Directionally sensitive weight functions
Roza Aceska
Hossein Hosseini Giv
Department of Mathematical Sciences
Ball State University
Muncie, Indiana
Email: roza.aceska@gmail.com
Department of Mathematics
University of Sistan and Baluchestan
Zahedan, Iran
Email: hossein.giv@gmail.com
Abstract—We study new properties of a recently introduced,
directionally sensitive short-time Fourier transform. Its specific
relationship to the classical short-time Fourier transform and
its quasi shift-invariance property motivate us to introduce
customized weight functions with directional sensitivity.
I. I NTRODUCTION
In the analysis and reconstruction of most real-life phenomena, best results are achieved when locally adaptive mathematical tools are involved. Time-frequency analysis (TFA) [9] has
the optimal mathematical tools in measuring the local qualities
of a sound-like function; these tools are based on transform
(1), often paired up with a so-called weight function which
satisfies inequality (3) and/or (4). The classical TFA tools are
thus fit to solve problems in audio signal processing; however,
they fail at detecting discontinuities (edges) of functions that
represent images, due to their lack of directional sensitivity.
In order to construct analysis tools that excel at edge
detection and are useful in image processing, it is practical
to employ a directionally sensitive (dirS) transform, paired
up with customized dirS weight functions. The purpose of
this report is twofold: we study the properties of a new dirS
transform, defined by (5) as a generalization of the STFT; then
we motivate and introduce dirS weight functions (section III)
that are necessary for future construction of a dirS TFA.
A. The Short-time Fourier transform
In TFA, locality properties are studied by localizing (windowing) the function of interest f and then applying the
Fourier transform, that is, by the short-time Fourier transform
(STFT). Given f, g ∈ L2 (Rd ), the STFT of f with respect to
a so-called window function g is the function Vg f , defined on
R2d as
Vg f (x, ω) = hf, Mω Tx gi = e−2πixω hf, Tx Mω gi,
formula for the STFT
ZZ
1
f=
Vg f (x, ω)Mω Tx γdxdω
hγ, gi
(2)
is well-defined in the weak sense for all f ∈ L2 (R) and windows satisfying hγ, gi =
6 0. Written
integrals,
as vector-valued
we have the adjointness relation Vγ∗ F, h = hF, Vγ hi for a
proper choice of F and h. The operator Vγ∗ is well-defined on
the (mentioned before) modulation spaces and is employed to
generate functions in these spaces. Modulation spaces are used
to classify functions by their time-frequency decay properties,
with the use of moderate weights.
B. Weight functions
In general, a weight is a positive function on a domain
of interest D. By v we denote a continuous, positive, even,
sub-multiplicative weight such that v(0) = 1 and
v(z1 + z2 ) ≤ v(z1 )v(z2 ), ∀z, z1 , z2 ∈ D.
(3)
A positive, even weight function m on R2d is called vmoderate if it satisfies for some C > 0:
m(z1 + z2 ) ≤ Cv(z1 )m(z2 ), ∀z, z1 , z2 ∈ D.
(4)
Two weights m1 and m2 are equivalent if there exists a
constant c > 0 so that for all z ∈ D, it holds c−1 m1 (z) ≤
m2 (z) ≤ cm1 (z). The standardly used weights in modulation spaces theory are the equivalent, sub-multiplicative
radial weights of polynomial type (1 + |x| + |ω|)s and
(1+x2 +ω 2 )s/2 , s ∈ R. Each is also moderate with respect to
v(x, ω) = (1+ω 2 )s/2 . The last one is of significant importance
in calculating the reproducing kernel of Sobolev spaces and
is in fact, equivalent to (1 + |ω|)s . Thus, weight functions
[10], [9] are useful to ensure certain smoothness and decay
properties of whole classes of functions.
(1)
C. Modulation Spaces
where Tx and Mω denote the operators of translation and
modulation. Since (1) is well-defined on time-frequency shiftinvariant, mutually dual function spaces, one can employ the
STFT in a variety of function spaces beyond L2 (Rd ), e.g.
modulation spaces ([6] and the references within).
For a fixed window g 6= 0, Vg is a norm-preserving isometry
i.e. it holds kVg f k2 = kf k2 kgk2 ; in addition, the inversion
c
978-1-4673-7353-1/15/$31.00 2015
IEEE
Moderate weights ensure a more precise classification of
functions; in particular, a time-frequency shift-invariance is
provided within a studied function class. The so-called modulation spaces [6] possess such shift-invariance property.
