Hexagonal QMF Banks and Wavelets

A chapter within Time-Frequency and Wavelet Transforms in Biomedical Engineering,

M. Akay (Editor), New York, NY: IEEE Press, 1997.

Sergio Schuler and Andrew Laine

Computer and Information Science and Engineering Department

UniversityofFlorida Gainesville, FL 32611

Hexagonal QMF Banks and Wavelets

Sergio Schuler and Andrew Laine

Department of Computer and Information Science and Engineering

University of Florida

Gainesville, FL 32611

Introduction

In this chapter weshalllay bare the theory and implementation details of hexagonal

sampling systems and hexagonal quadrature mirror lters (HQMF). Hexagonal sampling

systems are of particular interest b ecause they exhibit the tightest packing of all regular

two-dimensional sampling systems and for a circularly band-limited waveform, hexagonal

sampling requires 13.4 p ercentfewer samples than rectangular sampling [1]. In addition,

hexagonal sampling systems also lead to nonseparable quadrature mirror lters in which all

basis functions are lo calized in space, spatial frequency and orientation [2]. This chapter is

organized in two sections. Section I describ es the theoretical asp ects of hexagonal sampling

systems while Section I I covers imp ortant implementation details.

I. Hexagonal sampling system

This section presents the theoretical foundation of hexagonal sampling systems and

hexagonal quadrature mirror lters. Most of this material has app eared elsewhere in [1],

[3], [4], [5], [2], [6] but is describ ed here under a uni ed notation for completeness. In

addition, it will provide continuity and a foundation for the original material that follows

in Section I I. The rest of the section is organized as follows. Section I-A covers the

general formulation of a hexagonal sampling system. Section I-B intro duces up-sampling

and down-sampling in hexagonal sampling systems. Section I-C reviews the theory of

hexagonal quadrature mirror lters. Section I-D describ es redundant analysis/synthesis

lter banks in hexagonal systems. Finally, Section I-E covers the formulation of the discrete

Fourier transform in hexagonal sampling systems.

A. Hexagonal systems

Let x (t)=x (t ;t ) b e a 2-D analog waveform, then a sampling op eration in 2-D can

a a 1 2

b e represented by a lattice formed by taking all integer linear combinations of a set of two

T T

linearly indep endentvectors v =[v v ] and v =[v v ] . Using vector notation we

1 11 21 2 12 22

can represent the lattice as the set of all vectors t =[t t ] generated by

1 2

t = Vn; (1)

where n =[n n ] is an integer-valued vector and V =[v v ]isa2 2 matrix, known

1 2 1 2

as the sampling matrix. Because v and v are chosen to b e linearly indep endent, the

1 2

determinantof V is nonzero. Note that V is not unique for a given sampling pattern

and that two matrices representing the same sampling pro cess are related by a linear

transformation represented by a unimo dular matrix [7].

Sampling an analog signal x (t) on the lattice de ned by (1) pro duces the discrete signal

x(n)= x (Vn):

Figure 1(a) shows a hexagonal sampling lattice de ned by the pair of sampling vectors

2 3 2 3

2T T

1 1

6 7 6 7

= and = ; (2)

4 5 4 5

v v

1 2

0 T

3 1

and T = . The lattice is hexagonal since each sample lo cation has exactly where T =

2 1

2 2

six nearest neighb ors when T = T 3.

2 1

Let the Fourier transform of x (t) b e de ned by

x (t)exp(j t)dt; X ( )=

a a

where =[ ] . Similarly, let the Fourier transform of the sequence x(n) b e de ned

1 2

X (! )= x(n)exp(j ! n); (3)

where ! =[! ! ] . Mersereau [4] showed that the sp ectrum of the sequence x(n) and

1 2

the sp ectrum of the signal x (t) are related by

X (! )= X (V (! 2 k)); (4)

jdet V j k 4

v Ω 1 t1, n1 1

u1 v2 u2

n 2 t Ω

2 2

(a) (b)

Fig. 1. (a) A hexagonal sampling lattice in the spatial domain. (b) Hexagonal sampling lattice in the

frequency domain.

T 1 T

where k is an integer-valued vector and V denotes (V ) .

