Circulant Matrices and Convolution - Digital Image Processing

University of Ioannina - Department of Computer Science and Engineering

Filtering in the Frequency Domain (Circulant Matrices and Convolution)

Digital Image Processing

Christophoros Nikou

[email protected]

2

C. Nikou – Digital Image Processing (E12)

Toeplitz matrices

• Elements with constant value along the main

diagonal and sub-diagonals.

• For a NxN matrix, its elements are determined by

a (2N-1)-length sequence ( 1) 1|n N n Nt

1

( 1)0 1 2

1 0 1

2 2

1

2 1 0N

N

N N

t t t t

t t t

t t

t

t t t t

T

( , ) m nm n t T

3


Toeplitz matrices (cont.)

• Each row (column) is generated by a shift of the

previous row (column).

− The last element disappears.

− A new element appears.

1

( 1)0 1 2

1 0 1

2 2

1

2 1 0N

N

N N

t t t t

t t t

t t

t

t t t t

T

( , ) m nm n t T

4


Circulant matrices

• Elements with constant value along the main

diagonal and sub-diagonals.

• For a NxN matrix, its elements are determined by

a N-length sequence 0 1|n n Nc

1

1

2 1

1

0 1 2

0 1

0 2

1

1 2 0

N N

N

N

N

N N N N

c c c c

c c c

c c c c

c

c c c c

C

( )mod( , ) m n Nm n c C

5


Circulant matrices (cont.)

• Special case of a Toeplitz matrix.

• Each row (column) is generated by a circular shift

(modulo N) of the previous row (column).

( )mod( , ) m n Nm n c C

1

1

2 1

1

0 1 2

0 1

0 2

1

1 2 0

N N

N

N

N

N N N N

c c c c

c c c

c c c c

c

c c c c

C

6


Convolution by matrix-vector

operations

• 1-D linear convolution between two discrete

signals may be expressed as the product of a

Toeplitz matrix constructed by the elements of

one of the signals and a vector constructed by the

elements of the other signal.

• 1-D circular convolution between two discrete

signals may be expressed as the product of a

circulant matrix constructed by the elements of

one of the signals and a vector constructed by the

elements of the other signal.

• Extension to 2D signals.

7


1D linear convolution using Toeplitz

matrices

• The linear convolution g[n]=f [n]*h [n] will be of

length N=N1+N2-1=3+2-1=4.

• We create a Toeplitz matrix H from the

elements of h [n] (zero-padded if needed) with

− N=4 lines (the length of the result).

− N1=3 columns (the length of f [n]).

− The two signals may be interchanged.

1 2[ ] {1, 2, 2}, [ ] {1, 1}, 3, 2f n h n N N

8



matrices (cont.)

1 2[ ] {1, 2, 2}, [ ] {1, 1}, 3, 2f n h n N N

4 3

1 0 0

1 1 0

0 1 1

0 0 1

H

Length of f [n] = 3

Length of the result =4

Notice that H is not

circulant (e.g. a -1

appears in the second line

which is not present in the

first line.Zero-padded h[n] in the first column

9



matrices (cont.)

1 2[ ] {1, 2, 2}, [ ] {1, 1}, 3, 2f n h n N N

1 0 0 11

1 1 0 12

0 1 1 02

0 0 1 2

g = Hf

[ ] {1, 1, 0, 2}g n

10


1D circular convolution using

circulant matrices

• The circular convolution g[n]=f [n]h [n] will

be of length N=max{N1, N2}=3.

• We create a circulant matrix H from the

elements of h [n] (zero-padded if needed)

of size NxN.

− The two signals may be interchanged.

1 2[ ] {1, 2, 2}, [ ] {1, 1}, 3, 2f n h n N N

11



circulant matrices (cont.)

1 2[ ] {1, 2, 2}, [ ] {1, 1}, 3, 2f n h n N N

3 3

1 0 1

1 1 0

0 1 1

H

Zero-padded h[n] in the

first column

12




1 2[ ] {1, 2, 2}, [ ] {1, 1}, 3, 2f n h n N N

1 0 1 1 1

1 1 0 2 1

0 1 1 2 0

g = Hf

[ ] { 1, 1, 0}g n

13


Block matrices

• Aij are matrices.

