ISPRS Archives, Vol. XXXIV, Part 3/W8, Munich, 17.-19. Sept. 2003 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Registration of Remote Sensing Image Data Based on Wavelet Transform Mitsuyoshi Tomiya and Akihiro Ageishi Department of Applied Physics, Seikei University, Kichijoji-Kitamachi 3-3-1, Musasino-shi, Tokyo, 180-8633, Japan -
[email protected] Abstract: A registration method of remotely sensed image data which is base on the wavelet transform is proposed. The shape information of the regions in the images is analyzed by the Haar and Daubechies’ wavelets. Comparing the coefficients of the wavelets and minimizing the mean square error (MSE) which is defined by the coefficients, the regions are matched and the images are registered. Affine transformation is applied for the registration and its parameters are also determined by the minimum of the MSE. Our result shows that the wavelet method is remarkably stable, irrespective of various shapes of the regions. WG II/IV, III/4, III/5, III/6 Keywords: Registration, Haar wavelet transform, Daubechies’ wavelet transform,Affine transform, Fourier descriptor Normalizing the distance of the perimeter 2S , the coordinates
㧝㧚Introduction
(x, y) of the points on the curve can be considered as the periodic
Registration technique is necessary for geometrical matching of
functions
the remote sensing images. It makes possible to monitor natural
x(t 2S )
and artificial changes in land cover precisely. We have studied
x (t ) ,
y (t 2S )
y (t ) ( 0 d t 2S ) . (1)
the Fourier descriptor method that is applied to the feature of the
We present x(t) and y(t) of the closed curve in the x-y plane
regions on the data (Anzai and Tomiya, 2001). The Fourier
(image) as the function of t ( 0 d t d 2S ).
descriptors represent the shapes of the boundaries and are
transforms of the coordinates (x, y) are
compared to be matched and then registered. However this
§ x (t ) · ¸¸ ¨¨ © y (t ) ¹
method is very applicable only for round shaped regions. Here we propose an alternative method which is based on the wavelet transform, which has also been adopted for remote sensing data
1 2S
a0
last decade(e.g. (Simhadri, 1998)). For the image registration, it is necessary to extract and compare
ak
the image features. There are various kinds of image features: shapes, distributions (of brightness, colors,…), textures, etc. Among them the shape information must be the most robust and
ck
the most effective for the remote sensing data under various conditions. Generally the edge detection is applied to get the
1
S 1
S
§ a0 · f § a k ¨¨ ¸¸ ¦ ¨¨ © c0 ¹ k 1 © ck
bk · § cos kt · , ¸¨ ¸ d k ¸¹ ¨© sin kt ¸¹
2S
³ x(t ) dt , c 0 0
2S
S
0
2S
0
1 2S 1
³ x(t ) cos k t dt , b k
³ y(t ) cos k t dt ,
The Fourier
dk
(2)
2S
³ y(t ) dt , 0
2S
³ x(t ) sin k t dt , 0
1
S
2S
³ y (t ) sin k t dt , 0
pixels of the boundaries that from the shape features. In this
where a k , bk , ck , d k are the coefficients of the k-th Fourier
work, it is very important that the extracted boundaries are surely
series or called the k-th Fourier descriptors (Granlund, 1972, Zahn and Roskies, 1972), especially a0 , c0 are the center of
connected.
Therefore the region growing is applied to the
characteristic regions of the images after the segmentation.
gravity.
Then the edge detection is adopted to extract the boundaries of the regions as the shape information of the images.
2-2 Affine transform and mean square error To register the images, the geometrical distortion which is mainly due to the satellite attitude and altitude, etc. have to be detected
2㧚Fourier descriptor and affine transform
and adjusted.
In general it may cause parallel transform,
2-1 Fourier descriptor
rotation, expansion-reduction and also skew transform etc. of the
After the edge extraction, the image feature is represented as a
image.
one-dimensional close curve in two dimensional image. Setting
images. Appling the affine transform to the image A with the coordinates (x, y) to be matched to the image A c with xc, y c ,
a start point, a distance t is introduced along the curve.
145
Here we choose the affine transform to adjust the
ISPRS Archives, Vol. XXXIV, Part 3/W8, Munich, 17.-19. Sept. 2003 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
2)
we have § xc · ¨¨ ¸¸ © yc ¹
§D ¨¨ ©J
Admissibility condition
Integration of the mother wavelet in the range that includes its
E · § x· § 'x· ¸ ¨ ¸ ¨ ¸ , G ¸¹ ¨© y ¸¹ ¨© ' y ¸¹
DG EJ z 0
(4)
whole support must vanish
³
where D , E , J , G , 'x , 'y are the constants. The coefficients
f f
Ǡ ( t ) dt
0.