Alternatively,we can de ne the Fourier transform of the sequence x(n)as

X ( ) = x(n) exp(j Vn)

= X (V ); (5)

then Equation (4) may b e written as

X ( )= X ( Uk); (6)

V a

jdet V j

where

U =2 V : (7)

Thus, Equation (6) can b e interpreted as a p erio dic extension of X ( ) with p erio dicity

T T

vectors u =[u u ] and u =[u u ] , where U =[u u ]. The set of all vectors

1 11 21 2 12 22 1 2

generated by = Un de nes a lattice in the frequency domain known as the mo dulation or

recipro cal lattice. Thus, the sp ectrum of a sequence x(n) can b e viewed as the convolution

of the sp ectrum of x (t) with a mo dulation lattice de ned by U.

Figure 1(b) shows the recipro cal lattice corresp onding to the sampling vectors de ned

in Equation (2), that is the lattice de ned by the pair of mo dulation vectors

3 3 2 2

7 7 6 6

and : = =

5 5 4 4

u u

2 1

T T

2 2 5

Λ Λ K

1 23n1 1 2 3 n1

1 1

2 2

3 3

n2 n2

(a) (b)

Fig. 2. (a) Integer sampling lattice. (b) Sampling sublattice.

B. Up-sampling and down-sampling in hexagonal systems

Let denote the integer lattice de ned by the set of integer vectors n, and let

denote the sampling sublattice generated by the subsampling matrix K, that is the set of

integer vectors m such that m = Kn. Note that in order to prop erly de ne a sublattice

of , a subsampling matrix must b e nonsingular with integer-valued entries. In general,

a sublattice of is called separable if it can b e represented by a diagonal matrix K,

otherwise it is called nonseparable. Figure 2 shows an integer sampling lattice and a

sampling sublattice , for the separable subsampling matrix

3 2

2 0

7 6

K = : (8)

5 4

0 2

With and de ned in this way,we can view the op erations of up-sampling and

down-sampling as describ ed b elow.

The pro cess of up-sampling maps a signal on to a new signal that is nonzero only at

p oints on the sampling sublattice . The output of an up-sampler is related to the input

1 1

x(K n); if K n 2 ,

y (n)=

0; otherwise.

It is easy to show [5] that the Fourier transform relates the output and input of an

up-sampler by

Y (! )=X (K ! ); 6

K (a)

0 1 23n1 0 1 23n1 x x x x 0 x 0 00 10 20 x30 00 10

1 1 x x x 0 0 0 0 01 11 21 x31

2 2 x x x x 0 x 0 02 12 22 x32 01 11

3 3 x x x 0 0 0 0 03 13 23 x33

n2 n2

(b)

Fig. 3. (a) Up-sample op erator. (b) Mapping of samples under up-sampling: (left) input signal, (right)

output signal.

where X (! )isde nedby Equation (3). Figure 3 shows the blo ck diagram of an up-sampler

and the pro cess of up-sampling for the subsampling matrix de ned by Equation (8).

The pro cess of down-sampling maps p oints on the sublattice to according to

y (n)= x(Kn); (9)

and discards all other p oints.

The Fourier transform relation b etween the output and input of a down-sampler can

b e derived byintro ducing the concept of a sampling function s (n) asso ciated with the

sampling matrix K [5], that is,

1; if n 2 ,

s (n)=

0; otherwise.

Since s (n) can b e interpreted as a p erio dic sequence, with p erio dicity matrix K,i.e.

s (n)=s (n + Km), it may b e expressed as a Fourier series

K K

jdet Kj1

T 1

s (n)= exp(j 2 k K n); (10)

jdet Kj

l=0

where eachofthejdet Kj vectors k =[k k ] is asso ciated with one of the cosets of K .

l l l

1 2

Notice that a coset of a sublattice is de ned as the set of p oints obtained by shifting

the entire sublattice byaninteger shift vector k. There are exactly jdet Kj distinct cosets

of , and their union is the integer lattice . Each shift vector k asso ciated with a

K l

certain coset is known as a coset vector. For example, one choice for the k given the

sampling sublattice de ned by Equation (8) is

2 3 2 3 2 3 2 3

0 1 0 1

6 7 6 7 6 7 6 7

= ; = ; = ; and = : (11)

4 5 4 5 4 5 4 5

k k k k

0 1 2 3

0 0 1 1

Let

w (n)=x(n)s (n); (12)

then it is easy to see from (9) that a down-sampled signal y (n) can b e written as

y (n)=w (Kn);

since w (n) equals x(n)on . Therefore, the Fourier transform of the sequence y (n)may

b e written as

Y (! )= w (Kn )exp(j ! n): (13)

Since w (n) is zero for n not in wemay write (13) as

T 1

Y (! ) = w (n)exp(j ! K n)

= W (K ! );

where W (! ) is the Fourier transform of the sequence w (n). From (10) and (12) it is easy

to showthat

jdet Kj1

W (! )= X (! +2 K k );

jdet Kj

l=0

therefore, the Fourier transform relation b etween the output and input of a down-sampler

is given by

jdet Kj1

Y (! )= X (K (! +2 k )):

jdet Kj

l=0

Figure 4 shows the blo ck diagram of a down-sampler and the pro cess of down-sampling

for the subsampling matrix de ned on (8).