Modulation spaces are defined via weighted Lp,q -norms of
the STFT. Given a fixed Schwartz window function g 6= 0,
a submultiplicative weight v, a v-moderate weight m on R2
p,q
and 1 ≤ p, q < ∞, the modulation space Mm
(R) consists of
all tempered distributions f ∈ S 0 (R) for which Vg f has finite
weighted mixed-norm
Z Z
q/p
kf kqMm
|Vg f (x, ω)|p m(x, ω)p dx
dω < ∞
p,q :=
R
R
p,q
(with the usual adjustment for p, q = ∞). That is, Mm
(R) is
p,q
naturally embedded in Lm (R).
p,q
For all 1 ≤ p, q ≤ ∞, Mm
(R) is a Banach space,
independent of the choice of a nonzero window g. Also, given
p,q
p,q
any f ∈ Mm
(R), it shows that Tx Mω f is in Mm
(R) for all
x, ω ∈ R; in addition, the inversion formula (2) holds true in
the weak sense, which plays a key role in constructing atomic
decompositions [5] of modulation spaces.
D. Directional Sensitivity
II. I MPORTANT P ROPERTIES OF THE DIR STFT
We state several new properties of the dirSTFT itself, that
were not addressed in [7]. These properties indicate the desired
qualities of the respective dirS weight functions we further
introduce in section III.
A. Relationship between the two integral transforms
Theorem 1. Let (η, y, z) ∈ ∆ and (x, w) ∈ R2 . If f ∈
L1 (R2 ), then
Vg (Ty Rη Mz f )(x, w) = e−2πiyw DS g f (η, x − y, wη − z).
In particular,
Directionally sensitive frames, a directionally sensitive
wavelet transform and a ridged Gabor transform have been
studied in [4], [3] and [8] respectively. Here, we use a
directionally sensitive STFT (in short notation, dirSTFT) that
was recently introduced in [7] as a modification of (1). Given
δ = (ξ, x, t) ∈ ∆ = S 1 × R × R2 , a directional window is
defined by
gδ = gξ,x (t) = g(ξ · t − x),
1
In particular, if f , fˆ and g, ĝ are continuous, then (7) holds
point-wise.
∞
where g ∈ L (R) ∩ L (R) is a one-variable window. If f ∈
L1 (R2 ), then the dirSTFT of f with respect to g is welldefined by
Z
DS g f (ξ, x, ω) :=
f (t)g(ξ · t − x)e−2πiω·t dt. (5)
|Vg (Ty Rη Mz f )(x, w)| = |DS g f (η, x − y, wη − z)|.
Proof: If f ∈ L1 (R2 ), then h = Ty Rη Mz f ∈ L1 (R) as a
consequence of the Fourier slice theorem; so computing the
STFT of h makes sense. It shows that Vg (h)(x, w) equals
Z Z
f (t) e2πit·z dm(t) g(r − x) e−2πirw dr.
R
DSg f (ξ, x, ω) = hRξ M−ω f, Tx gi,
(6)
where Rξ denotes the Radon transform [11] evaluated at ξ.
As a consequence, a weak-sense reconstruction formula holds
true on a restricted set of functions: for a fixed g ∈ L∞ (R)
and all f ∈ L1 (R2 ) ∩ L2 (R2 ), it holds
Z
1
DS g f (δ)gδ dδ.
(7)
f=
kgk22 ∆
η·t=r−y
With a simple change of variables s = r − y, we obtain the
stated result.
Due to Theorem 1, if η = (1, 0) and (x, ω) ∈ R × R, then
DS g f (η, x, (ω, 0)) = Vg (Rη )f (x, ω).
R2
Note that the integral transform G used in [8] is defined on
S 1 × R × R and is somewhat similar to (5); in fact, it can
be described by (5) as G(ξ, x, c) = DSg (ξ, x, cξ). Therefore,
transform (5) is more general and has the potential to define
larger function spaces than the spaces introduced in [8].
The dirSTFT is evaluated on ∆ = S 1 × R × R2 ; we make
use of notation δ = (ξ, x, ω) ∈ ∆ and dδ = dωdxdξ. The
operations of addition/subtraction and multiplication by scalars
on R and R2 are the standard ones, while on S 1 we have: given
ξ = (cos α, sin α), ν = (cos β, sin β) ∈ S 1 and c ∈ R, let
ξ ⊕ ν := (cos(α + β), sin(α + β)),
ξ ν := (cos(α − β), sin(α − β)) and
c · ξ := (cos(cα), sin(cα)).