• If the structure of A, with respect to its sub-matrices, is

Toeplitz (circulant) then matrix A is called block-Toeplitz

(block-circulant).

• If each individual Aij is also a Toeplitz (circulant) matrix then

A is called doubly block-Toeplitz (doubly block-circulant).

11 12 1

21 22 2

1 2

N

N

M M MN

A A A

A A AA

A A A

14


2D linear convolution using doubly

block Toeplitz matrices

m

n

1 4 1

2 5 3

f [m,n]

m

n

1 11 -1

h [m,n]

M1=2, N1=3 M2=2, N2=2

The result will be of size (M1+M2-1) x (N1+N2-1) = 3 x 4

15



block Toeplitz matrices (cont.)

• At first, h[m,n] is zero-padded

to 3 x 4 (the size of the

result).

• Then, for each row of h[m,n],

a Toeplitz matrix with 3

columns (the number of

columns of f [m,n]) is

constructed.

m

n

1 1 0

1 -1 0

h[m,n]

00

0 0 0 0

m

n

1 4 1

2 5 3

f [m,n]

m

n

1 11 -1

h [m,n]

16




• For each row of h[m,n], a

Toeplitz matrix with 3 columns

(the number of columns of f

[m,n]) is constructed.

m

n

1 1 0

1 -1 0

h[m,n]

00

0 0 0 0

1

1 0 0

1 1 0

0 1 1

0 0 1

H2

1 0 0

1 1 0

0 1 1

0 0 1

H 3

0 0 0

0 0 0

0 0 0

0 0 0

H

17




• Using matrices H1, H2

and H3 as elements, a

doubly block Toeplitz

matrix H is then

constructed with 2


rows of f [m,n]).

m

n

1 4 1

2 5 3

f [m,n]

m

n

1 11 -1

h [m,n]

1 3

2 1

3 2 12 6

H H

H H H

H H

18




• We now construct

a vector from the

elements of f [m,n].

m

n

1 4 1

2 5 3

f [m,n]

m

n

1 11 -1

h [m,n]

2

5

(2 5 3)3

(1 4 1)1

4

1

T

T

f

19



block Toeplitz matrices (cont.) m

n

1 4 1

2 5 3

f [m,n]

m

n

1 11 -1

h [m,n]

1 3

2 1

3 2

2

5

3

1

4

1

H H

g = Hf H H

H H

20



block Toeplitz matrices (cont.) 1 0 0 0 0 0 2

1 1 0 0 0 0 3

0 1 1 0 0 0 2

0 0 1 0 0 0 2 3

1 0 0 1 0 0 5 3(

1 1 0 1 1 0 3 10

0 1 1 0 1 1 1 5

0 0 1 0 0 1 4 2

0 0 0 1 0 0 1 1

0 0 0 1 1 0 5

0 0 0 0 1 1 5

0 0 0 0 0 1 1

g = Hf

2 3 2 3)

(3 10 5 2)

(1 5 5 1)

T

T

T

21




*

m

n

1 4 1

2 5 3

f [m,n]

m

n

1 11 -1

h [m,n]

=3

2

m

n

3 10 5

2 3 -2

g [m,n]

1 5 5 1 (2 3 2 3)

(3 10 5 2)

(1 5 5 1)

T

T

T

g =

22




m

n

3 4

1 2

f [m,n]

m

n1 -1

h [m,n]

M1=2, N1=2 M2=1, N2=2

The result will be of size (M1+M2-1) x (N1+N2-1) = 2 x 3

Another example

23




• At first, h[m,n] is zero-padded

to 2 x 3 (the size of the

result).

• Then, for each line of h[m,n],

a Toeplitz matrix with 2


columns of f [m,n]) is

constructed.

m

n

0 0 0

1 -1 0

h[m,n]

m

n

3 4

1 2

f [m,n]

m

n1 -1

h [m,n]

24




• For each row of h[m,n], a Toeplitz

matrix with 2 columns (the

number of columns of f [m,n]) is

constructed.

m

n

0 0 0

1 -1 0

h[m,n]

1

1 0

1 1

0 1

H2

0 0

0 0

0 0

H

25




• Using matrices H1 and

H2 as elements, a

doubly block Toeplitz

matrix H is then

constructed with 2


rows of f [m,n]).