Note that the relations derived ab ove are based on the Fourier transform de ned in

Equation (3). However, a more general de nition is describ ed in Equation (5). This

formulation takes into account the lattice structure used to sample the original 2-D analog 8

K (a)

0 1 23n1 0 1 23n1 x x x x x x 00 10 20 x30 00 20 40 x60

1 1 x x x x x x 01 11 21 x31 02 22 42 x62

2 2 x x x x x x 02 12 22 x32 04 24 44 x64

3 3 x x x x x x 03 13 23 x33 06 26 46 x66

n2 n2

(b)

Fig. 4. (a) Down-sample op erator. (b) Mapping of samples under down-sampling: (left) input signal,

(right) output signal.

waveform and allows the Fourier transform relation b etween the input and output of an

up-sampler and a down-sampler to b e written as

T T

Y ( )=X (K V ); (14)

and

jdet Kj1

T T

Y ( )= X (K (V +2 k )): (15)

V l

jdet Kj

l=0

Therefore, if we assume K as de ned in (8) wemay write Equations (14) and (15) as

Y ( )=X (K ); (16)

V V

and

jdet Kj1

X (K + k ); (17) Y ( )=

V l V

jdet Kj

l=0

resp ectively, where

k = UK k : (18)

l l

C. Analysis/synthesis lter banks in hexagonal systems

This section fo cuses on p erfect reconstruction lter banks in hexagonal sampling systems

and wavelets that can b e obtained by iterating such lter banks. Parts of this material

are describ ed in Simoncelli [2], but are reviewed here for completeness of presentation. 9

y(n) F(Ω) K 0 K G(Ω) 0 0

y(n) F(Ω) K 1 K G(Ω) 1 1 x(n) x(n)^ y(n) F(Ω) K 2 K G(Ω) 2 2

y(n) F(Ω) K 3 K G(Ω)

3 3

Fig. 5. A two-dimensional 4-channel analysis/synthesis lter bank.

There are a wide variety of analysis/synthesis (A/S) lter banks for two dimensional

systems. We restrict our fo cus to analysis/synthesis lter banks in which eachchannel

shares the same subsampling matrix K and the number of channels equals jdet Kj. Figure

5 shows a 4-channel analysis/synthesis lter bank. We further restrict our study to the

separable sublattice de ned in Equation (8) since this choice will enable us to apply the

A/S lter bank recursively to each of the subband signals y (n)shown in Figure 5 as

describ ed in [2].

Consider a 4-channel analysis/synthesis lter bank with K de ned by Equation (8), then

using (17) we can showthattheFourier transform of y (n)may b e written as

jdet Kj1

T T

~ ~

Y ( )= F (K + k )X (K + k ); (19)

i i l l

jdet Kj

l=0

where the subindex V has b een suppressed for simplicity. Similarly, using (16) wehave

that the Fourier transform ofx ^(n)isgiven by

jdet Kj1

X ( )= G ( )Y (K ): (20)

i i

i=0

Therefore, combining Equations (19) and (20) weobtainanoverall lter bank resp onse of

jdet Kj1 jdet Kj1

X X

~ ~ ^

G ( ) F ( + k )X ( + k ) X ( ) =

i i l l

jdet Kj

i=0

l=0

2 3

jdet Kj1 jdet Kj1

X X

~ ~

4 5

X ( + k ) = G ( )F ( + k ) : (21)

l i i l

jdet Kj

i=0

l=0

Combining Equations (7), (11) and (18) for the values of T and T in Equation (2) yields

1 2

the following set of vectors k , that is,

3 3 3 2 2 2 3 2

0 0

7 7 7 6 6 6 7 6

~ ~ ~ ~

; ; and : (22) = = = ; =

5 5 5 4 4 4 5 4

k k k k

2 3 0 1

2 3

p p p

3 3 3

From Equation (22) it is clear that one term of the sum in Equation (21) corresp onds

to the linear shift invariant (LSI) system resp onse, and the remaining terms corresp ond to

the system alias. The analysis/synthesis lter bank for which the system aliasing terms in

Equation (21) are canceled is generally known as a quadrature mirror lter (QMF) bank.

