Figure 1: Essential support of a wave packet with parameters (α, β), in space (left), and in frequency
(right). The parameter α indexes the multiscale nature of the transform, from 0 (uniform) to 1
(dyadic). The parameter β measures the wave packet’s directional selectivity, from β = 0 (best
selectivity) to β = 1 (poor selectivity). Wave atoms are the special case α = β = 1/2. On the left,
the dots indicate the grid over which wave packets corresponding to the same wave number should
be translated. On the right, the different wedges correspond to wave vectors at different scales and
angles.
• in x, the essential support of ϕ
µ
(x) is of size ∼ 2
−αj
vs. 2
−βj
as scale j ≥ 0, with oscillations
of wavelength ∼ 2
−j
transverse to the ridge; and
• in frequency ω, the essential support of ˆϕ
µ
(ω) consists of two bumps, each of size ∼ 2
αj
vs.
2
βj
as scale j, at opposing angles and distance ∼ 2
j
from the origin.
These conditions are perhaps more easily grasped from Figure 1. These are the requirements we
put on each single wave packet. Then the collection of wave packets, corresponding to a choice
of α and β, should be indexed by various wave numbers and positions (put together in the single
index µ), so that the tiling of phase-space is compatible with the estimates above.
We hope that a description in terms of α and β will clarify the connections between various
transforms of modern harmonic analysis. Curvelets [8] correspond to α = 1, β = 1/2, wavelets
(including MRA [26], directional [2], complex [29]) are α = β = 1, ridgelets [4] are α = 1, β = 0,
and the Gabor transform is α = β = 0. Wave atoms are defined as the point α = β = 1/2. The
situation is summarized in Figure 2.
1.2 Trade-off: Warping Invariance vs. Directionality
Let us consider the following very simple model, a toy for anisotropic textures. We say that a
function is an oscillatory pattern if it is the image under a smooth diffeomorphism of a function
that oscillates only in one direction, say along the coordinate x
1
. Mathematically, we can put
x = (x
1
, x
2
) and formulate our model as
f(x) = sin(Ng(x)) h(x), (1)
3
