IMAGE COMPRESSION WITH GEOMETRICAL WAVELETS
Erwan
Le
Pennec
Ecole Polytechnique
Centre de MathCmatiques AppliquCes
91 128 Palaiseau Cedex
France
ABSTRACT
We introduce a sparse image representation that takes ad
vantage of the geometrical regularity of edges in images. A
new class of onedimensional wavelet orthonormal bases,
called foveal wavelets, are introduced to detect and recon
struct singularities. Foveal wavelets are extended in two di
mensions, to follow the geometry of arbitrary curves. The
resulting two dimensional “bandelets” define orthonormal
families that can restore close approximations of regular
edges with few nonzero coefficients. A double layer im
age coding algorithm is described. Edges are coded with
quantized bandelet coefficients, and a smooth residual im
age is coded in a standard twodimensional wavelet basis.
1. GEOMETRICAL COMPRESSION
Currently, the most efficient image transform codes are ob
tained in orthonormal wavelet bases. For a given distor
tion associated to a quantizer, at high compression rates the
bit budget is proportional to the number of nonzero quan
tized coefficients
[
11. For images decomposed in wavelet
orthonormal bases, these nonzero coefficients are created
by singularities and contours. When the contours are along
regular curves, this bit budget can be reduced by taking ad
vantage of this regularity [2]. Many image compression
with edge coding have already been proposed [3,4,5,6],
but they rely on adhoc algorithms to represent the edge
information, which makes it difficult to compute and opti
mize the distortion rate. In this paper, we construct “ban
delet” orthonormal bases that carry all the edge informa
tion and take advantage of their regularity by concentrating
their energy over few coefficients. An application to image
compression is studied.
2. FOVEAL WAVELET BASES
Contours are considered here as onedimensional singular
ities that move in the image plane. We first construct a
new family of orthonormal wavelets, all centered as the
same location, which can “absorb” the singular behavior
‘Support in parts by
an
AlcatelEspace
grant and
a DARPAFastvideo
grant
25741OOFo945
Stkphane Mallat*
Ecole Polytechnique
Centre de MathCmatiques AppliquCes
New York University
Courant Institute of Mathematical Sciences
of a signal. We define two mother wavelets
Q’(t)
and
Q2
(t),
which are respectively antisymmetric and symmet
ric with respect to
t
=
0, and such that
J
@(t)dt
=
0 for
IC
= {
1,2}. For any location
U
we denote
Q?
39u
(t)
=
2j/’
qk(2j(t

U))
for
L
=
1,2.
There exists such mother wavelets, which are
C’
and such
that for any
U
E
R and
J
E
Z,
the family
I
is orthonormal [7]. These wavelets zoom on a single posi
tion
U
and are thus called
foveal
wavelets,
by analogy with
the foveal vision. To reconstruct discontinuities, we insure
that left and right indicator functions,
l[u,+m)
and
l(m,u~
can be written as linear combinations of foveal wavelets.