(10)
D , E , J , G form the affine transform and include rotation,
It is the fundamental condition for the wavelet and also called the
expansion-reduction, skew transform and 'x , 'y are the
wavelet condition.
displacement to be parallel transformed.
3)
Orthogonal condition
The mother wavelet and the scaling function own the orthogonal relation,
Applying the Fourier transform (2) to Eq.(4), the transform rules of the Fourier descriptors for (x,y) to the descriptor a ck , bkc , c ck , d kc for (x’,y’) are derived as
§ akc ¨¨ © ckc
bkc · ¸ d kc ¸¹
§D E · § ak ¸¸ ¨¨ ¨¨ © J G ¹ © ck
cG m,n ޓޓ ( m, n : integer )
Ǿ(t m),Ǿ(t n)
bk · (5) ¸ d k ¸¹
Ǡ(t m),Ǡ(t n)
cG m,n
Ǡ(t m),Ǿ(t n)
0 ,
(11)
where , denotes the inner product of the wavelet function
§¨ a0c ·¸ §¨D E ·¸ §¨ a0 ·¸ §¨ ' x ·¸ ¨ cc ¸ ¨ J G ¸ ¨ c ¸ ¨ ' y ¸ ¹ ¹ © 0¹ © © 0¹ ©
(6)
space
D t , E t
Actually the coarse feature of the shape which belongs to the
³
f
f
D * t E t dt .
(12)
coefficients k d 2 is sufficient for the matching (Tseng et.al.,
In Eq.(11), m and n represent the displacement in t. The scaling
1997; Anzai and Tomiya, 2001). For realistic digital image,
functions with the different displacements are orthogonal. The
Eq.(5) cannot be exactly satisfied. There are always errors in
mother wavelets with the different displacements are also
the calculated coefficients. The mean square error (MSE) for
orthogonal. The mother wavelet and the scaling function are
the affine coefficients is defined as
always orthogonal. It leads us to the relation for the sequences { pk } and { qk }
n
1 2 2 2 2 ¦ H a ,k H b , k H c , k H d , k , nk1
MSE F
(7)
§ H a ,k H b,k · ¨¨ ¸¸ © H c,k H d ,k ¹
§ akc ¨¨ © ckc
bkc · §D E · § ak ¸ ¨ ¸¨ d kc ¸¹ ¨© J G ¸¹ ¨© ck
1 k
qk
where the errors for the coefficients are
bk · . ¸ d k ¸¹
p 1 k .
(13)
Setting c=1, the mother wavelet and the scaling function form the
(8)
orthonormal system. Then the system of Eq.(11) is called the orthonormal wavelet.
Varying the affine parameters, the minimum of the MSE F for the fixed Fourier descriptors determines the matching transform.
3-2 Haar wavelet
The wavelet made of the Haar function is the simplest one. The
3.
Wavelet transformation
mother wavelet and the scaling function (Fig.1) are defined as
3-1 Wavelet
The wavelet is defined as the superposition of the wave which has local support (Stollnitz et.al., 1996). Its fundamental waves
Ǡ (t )
1 ° ® 1 ° 0 ¯
0 d t d 1 2 1 2 d t d 1 otherwise
I (t )
® ¯
0 d t d 1 otherwise
are the mother wavelet < t and the scaling function (or the father wavelet) I t . The wavelet transform is to expand the original wave into these two kinds of waves.
The mother
wavelet and the scaling function obey three following rules. 1)
¦
c k( j 1 )
q kǾ ( 2 t k ) , (9)
¦
,
(15)
The decomposition algorithm of the Haar wavelet is
k
Ǿ(t )
0
(14)
The system obeys the three conditions (9), (10), (11) and c=1.
Two-scale relation
The mother wavelet and the scaling function are defined by the reconstruction sequences { pk } and { qk }, Ǡ (t )
1
,
d
p kǾ ( 2 t k ) 㧔k: integer㧕.
k
146
( j 1) k
1 ( j) ( c 2 k c 2( kj ) 1 ) , 2 1 ( c 2( kj ) c 2( kj ) 1 ) , 2
(16)
ISPRS Archives, Vol. XXXIV, Part 3/W8, Munich, 17.-19. Sept. 2003 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
where j t 0 . At first ^ck 0 ` is set the consecutive coodinate values on the boundary. The wavelet coefficient ck j 1 after the
ck( j 1)
d
( j 1) k
To reconstruct the original data, we can use the
c k( j 1) d k( j 1) ,
1 ޓ (1 3 )c2( kj ) (3 3 )c2( kj )1 (3 3 )c2( kj ) 2 (1 3 )c2( kj ) 3 . 8
^
`
I (t ) 1
1
(17)
c 2( kj )1
`