1 2

2 1 6 4

H HH

H H

m

n

3 4

1 2

f [m,n]

m

n1 -1

h [m,n]

26




• We now construct a

vector from the elements

of f [m,n].

1

(1 2)2

(3 4)3

4

T

T

f

m

n

3 4

1 2

f [m,n]

m

n1 -1

h [m,n]

27




m

n

3 4

1 2

f [m,n]

m

n1 -1

h [m,n]

1 2

2 1

1

2

3

4

H Hg = Hf

H H

28




1 0 0 0 1

1 1 0 0 1 1

(1 1 2)0 1 0 0 2 2

(3 1 4)0 0 1 0 3 3

0 0 1 1 4 1

0 0 0 1 4

T

T

g = Hf

29




*

=

m

n

3 1 4

1 1 -2

g [m,n]

m

n

3 4

1 2

f [m,n]

m

n1 -1

h [m,n]

(1 1 2)

(3 1 4)

T

T

g =

30


2D circular convolution using doubly

block circulant matrices

where H is a doubly block circulant matrix

generated by h [m,n] and f is a vectorized form

of f [m,n].

0 1, 0 1,m M n N

The circular convolution g[m,n]=f [m,n]h [m,n]

may be expressed in matrix-vector form as:

g = Hf

with

31



block circulant matrices (cont.)

Each Hj, for j=1,..M, is a circulant matrix with N

columns (the number of columns of f [m,n])

generated from the elements of the j-th row of

h [m,n].

0 1 2 1

1 0 1 2

2 1 0 3

1 2 3 0

M M

M

M M M

H H H H

H H H H

H H H H H

H H H H

32




[ ,0] [ , 1] [ ,1]

[ ,1] [ ,0] [ , 2]

[ , 1] [ , 2] [ ,0]

j

N N

h j h j N h j

h j h j h j

h j N h j N h j

H

Each Hj, for j=1,..M, is a NxN circulant matrix

generated from the elements of the j-th row of

h [m,n].

33




m

n

1 3 -1

1 2 1

f [m,n]

m

n

1 0

1 -1

h [m,n]

0 1 0

0

0

0 0 0

0

1 0 1

1 1 0

0 1 1

H 1

1 0 0

0 1 0

0 0 1

H 2

0 0 0

0 0 0

0 0 0

H

34




m

n

1 3 -1

1 2 1

f [m,n]

m

n

1 0

1 -1

h [m,n]

0 1 0

0

0

0 0 0

0 2 1

1 0 2

2 1 0

1 2 1

1 3 1

0 1 0

T

T

T

H H H

g = Hf H H H

H H H

35




1 0 1 0 0 0 1 0 0 1 0

1 1 1 0 0 0 0 1 0 2 2

0 1 0 0 0 0 0 0 1 1 1

1 0 0 1 0 1 0 0 0 1 3

0 1 0 1 1 1 0 0 0 3 4

0 0 1 0 1 0 0 0 0 1 3

0 0 0 1 0 0 1 0 1 0 1

0 0 0 0 1 0 1 1 1 1 4

0 0 0 0 0 1 0 1 0 0 2

g = Hf

36




m

n

1 0

1 -1

h [m,n]

m

n

3 4 -3

0 2 -1

g [m,n]

1 4 -2

0

0

0 0 0

=

m

n

1 3 -1

1 2 1

f [m,n]

0 1 0

0 2 1

3 4 3

1 4 2

T

T

T

g =

37


Diagonalization of circulant matrices

Theorem: The columns of the inverse DFT matrix

are eigenvectors of any circulant matrix. The

corresponding eigenvalues are the DFT values of

the signal generating the circulant matrix.

Proof: Let

2 2 nj j k

nk

N Nw e w e

be the DFT basis elements of length N with:

0 1, 0 1,k N n N

38



(cont.)