We can cho ose the lters to eliminate the aliasing terms in Equation (21) as follows

F ( ) = G ( ) = H ( )=H ( );

0 0

F ( ) = G ( ) = exp(j s )H ( + k );

1 1 1 1

(23)

F ( ) = G ( ) = exp(j s )H ( + k );

2 2 2 2

F ( ) = G ( ) = exp(j s )H ( + k );

3 3 3 3

where s , s and s must satisfy the following equations

1 2 3

T T T

~ ~ ~

s s j k s j k j k

3 2 1

1 1 1

=0; + e =0; e 1+e

T T T

~ ~ ~

j k s j k s j k s

2 1 3

2 2 2

1+e =0; e + e =0;

T T T

~ ~ ~

s s j k s j k j k

2 1 3

3 3 3

=0: + e =0; e 1+e

Therefore, a suitable choice for the vectors s given the vectors k in Equation (22) is

l l

2 3 2 3 2 3

1 1=2 1=2

6 7 6 7 6 7

= ; = ; and = : (24)

4 5 4 5 4 5

s s s

p p

1 2 3

0 3=2 3=2

After canceling all of the aliasing terms in Equation (21) the remaining LSI system

resp onse b ecomes

X ( ) = X ( ) G ( )F ( )

i i

i=0

~ ~

= X ( ) k )H ( + k ) H ( +

i i

i=0

jH ( + k )j : X ( ) =

i=0 11

j k l l k j

k g h i h g k

l h e f f e h l

l i f c d c f i l

k h f d b b d f h k

j g e c b a b c e g t1, n1

k h f d b b d f d k

l i f c d c f d l

l h e f f e d l

k g h i h g k

k l l k j

n2 t2

Fig. 6. Region of supp ort of a 5-ring hexagonally symmetric lter. The parameters a through l refer to

the low-pass lter co ecients h(n).

Note that the aliasing cancellation is exact, and indep endentofthechoice of H ( ), and

the design problem is reduced to nding a lter satisfying the constraint

jH ( + k )j =4: (25)

i=0

Alow-pass solution for H ( ) in the ab ove equation results in a band-splitting system

whichmay b e cascaded hierarchically through the low-pass band of the QMF bank to

pro duce a multiresolution decomp osition in two dimensions. Simoncelli [2] describ es a

simple frequency-sampling design metho d that pro duces hexagonally symmetricQMFs

with small regions of supp ort for which p erfect reconstruction was well approximated.

Figure 6 shows the region of supp ort of a 5-ring hexagonally symmetric lter. Notice that

the size of the lter is measured in terms of the numb er of hexagonal rings it contains. The

parameters a through l in Figure 6 refer to the lter co ecients of the low-pass solution

h(n) computed in [2]. Figure 7(a) shows an idealized diagram of the partition of the

frequency domain resulting from a 2-level hexagonal multiresolution decomp osition. 12

2 y(n) F(Ω) K 0 K G(Ω) 0 0

2 y(n) F(Ω) K 1 K G(Ω) 1 1

2 y(n) F(Ω) K 2 K G(Ω) 2 2

2 Level 1 y(n) F(Ω) K 3 K G(Ω) 3 3

F(Ω) K K G(Ω) 0 0

1 y(n) F(Ω) K 1 K G(Ω) 1 1 Level 2 x(n) x(n)^ 1 y(n) F(Ω) K 2 K G(Ω) 2 2

1 y(n) F(Ω) K 3 K G(Ω)

3 3

(a) (b)

Fig. 7. (a) Partitions of the frequency domain resulting from a 2-level multiresolution decomp osition of

hexagonal lters. The upp er left frequency diagram represents the sp ectrum of the original image.

(b) A two-stage 4-channel analysis/synthesis lter bank.

D. Redundant analysis/synthesis lter banks in hexagonal systems

In this section we discuss the mathematical formulation of redundant analysis/synthesis

lter banks in hexagonal systems. In particular, wewould like to nd equivalent lters for

the i stage of the traditional A/S system shown in Figure 7(b).

It can b e easily shown that subsampling by K followed by ltering with F ( ) is equiva-

lent to ltering by F (K ) followed by subsampling. Hence, the rst two steps of low-pass

ltering in Figure 7(b) can b e replaced bya lterwithFourier transform F ( )F (K ),

0 0

followed by subsampling by K .

