......Page 232
37 Limits of admissible pairs......Page 237
38 Limits of local almost-minimizers are local topological almost-minimizers......Page 244
39 Limits of almost-minimizers up to the boundary......Page 267
40 Blow-up limits......Page 276
E Pieces of C[sup(1)] Curves for Almost-Minimizers......Page 280
41 Uniform rectifiability, the Peter Jones numbers β(x, r), and why K is flat in many places......Page 281
42 Lower bounds on the jump of u when K is flat and ∇u is small......Page 287
43 Normalized jumps often tend to get larger......Page 291
44 On the size of the holes in K......Page 293
45 ω*[sub(p)](x,r) (sometimes with flatness) controls the surface and energy inside......Page 297
46 Simple consequences when n=2, and the existence of good disks......Page 302
47 Bonnet’s lower bound on the derivative of energy......Page 306
48 The main estimate on the decay of the normalized energy ω[sub(2)](x,r) (n=2)......Page 319
49 Self-improving sets of estimates (n=2)......Page 324
50 Three conditions for K to contain a nice curve (n=2)......Page 330
51 K∩B(x,r) is often a nice C[sup(1)] curve (n=2)......Page 341
52 Jump and flatness control surface and energy......Page 347
53 Similar results with spiders and propellers (n=2)......Page 353
54 Global minimizers in R[sup(n)] and a variant of the Mumford-Shah conjecture......Page 364
55 No bounded connected component in R[sup(2)]\K......Page 371
56 No thin connected component in R[sup(2)]\K either; John domains......Page 376
57 Every nontrivial connected component of K is a chord-arc tree......Page 379
58 The case when u is locally constant somewhere (n=2)......Page 389
59 Simple facts about lines and propellers; blow-in limits......Page 400
60 Another regularity result for local almost-minimizers; points of low energy......Page 405
61 Bonnet’s theorem on connected global minimizers......Page 410
62 Cracktip is a global minimizer......Page 423
63 The magic formula with [equation omitted] in terms of K......Page 440
64 The case when K is contained in a countable union of lines......Page 451
65 Another formula, with radial and tangential derivatives......Page 456
66 R[sup(2)]\K is connected when (u,K) is an exotic global minimizer......Page 462
67 Points of high energy, the singular singular set K[sup(#)]......Page 472
68 No small loop in K, the chord-arc and the John conditions......Page 478
69 Spiral points......Page 484
70 Two last regularity results......Page 498
71 From the Mumford-Shah conjecture in the plane to its local version......Page 502
72 Local Ahlfors-regularity for quasiminimizers......Page 504
73 Uniformly rectifiable sets of all dimensions......Page 521
74 Uniform rectifiability for quasiminimizers......Page 527
75 C[sup(1)] regularity almost-everywhere for almost-minimizers......Page 532
76 What is a Mumford-Shah conjecture in dimension 3?......Page 537
77 Ahlfors-regularity near the boundary......Page 548
78 A reflection trick......Page 560
79 Boundary regularity for almost-minimizers in dimension 2......Page 568
80 A few questions......Page 579
References......Page 582
E......Page 588
R......Page 589
W......Page 590