Specific relations, typical of the STFT, hold true for transform (5) as well e.g. the orthogonality relations. Also, a normequivalence relation holds true for f ∈ L1 (R2 ) ∩ L2 (R2 ),
g ∈ L∞ (R) ∩ L2 (R); that is, kDS g f k2 = kf k2 kgk2 . In
addition,
(8)
(9)
This motivates the specific design of the weight functions
studied in section III: those weight functions are invariant
when getting close to directions1 (1, 0), (0, 1) ∈ S 1 .
B. Quasi Shift-Invariance of dirSTFT
By rβ (ω) we denote the counter-clockwise rotation by β
for any ω ∈ R2 . Given a function of interest f , defined on
R2 , we have f−β (t) := f (r−β (t)) for every t ∈ R2 .
Theorem 2. Let f ∈ L1 (R2 ) and g ∈ L∞ (R). For arbitrary
(ξ, x, ω), (ν, y, η) ∈ ∆, it holds
DS g f (ξν, x−y, ω−η) = DS g (Mrβ (η) f−β ) (ξ, x−y, rβ (ω)),
where β is such that ν = (cos β, sin β) ∈ S 1 .
Proof: By the basic properties of dirSTFT, it can be easily
shown that DS g f (ξ ν, x − y, w − η) equals
Z
Z
g(s + y − x)
e2πi(η−w)·t f (t) dm(t) ds. (10)
R
(ξν)·t=s
Since (ξ ν) · t = r−β (ξ) · t = ξ · rβ (t), after a change of
variables l = rβ (t), it shows that the inner integral in (10)
R
equals ξ·l=s e2πi rβ (η−ω)·l f (r−β (l)) dm(l). Then the stated
result follows after employing the basic rules of integration.
1a
formula similar to (9) holds true for η = (0, 1)
An easy consequence of Theorem 2 is the precise norm
change for different choices of a window; also, translated
windows will have no significant norm effect:
Corollary 1.
cos2 ((α + β)γ ) ≤ Cγ cos2 αγ , sin2 ((α + β)γ ) ≤ Cγ sin2 αγ .
(16)
1) If g 6= h ∈ L1 (R) ∩ L∞ (R), then
kDS h f k2 =
khk2
kDS g f k2 .
kgk2
(11)
2) If g̃ := T−y g, then kDS g̃ f k2 = kDS g f−β k2 .
III. W EIGHT F UNCTIONS WITH D IRECTIONAL
S ENSITIVITY
In [8], the authors weight the ridged window and then
introduce weighted function spaces, which are all subspaces of
L1 ∩ L2 . In Section II we study the properties of a transform
that makes use of a weight-free dirS window. In this section we
offer a moderate (or alternatively, bounded) weight to pair-up
with the dirS transform, and emphasize functions smoothness
properties as they vary with direction. Once we have defined
dirS weights with good quality properties, we can introduce
various function classes using weighted norms. For instance,
as an upcoming, natural generalization of modulation spaces
(subsection I-C), we can categorize the behavior of a function
f i.e. measure its decay/smoothness in various directions,
weighted by weight m, by checking the existence of the
following integral:
Z
r1
p
r
.
(12)
kDSg f (ξ, ·, ·)| m(ξ, ·, ·)kLp,q (R×R2 ) dξ
S1
We introduce a weight function on ∆ that is moderate and
sensitive to direction as follows:
Let ξ = (cos α, sin α), ν = (cos β, sin β) ∈ S 1 . Let
δ = (ξ, x, ω), δ 0 = (ν, y, η) ∈ ∆. For some s > 0, the submultiplicative weight to be employed is defined by
v(ν, y, η) = (1 + |y|2 )s1 (1 + η12 + η22 )s2 .
(13)
Take γ ∈ (0, π/10) be a small positive control angle. Let
cos2 (α + β) ≤ cot2 γ cos2 α, sin2 (α + β) ≤ cot2 γ sin2 α.
Let Cγ = cot2 γ > 1, which is a good choice for all cases as
cos2 γ
≤ Cγ cos2 γ,
cos2 (π/2 − γ) ≤ cos2 γ ≤ Cγ cos2 γ,
cos2 γ
≤ Cγ cos2 (π/2 − γ).
(17)
Proof of Theorem 3: Due to Lemma 1, it shows that
m(δ + δ 0 )
(18)
(1 + |x + y|2 )
= 1 + (ω1 + η1 )2 cos2 (α + β)γ + (ω2 + η2 )2 sin2 (α + β)γ
≤ 2Cγ 1 + ω12 cos2 αγ + ω22 sin2 αγ 1 + η12 + η22 .
