Image Retrieval from a Forensic Cartridge Case Database - Multimedia Database Retrieval: Technology and Applications - page 150

{ [ , ] } = [ + − , − + − ]

g i k f i cos ʸ k sin ʸ a i sin ʸ k cos ʸ b (6.15)

0 0 0 0

where index 0 1 for an image of size M row-pixels and

N column-pixels. Following the property of Fourier transformation [ 173 ], we can
obtain the Fourier transformation of

≤ i ≤ M − 1 and 0 ≤ k ≤ N −

{ g [ i , k ] } that is related to the Fourier transform

of { f [ i , k ] } by:

F m cos ʸ 0

(6.16)

nM
N

mN
M

e − j 2ˀ (

M + n N

) ×

m

G [ m , n ]= ʸ 0 + sin ʸ 0 ,− sin ʸ 0 + n cos

[ , ]

with the corresponding magnitude of G m n which is equal to:

F m cos ʸ 0

nM
N

mN
M

ʸ 0 + sin ʸ 0 ,− sin ʸ 0 + n cos (6.17)

=

When M N , we have the following relationship:

| [ , ] | = | [ + ,− + ] |

G m n F m cos ʸ n sin ʸ m sin ʸ n cos ʸ (6.18) 0 0 0 0

Let | G [ m , n ] | = M G [ m , n ] and | F [ m , n ] | = M F [ m , n ] , and then the relationship

shown in Eq. ( 6.18 ) can be rewritten as:

M G [ m , n ]= M F [ m cos ʸ 0 + n sin ʸ 0 ,− m sin ʸ 0 + n cos ʸ 0 ] (6.19)

From the relationship in Eq. ( 6.19 ), the translation parameters disappear, leaving
only rotation parameters. The estimation of the parameters for rotation starts with
changing the magnitude of the Fourier transform from rectangular coordinates [ m , n ]

to polar coordinates [ r , ʸ ] by substituting index m = r cos ʸ and index n = r sin ʸ into

Eq. ( 6.5 ). The Fourier transform in polar coordinates can be obtained by:

M − 1
i = 0

N − 1
k = 0 g [ i , k ] e − j 2ˀ ( r cos ʸ

1
MN

)

M + N

r sin ʸ

G [ r , ʸ ]= (6.20)

Similarly, the Fourier transform in polar coordinates of f [ i cos ʸ 0 + k sin ʸ 0 −

a , − i sin ʸ 0 + k cos ʸ 0 − b ] can be obtained by:

1
MN

e − j 2ˀ ( ( m cos ʸ 0 + n N sin ʸ 0 )

M + ( − m M sin ʸ 0 + ncos ʸ 0 )

u v

N

)

u ∑ v

f [ u , v ] (6.21)

Next, substituting m = rcos ʸ and n = rsin ʸ into Eq. ( 6.21 ) produces the Fourier

transform in polar coordinates:

f [ u , v ]

e − j 2ˀ (

u
M +

v
N

) =

u ∑ v

r cos ( ʸ − ʸ 0 ) r sin ( ʸ − ʸ 0 )

F [ r , ʸ − ʸ 0 ] (6.22)

MN