In general, equivalent lters for the i stage (i 1) of a cascade of analysis lters are

given by

i1 i

F (K ); ( )=F (K ) F

0 0

l=0

i i1

F (K ); F ( )=F (K )

0 1

l=0

(26)

i i1

F (K ); F ( )=F (K )

0 2

l=0

i i1

F ( )=F (K ) F (K );

3 0

l=0

followed by subsampling by K . The synthesis lters are obtained in a similar way.

j=0 j=1 j=2 j=3

i=1

i=2

Fig. 8. Analyzing lters F for levels 1 and 2.

By removing the op erations of down-sampling and up-sampling from the resulting equiv-

alent A/S system we obtain an overcomplete hexagonal multiresolution representation.

From Equation (25), p erfect reconstruction is also accomplished in this case. Figure 8

shows the magnitude of the equivalent hexagonal lters F for levels 1 and 2 for the 4-ring

lter co ecients computed in [2].

E. The discrete Fourier transform in hexagonal systems

Letx ~ (n) b e a p erio dic sequence with p erio dicity matrix N, that isx ~ (n)=x ~(n + Nr) for

anyinteger vector r. Thenitiseasytoverify [4] that the following Fourier series relations

hold,

T 1

X (k)exp(j k (2 N )n); (27) x~(n)=

j det Nj

k2J

and,

T 1

X (k)= x~ (n) exp(j k (2 N )n); (28)

n2I

where, I and J denote nite-extent regions (consisting of j det Nj samples) in the n-domain

and k-domain, resp ectively.

Let x(n)beany nite-extent sequence con ned to a region I containing S samples. We

say that the sequence x(n) admits a p erio dic extensionx ~(n) with p erio dicity matrix N if

x~(n) = x~ (n + Nr);

x~(n) = x(n); for n 2 I;

and S = j det Nj.

If x(n) admits a p erio dic extension with p erio dicity matrix N then the Fourier series

relation in Equation (28) can b e used to de ne its discrete Fourier transform (DFT) as

follows,

^ ~

X (k) = X (k); k 2 J;

(29)

T 1

= x(n)exp(j k (2 N )n); k 2 J:

n2I

Similarly, the Fourier series relation in Equation (27) can b e used to recover the sequence

x(n) from its DFT as follows,

x(n) = x~ (n); n 2 I;

(30)

T 1

= X (k) exp(j k (2 N )n); n 2 I:

j det Nj

k2J

Supp ose x(n) is a hexagonally sampled signal with supp ort con ned to a region I con-

taining (2N + N )N samples. In addition supp ose that x(n) admits a p erio dic extension

1 2 2

with p erio dicity matrix

2 3

N + N N

1 2 2

6 7

N = : (31)

4 5

N 2N

2 2

Then, wesaythat x(n) admits a hexagonally p erio dic extension. Notice that x(n)may

admit more than one p erio dic extension and that each p erio dic extension de nes a di erent

DFT. It is easy to show that the DFT of each admissible p erio dic extension of x(n)

corresp onds to a sampled version of its Fourier transform, and that the sampling lattice is

controlled through the p erio dicity matrix N. This result is a consequence of the following

relation b etween the Fourier transform (3) and the discrete Fourier transform (29) of x(n),

X (k)=X (! )j T :

! =2 N k

Using the more general de nition of the Fourier transform (5) that accounts for the

sampling lattice used to sample the original 2-D analog signal we can write

X (k)=X ( )j T : (32)

=2 ( VN) k

It then follows from Equations (31) and (32) that for N = N = N , that is, for

1 2

3 2

2N N

7 6

; (33) N =

5 4

N 2N

the DFT of x(n) corresp onds to a hexagonal sampled version of its Fourier transform. In

this case Equation (29) is referred to as the hexagonal discrete Fourier transform (HDFT)

of x(n). It is easy to verify from (29) and (33) that the HDFT of x(n)isgiven by

X (k ;k ) = x(n ;n ) exp j ((2n n )k +(2n n )k ) ;

1 2 1 2 1 2 1 2 1 2

(n ;n )2I

1 2

(34)

= x(n ;n ) exp j (2n n )(2k k )+ n k :

1 2 1 2 1 2 2 2

3N N

(n ;n )2I

1 2

Mersereau [4] showed that ecient algorithms for the implementation of the discrete

Fourier transform (29) exist if the p erio dicity matrix N is comp osite (i.e., if N can b e

factored into a nontrivial pro duct of integer matrices.) For N =2 ;l 0we observe that

the p erio dicity matrix N in (33) can b e factored as,

3 3 2 2

2 0 2 1

7 7 6 6

: (35) N =

5 5 4 4

0 2 1 2

This factorization leads to an ecient implementation of (34) known as the hexagonal fast

Fourier transform (HFFT). Implementation details of the HFFT can b e found in [1].