B. Controlled Weights
Working with weights that are not moderate can allow for
enlarging or narrowing down the specific class of functions
of interest. For this reason we introduce so-called controlled
weights i.e. weights bounded by a sub-multiplicative weight.
Let
M (ξ, x, ω) := (1 + |x|2 )s1 (1 + |ξ · ω|2 )s2 .
R1
:=
(−γ, γ) ∪ (π − γ, π + γ),
R2
:=
(π/2 − γ, π/2 + γ) ∪ (3π/2 − γ, 3π/2 + γ),
R3
:=
[0, 2π] \ (R1 ∪ R2 ).


γ
αγ = π/2 − γ


α
Proof: It suffices to prove the lemma for si = 1, i = 1, 2.
Notice that, due to the oriented angles (14) and the symmetry
involved, only angles in [γ, π/2 − γ] are employed in (15).
Given α, α + β ∈ [γ, π/2 − γ], it holds that sin2 α, cos2 α,
sin2 (α + β), cos2 (α + β) ∈ [sin2 γ, cos2 γ], thus
This means that (15) is a moderate weight.
A. Moderate Weights
Let
Lemma 1. There is γ ∈ (0, π/4) and Cγ = cot2 γ > 1 such
that for all α, α + β ∈ [0, 2π] it holds
(19)
Immediately, the following result holds true:
Lemma 2. The weight (19) is bounded by the submultiplicative weight
if α ∈ R1
if α ∈ R2
if α ∈ R3
(14)
and we introduce an oriented weight function by
m(δ) = (1 + |x|2 )s1 (1 + ω12 cos2 αγ + ω22 sin2 αγ )s2 . (15)
Theorem 3. The oriented weight (15) is moderate with respect
to the submultiplicative weight (13).
To prove this theorem, we introduce a new lemma:
V (ξ, x, ω) = (1 + |x|2 )s1 (1 + |ω|2 )s2 .
(20)
Note that instead of using (19), if it is of interest, we can
work with a weight function of form
(1 + |x|2 )s1 (1 + |ξ1 ω1 |2 )s2 (1 + |ξ2 ω2 |2 )s3 ,
(21)
and weight (21) delivers more precise control in different
directions. It is easy to see that the following holds true:
Lemma 3. For s2 = s3 , (19) and (21) are equivalent weights.
C. Directionally Sensitive Variable Bandwidth Weight
The notion of variable bandwidth (VB) has been precisely
defined in [1], [2] for sound-like functions via a strip STb =
{(x, ω) ∈ R × R̂ : |ω| ≤ b(x)}, where b(x) ≥ 0 describes
the time-varying broadness of STb . A function with VB has
essential time-frequency support on a set of type STb i.e. most
of its STFT content is supported within STb . Functions with
VB have been extensively studied in applications ([1], [2] and
references within).
A VB weight mb,s is defined with the use of a vertical
distance function db
(
0,
if z ∈ STb ,
.
db (z) =
|ω| − b(x), if z ∈
/ STb .
functions b1 , b2 will still produce moderate weights under
some mild conditions.
To give a more precise classification of a dirVB function of
interest, it is beneficial to use multiplication by (25); this adds
graded weight to the exterior of STb and, as an effect, the local
dirSTFT decay is better quantified. In particular, if a weighted
dirSTFT of a function f is integrable for some s1 , s2 > 0, the
dirSTFT itself is ensured to decay outside STb , faster than a
polynomial determined by the chosen weight.
IV. C ONCLUSION AND FUTURE WORK
We introduce and study dirS, locally adaptive mathematical tools: a transform, defined by (5) as a generalization
of the STFT, and several classes of dirS weight functions
(section III). These tools are dirS, and have the potential to
deliver a very precise classification of functions by the use
The following weight is moderate under mild conditions
of norms of type (12). Properties stated in (7), Theorem 1
mb,s (z) = (1 + db (z))s , s > 0.
(22) and Theorem 2 can be seen as generalizations of the inversion
formula and the shift-invariance property of the STFT.
For instance, the constant bandwidth weight
We introduced oriented, dirS moderate weights (III-A), and
(
as an alternative, dirS controlled weights (19) and (21). The
1,
if |ω| ≤ a,
ma,s (z) =
for some
s > 0,
local dirSTFT decay is better quantified when using weight
(1 + |ω| − a)s , otherwise,
(23) functions in the norm estimates of type (12). Controlled
s
is moderate with respect to (1 + |ω|) ; generalizations of (23) weights allow for enlarging or narrowing down a specific class
by slowly varying the values for a are possible and are used of functions of interest. We also give an example of a moderate
to define a function with VB as an element of a weighted dirS variable bandwidth weight in subsection III-C, and conjecture that under some conditions, more complex bandwidth
modulation space.