We know that the DFT F [k] of a 1D signal f [n]

may be expressed in matrix-vector form:

where

F = Af

[0], [1],..., [ 1] , [0], [1],..., [ 1]T T

f f f N F F F N f = F =

0 1 2 10 0 0 0

0 1 2 11 1 1 1

0 1 2 11 1 1 1

N

N N N N

N

N N N N

NN N N N

N N N N

w w w w

w w w w

w w w w

A

39



(cont.)

The inverse DFT is then expressed by:

where

-1f = A F

*0 1 2 1

0 0 0 0

0 1 2 11 1 1 1

1 *

0 1 2 11 1 1 1

1 1

TN

N N N N

NT

N N N N

NN N N N

N N N N

w w w w

w w w w

N N

w w w w

A A

The theorem implies that any circulant matrix has

eigenvectors the columns of A-1.

40



(cont.)

Let H be a NxN circulant matrix generated by the 1D

N-length signal h[n], that is:

mod( , ) ( ) [ ]N Nm n h m n h m n H

Let also αk be the k-th column of the inverse DFT

matrix A-1.

We will prove that αk, for any k, is an eigenvector of H.

The m-th element of the vector Hαk, denoted by

is the result of the circular convolution of the signal

h[n] with αk.

k mHα

41



(cont.)

1

0

[ ] [ ]N

k N kmn

h m n n

Hα1

0

1[ ]

Nk n

N N

n

h m n wN

( 1)( )1

[ ]m Nl m n

k m l

N N

l m

h l wN

( 1)1

[ ]m N

k m k l

N N N

l m

w h l wN

1

( 1) 0

1[ ] [ ]

mk m k l k l

N N N N N

l m N l

w h l w h l wN

We will break it into two parts

42



(cont.)

1 1 1

( 1) 0 1

1[ ] [ ] [ ]

N Nk m k l k l k l

N N N N N N N

l m N l l m

w h l w h l w h l wN

Periodic with respect to N.

1 1 1

( 1) 0 1

1[ ] [ ] [ ]

N N Nk m k l k l k l

N N N N N N N

l N m N l l m


1 1 1

1 0 1

1[ ] [ ] [ ]

N N Nk m k l k l k l

N N N N N N N

l m l l m


43



(cont.)

1

0

1[ ]

Nk m k l

k N N Nml

w h l wN

Hα [ ] k m

H k α

DFT of h[n] at k.

This holds for any value of m. Therefore:

[ ]k kH kHα α

which means that αk, for any k, is an eigenvector of

H with corresponding eigenvalue the k-th element

of H[k], the DFT of the signal generating H .

44



(cont.)

The above expression may be written in terms of

the inverse DFT matrix:

1 1 1 HA A Λ H A ΛA

Based on this diagonalization, we can prove the

property between circular convolution and DFT.

=diag [0], [1],..., [ 1]H H H N Λ

or equivalently: 1Λ ΑHA

45



(cont.)

g = Hf

DFT of g[n]

-1g = HA Af -1Ag = AHA Af G = ΛF

DFT of f [n]DFT of h [n]

[0] [0] 0 0 [0]

[1] 0 [1] 0 [1]

[ 1] 0 0 [ 1] [ 1]

G H F

G H F

G N H N F N

=

46


Diagonalization of doubly block

circulant matrices

• These properties may be generalized in 2D.

• We need to define the Kronecker product:

11 12 1

21 22 2

1 2

N

N

M M MN MK NL

a a a

a a a

a a a

B B B

B B BA B

B B B

,M N K L A B

47




• The 2D signal f [m,n],

may be vectorized in lexicographic ordering

(stacking one column after the other) to a

vector:1MNf

0 1, 0 1,m M n N

• The DFT of f [m,n], may be obtained in

matrix-vector form:

F A A f

48




1

Λ A A H A A

Theorem: The columns of the inverse 2D DFT

matrix

are eigenvectors of any doubly block circulant

matrix. The corresponding eigenvalues are the 2D

DFT values of the 2D signal generating the doubly

block circulant matrix:

1

A A

Doubly block circulantDiagonal, containing the 2D DFT

of h[m,n] generating H

Circulant Matrices and Convolution - Digital Image Processing

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