I I. Implementation

Next, we present implementation details of hexagonal multiresolution representations.

Section I I-A describ es the selection of image supp ort for ecient signal pro cessing with

hexagonal systems. Section I I-B describ es ltering in hexagonal sampling systems using a

HFFT based strategy and the computation of hexagonal multiresolution representations.

Finally, Section I I-C describ es the computation of overcomplete hexagonal multiresolution

representations. A listing of Matlab functions implementing hexagonal multiresolution

representations is available at http://www.iprg.cise.ufl.edu/.

A. Image support in hexagonal systems

Ecient discrete signal pro cessing in hexagonal sampling systems can b e achieved by

using the hexagonal fast Fourier transform but at the cost of restricting sequences to

those that admit a hexagonally p erio dic extension with N = N = N and N =2;l 0.

1 2

Systematic application of the HFFT to image pro cessing further restricts sequences to

rectangular regions. It is easy to verify that for any N =2;l 1 it is p ossible to

nd a rectangular region I suchthatany sequence with supp ort con ned to I admits a

l l

hexagonally p erio dic extension. Figure 9 shows such a sequence con ned to region I and

its hexagonally p erio dic extension. For any l 1, region I may b e obtained from I by

l+1 l

doubling the number of rows and the number of samples per rows in I . Notice that given

the p erio dic sequence shown in Figure 9 there exists more than one fundamental p erio d

such that when extended p erio dically in a hexagonal fashion results in the same p erio dic

sequence. In particular, Figure 9 shows a parallelogram P containing an alternative set

of samples that could b e used to de ne the same p erio dic sequence but greatly simpli es

the HFFT computation. The simpli cation comes from the fact that samples contained in

the parallelogram can b e stored in an arrayofsizeN 3N where the indices of the array

directly corresp ond to the co ordinates (n ;n ) of the samples. It is straight forward to

1 2

verify that for any sequence con ned to I ;l 1 there exists a parallelogram P containing

l l

an alternative set of samples that de nes the same p erio dic sequence. For any l 1, P

l+1

is obtained from P in the same way I was obtained from I .Itfollows that computation

l l+1 l

of the HFFT for a sequence con ned to I can b e accomplished by using Equation (34)

where I corresp onds to a region de ned by P .

Although there are certain image pro cessing applications in which image samples are

acquired in a hexagonal fashion, most digital detectors (e.g., CCDs) sample images rect-

angularly with the same sampling rate along b oth directions. Pro cessing a rectangularly

sampled image with a hexagonal sampling system requires that the rectangularly sampled

image b e mapp ed into a hexagonal sampling lattice [3], [2]. For square images whose size is

apower of twowe describ e a strategy that maps a rectangularly sampled image con ned to

a square region consisting of 2N 2N samples (N =2 ;l 1) into a hexagonally sampled

image con ned to a rectangular region for which it is p ossible to compute its HFFT. This

mapping may b e accomplished in twodistinctways. If the original image is oversampled,

we rst interp olate horizontally by a factor of 3 and then mask the result with the masking

n n =2+n

h h

function M (n ;n )= (1 + (1) )(1 + (1) )asshown in Figure 10. The resulting

1 h v 4 17

x x x 00 10 20 t1, n1

x11 x21 x31

P1 N1 x12 x22 x32

x23 x33 x43 I1

n2 t2

Fig. 9. Hexagonally sampled sequence con ned to a rectangular region I and its hexagonally p erio dic

extension. Parallelogram P contains an alternative set of samples that de nes the same p erio dic

1 sequence. 18

0 1 23 0 3 69 0 2 41068 0 0 0 tx, nx tx, nh tx, nh

1 1 1

2 2 2

3 3 3

ty, ny ty, nv ty, nv

(a) (b) (c)

Fig. 10. Mapping a rectangularly sampled image into a hexagonal sampling lattice. (a) Rectangular

lattice, (b) Intermediate lattice with interp olated samples, (c) Sampling function.

(a) (b) (c)

Fig. 11. (a) A 512 512 rectangularly sampled radiograph of the breast, (b) Image with interp olated

samples, (c) Hexagonally sampled image con ned to P { Axes n and n are shown at 90 degrees.