The notion of variable bandwidth (VB) can be expanded functions b1 , b2 will still produce moderate weights. Note that
on ∆ in the following manner: let the pair of non-negative the purpose of weight (25) is to set graded importance to the
functions b = (b1 , b2 ) describe a directional VB (dirVB) strip exterior of STb and to locally describe the dirSTFT decay.
If a weighted dirSTFT of a function f is integrable (12),
STb , which is defined as the set
then the dirSTFT itself is decaying outside STb , faster than a
{(ξ, x, ω) ∈ ∆ : |ω1 ξ1 | ≤ b1 (ξ, x), |ω2 ξ2 | ≤ b2 (ξ, x)}.
polynomial determined by the chosen weight. This helps with
giving a more precise classification of a function of interest,
Let δ = (ξ, x, ω) ∈ ∆, i = 1, 2 and define a distance function
and is suitable for filtered image analysis and reconstruction.
(
With this report we have set the ground for creating a natural
0,
if δ ∈ STb ,
(24) generalization of modulation spaces. As future work, we aim
db,i (δ) :=
|ωi ξi | − bi (ξ, x), if δ ∈
/ STb .
to construct a dirS TFA, that succeeds at edge detection by
ensuring certain smoothness and decay properties in certain
A VB weight on ∆ for some s1 , s2 > 0 is defined by
directions, and can be beneficial in image processing.
mb,s1 ,s2 (δ) := (1 + db,1 (δ)2 )s1 /2 (1 + db,2 (δ)2 )s2 /2 . (25)
R EFERENCES
Notice that this definition is reducible to (22); in addition, for
[1] R. Aceska and H. Feichtinger. Functions with variable bandwidth via
bi ≡ 0, (25) is reduced to (21).
time-frequency analysis tools.
J. of Mathematical Analysis Appl.,
Corollary 2. Let a > 0, s= (s1 , s2 ) and choose bi = a. Set
s2
s1
h(δ) = 1 + (|ω1 ξ1 | − a)2
1 + (|ω2 ξ2 | − a)2 . Then

1
if both |ωi ξi | ≤ a



 1 + (|ω ξ | − a)2 s1 , if only |ω ξ | ≤ a
1 1
2 2
m2a,s (δ) =
2 s2

, if only |ω1 ξ1 | ≤ a
1 + (|ω2 ξ2 | − a)



h(δ),
otherwise
is a moderate weight with respect to (20).
The weight function in the last example behaves in a
manner similar to weight (23) and proving it is moderate is a
simple exercise. This gives hope that more complex bandwidth
vol. 382(1): pp. 275-289, Elsevier, 2011.
[2] R. Aceska and H. G. Feichtinger. Reproducing kernels and variable
bandwidth. J. Funct. Spaces Appl., (art. no. 469341), 2012.
[3] E. Candes. Ridgelets: theory and applications.. PhD thesis, Standford
University, 1998.
[4] O. Christensen, B. Forster and P. Massopust. Directional time frequency analysis via continuous frames. arXiv:1402.3682.
[5] H. G. Feichtinger and K. Gröchenig. Banach spaces related to
integrable group representations and their atomic decompositions, I.
J. Funct. Anal., 86(2):307–340, 1989.
[6] H. G. Feichtinger. Modulation spaces: looking back and ahead.
Sampling Theory, Signal and Image Processing, 5(2): 109-140, 2006.
[7] H. Hosseini Giv. Directional short-time Fourier transform. J. Math.
Anal. Appl., vol.399(1): pp. 100 - 107, 2013.
[8] L. Grafakos and C. Sansing. Gabor frames and directional timefrequency analysis. Appl. Comput. Harmon. Analysis, 25(1): pp.47–
67, 2008.
[9] K. Gröchenig. Foundations of time-frequency analysis. Applied and
Numerical Harmonic Analysis. Birkhäuser Boston, MA, 2001.
[10] K. Gröchenig. Weight functions in time-frequency analysis. In
Luigi Rodino and et al., editors, Pseudodifferential operators: Partial
differential equations and time-frequency analysis, volume 52, pages
343 – 366, 2007.
[11] S. Helgason. The Radon Transform. Birkhäuser, 1980.