9 1 2

image is con ned to region I and consists of 3N samples (25% fewer samples than the

original). In this case, it is assumed that oversampling of the original image accounts for

the reduction of the numb er of samples in the resulting image. Alternatively, if the original

image is critically sampled, weinterp olate horizontally by a factor of 3 and vertically bya

n +n

(1 + (1) ). factor of 2 and mask the result with the masking function M (n ;n )=

2 h v

In this case the resulting image is con ned to region I and consists of 3(2N ) samples.

l+1

In each case the resulting sampling lattice gives a reasonable geometric approximation to

a hexagonal sampling lattice. Note that the oversampled metho d or the critically sampled

metho d, together with the equivalence b etween I and P can b e used to map a 2N 2N

l l

rectangularly sampled image into P or P , resp ectively. Figure 11 shows a 512 512

l l+1

rectangularly sampled image and its mapping into P using the critically sampled strategy

describ ed ab ove.

Mapping from Rectangular

to Hexagonal lattice

Interp olate

Masking by Mapping from

- - - -

Input Image

by 3 horizontally HFFT

M (n ;n ) I to P

2 h v l+1 l+1

and by2 vertically

Filter HFFT

Mapping from

Hexagonal to IHFFT Output Image

P to I

l+1 l+1

Rectangular lattice

Fig. 12. Filtering a rectangularly sampled image of size 2N 2N; N =2 with a hexagonally sampled

system by means of the HFFT. Critically sampled case shown.

B. Multiresolution representations in hexagonal systems

Filtering an image with a hexagonal lter may b e accomplished by computing the pro d-

uct of the HFFTs of the image and the lter kernel and taking the inverse hexagonal fast

Fourier transform (IHFFT) of the result. This approach requires that b oth the image and

the lter kernel b e supp orted in the same region I . Again, computation of the HFFT can

b e accomplished using P instead of I due to p erio dicity. Figure 12 shows a blo ck diagram

l l

of the ltering pro cess describ ed ab ove for a rectangularly sampled signal con ned to a

square region of size 2N 2N; N =2.

Here, we are interested in ltering an image with the quadrature mirror lters in equation

(23) given the low-pass solution h(n ;n ) computed in [2]. Replacing equations (22) and

1 2

(24) in (23) and using relations (5) and (32) we obtain the following HDFT relations for

the lters,

T 1

^ ^

F (k) = G (k) = h (n) exp(j k (2 N )n);

0 0 0

n2I

T 1

^ ^

(2k k ) h (n)exp(j k (2 N )n); F (k) = G (k) = exp j

1 2 1 1 1

n2I

(36)

T 1

^ ^

h (n) exp(j k (2 N )n); (k + k ) F (k) = G (k) = exp j

2 1 2 2 2

n2I

T 1

^ ^

F (k) = G (k) = exp j h (n)exp(j k (2 N )n); (k 2k )

3 3 3 1 2

n2I

n n

1 2

where h (n)= h(n ;n ), h (n)=(1) h(n ;n ), h (n)=(1) h(n ;n ), h (n)=

0 1 2 1 1 2 2 1 2 3

n +n

1 2

(1) h(n ;n ), and I is the region of supp ort of the lter kernels. It follows from the

1 2

equations ab ove that the HFFT of lters f (n) can b e obtained by mo dulating the HFFTs

of the kernels h (n) with complex exp onentials. Notice that ltering an image con ned to

a region I with an r -ring lter kernel f (n) using the HFFT- ltering strategy describ ed

l k

ab ove requires that the region of supp ort of the kernel b e con ned to I (or equivalently

to P ). Indeed, this requirement is satis ed as long as l dlog (2r +1)e1.

A one-level multiresolution decomp osition of an image can b e obtained by ltering the

image with the lters kernels f (n) followed bydown-sampling (as shown in Figure 5.)

This can b e accomplished using the HFFT- ltering strategy describ ed ab ove if the image

is con ned to I or P . Notice that if wework with sequences con ned to P then the down-

l l l

sampling op eration is equivalent to taking every other rowandevery other column of the

array storing the sequence. An (L +1)-level multiresolution decomp osition can then b e

obtained recursively by cascading an analysis section through the low-pass branchofan L-

level multiresolution decomp osition. Notice that the maximum number of levels is limited

by the smallest P supp orting the lter kernels. Figure 13 shows a two-level hexagonal

multiresolution decomp osition and reconstruction using the 3-ring low-pass solution to

(23) computed in [2]. An algorithm for the multiresolution reconstruction follows directly

from its decomp osition.

C. Overcomplete multiresolution representations in hexagonal systems

An overcomplete hexagonal multiresolution representation is computed by ltering an

image with the equivalent lters intro duced in Section I-D. Using the HFFT- ltering

strategy describ ed previously it follows that the equivalent lters F can b e computed k

(a) (b)

Fig. 13. Two-level hexagonal multiresolution representation. (a) Decomp osition. (b) Reconstruction.

Both images are displayed on their original sampling lattice.

taking the pro duct of the HFFTs of lter kernels f (n) where F ( )=F (K ). It is

k;l k;l k

easy to show that the following HDFT relation holds for f (n),

k;l

^ ^

F (k)= F (K k):

k;l k

This result combined with equation (36) leads to the following HDFT relations for the

lters,

T 1

^ ^

F (k) = G (k) = h (n)exp(j k (2 N )n);

0;l 0;l 0;l

n2I

l+1

T 1

^ ^

F (k) = G (k) = exp j (2k k ) h (n)exp(j k (2 N )n);

1;l 1;l 1 2 1;l

n2I

l+1

T 1

^ ^

(k + k ) h (n)exp(j k (2 N )n); F (k) = G (k) = exp j

1 2 2;l 2;l 2;l

n2I

l+1

T 1

^ ^

(k 2k ) h (n)exp(j k (2 N )n); F (k) = G (k) = exp j

1 2 3;l 3;l 3;l

n2I

(37)

where

l 1 l 1

h (K ) n ; if (K ) n is an integer vector,

h (n)=

k;l

0; otherwise.

Note that in the ab ove derivation we used the fact that K de ned a separable subsam-

pling matrix. It follows from the equation ab ove that the HFFT of the lter kernels f (n)

k;l

can b e obtained by mo dulating the HFFTs of the kernels h (n) with complex exp onen-

k;l

tials. The lters h (n) can b e constructed by up-sampling h (n) with sampling matrix

k;l k

l i

K . The lters F can then b e computed following equation (26) with the lters given in

(37). Notice that it is p ossible to discard the complex exp onentials and still obtain p erfect

reconstruction. However, this will de ne a di erent set of lters. A hexagonal overcom-

plete multiresolution representation of an image can b e obtained by ltering the image

with the lters f derived ab ove. Figure 14 shows an example of hexagonal overcomplete

multiresolution representations applied to a digitized radiograph of the breast. Contrast

enhancementwas accomplished adaptively based on the lo cation of multiscale edges de-

rived from the hexagonal overcomplete multiresolution representation. Figure 15 shows

another example of this technique applied to a region of interest of a digitized radiograph

of the chest. Please refer to [8] for a complete description of this enhancement algorithm

and other p ossible metho ds of enhancement.

Acknow ledgments

Original digitized mammogram shown in Figure 11 was provided courtesy of the Center

for Engineering and Medical Image Analysis and the H. Lee Mott Cancer Center and

Research Institute at the University of South Florida, Tampa.

Original digitized radiograph shown in Figure 15 was provided courtesy of the Depart-

ment of Radiology and the J. Hillis Miller Health Science CenterattheUniversityof Florida, Gainesville.

(a) (b)

Fig. 14. (a) Mathematical phantom. (b) Mammogram blended with phantom. (c) Combined orientations

of hexagonal edges obtained from level 3 co ecients. (d) Contrast enhancementbymultiscale edges

obtained from a hexagonal overcomplete multiresolution representation.

(a)

(b) (c)

Fig. 15. (a) Original image. (b) Region of interest. (c) Enhanced region of interest via multiscale analysis.

References

[1] R.M. Mersereau, \The pro cessing of hexagonally sampled two-dimensional signals", Proceedings of the IEEE,

vol. 67, no. 6, pp. 930{949, June 1979.

[2] E.P. Simoncelli and E.H. Adelson, \Non-separable extensions of quadrature mirror lters to multiple dimen-

sions", Proceedings of the IEEE,vol. 78, no. 4, pp. 652{663, Apr. 1990.

[3] R.M. Mersereau and T.C. Sp eake, \The pro cessing of p erio dicall y sampled multidimensi onal signals", IEEE

Transactions on Acoustics, Speech, and Signal Processing,vol. 31, no. 1, pp. 188{194, Feb. 1983.

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[7] M. Newman, Integral Matrices, Academic Press, New York, 1972.

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