Method and system for increasing signal-to-noise ratio
Methods and systems for increasing the signal-to-noise ratio for satellite sensor data or signals, such as hyperspectral imageries (also referred to as datacubes due to their 3-dimensional nature). This is done by reducing the noise in the data or signals by first elevating the noise level temporarily for effective denoising. The denoising process is then performed in this condition and the noise level is then reversibly de-elevated after denoising. The denoising process comprises noise removal in both the spectral and the spatial domains. Once the denoising process is complete, the data is converted back from the spectral and spatial domains. Since this reconstruction process introduces errors, these errors are compensated for using the components from both the original data and denoised data filtered by the low pass filters.
1. A method for improving a signal to noise ratio of data in a multidimensional datacube, said data being in an original domain, the method comprising the steps of:
a) elevating a noise level of said data
b) removing noise from said data in a spatial domain
c) removing noise from said data in a spectral domain to result in denoised data
d) converting denoised data into said original domain
e) correcting errors introduced to said data by step d).
2. A method according to claim 1 wherein step a) is accomplished by transforming said datacube into a spectral derivative domain.
3. A method according to claim 2 wherein said datacube is transformed to said spectral derivative domain by using a formula
θ
(
λ
,
p
,
l
)
=
∂
y
(
λ
,
p
,
l
)
∂
λ
=
y
(
λ
+
δ
λ
,
p
,
l
)
-
y
(
λ
,
p
,
l
)
δ
λ
where
λ is a spectral band center,
p is a cross-track pixel number,
l is an along-track line number of said datacube y(λ,p,l) λ=1, 2, . . . , N b ; p=1, 2, . . . , N c ; l=1, 2, . . . , N r represents said datacube in said original domain and
δ λ is a small displacement in the spectral dimension
N b , is a total number of bands
N c , is a total number of pixels per line
N r is a total number of cross-track lines of the datacube.
4. A method according to claim 1 wherein step b) is accomplished by a series of steps including computing a 2-d wavelet transform for each spectral band image of a spectral derivative of said datacube.
5. A method according to claim 4 wherein said series of steps further comprises:
estimating a threshold value for each spectral band image
performing a soft threshold wavelet shrinkage operation
computing an inverse 2-D wavelet transform.
6. A method according to claim 5 wherein said inverse 2D wavelet transform comprises:
{tilde over (θ)}(λ)=IDWT2{η spatial (DWT2{θ(λ)})} λ=1,2,3, . . . , N b
wherein
θ(λ) is a spectral derivative of a band image at band λ of said datacube
{tilde over (θ)}(λ) is a spatially denoised derivative spectral band image of said datacube
DWT2 is a 2-D discrete wavelet transform applied to said spectral derivative of said band images at two spatial dimensions (along-track and across-track dimensions)
IDWT2 is an associated 2-D inverse discrete wavelet transform.
η spatial is a threshold function applied to said band image on a band-by-band basis for the entire said datacube.
7. A method according to claim 1 wherein step c) is accomplished by a series of steps including computing a 1-D wavelet transform for each spectrum corresponding to a spatial pixel at each location at a spatially denoised datacube resulting from step b).
8. A method according to claim 7 wherein said series of steps further comprises:
estimating a threshold value for each spectrum
performing a soft threshold wavelet shrinkage operation
computing an inverse 1-D wavelet transform.
9. A method according to claim 8 wherein said inverse 1-D wavelet transform comprises:
{circumflex over (θ)}( p,l )=IDWT2{η spatial (DWT{{tilde over (θ)}( p,l )})} p= 1,2, . . . , N c ; l= 1,2, . . . , N r
wherein
{tilde over (θ)}(p,l) is a spectrum derivative of a spatially denoised datacube at spatial location (p,l)
{tilde over (θ)}(p,l) is a spatially-spectrally denoised spectrum derivative of said datacube at location (p,l)
DWT is a 1-D discrete wavelet transform applied to a spectrum
IDWT is an associated 1-D inverse discrete wavelet transform
η spectral is a threshold function applied to said spectrum on a pixel-by-pixel basis for the entire said datacube.
10. A method according to claim 1 wherein said denoised data is converted into said original domain by spectral integration.
11. A method according to claim 10 wherein said spectral integration is accomplished by using a formula:
x
^
(
λ
j
,
p
,
l
)
=
{
x
^
1
(
p
,
l
)
,
j
=
1
x
^
1
(
p
,
l
)
+
∑
i
=
1
j
-
1
θ
^
(
λ
i
,
p
,
l
)
·
δ
λ
,
j
>
1
wherein
λ i is a center wavelength of an i th spectral band
λ j is a center wavelengths of a j th spectral band
and {circumflex over (x)} 1 (p,l)=y(λ 1 ,p,l) represents said datacube in said original domain.
12. A method according to claim 9 wherein step e) is accomplished by cancelling error accumulated by step d) and by compensating a cancelled signal.
13. A method according to claim 12 wherein compensating said cancelled signal is accomplished by filtering.
14. A method according to claim 13 wherein at least two identical low pass filters are used to compensate said cancelled signal.
15. A method according to claim 14 wherein said low pass filters are moving average filters.
16. A method according to claim 15 wherein said filters replace low pass frequency components of a denoised signal from step d) with low frequency components of an original signal derived from said datacube such that
x
~
(
λ
j
,
p
,
l
)
=
x
^
(
λ
j
,
p
,
l
)
-
∑
i
=
j
-
Δ
2
j
+
Δ
2
x
^
(
λ
i
,
p
,
l
)
Δ
+
∑
i
=
j
-
Δ
2
j
+
Δ
2
y
(
λ
i
,
p
,
l
)
Δ
(
29
)
wherein
Δ+1 is a width of a correction window used by said filters
{circumflex over (x)}(λ,p,l) is a denoised signal before correction
{tilde over (x)}(λ,p,l) is a denoised signal after correction.
17. A system for increasing a signal to noise ratio of multidimensional data, said data being in an original domain, the system comprising:
means for elevating a noise level of said data
means for removing noise from said data in at least one domain to result in denoised data
means for converting denoised data into said original domain
means for correcting errors introduced to said data by a conversion of said denoised data.
18. A system according to claim 17 wherein said means for removing noise comprises:
means for removing noise from said data in a spatial domain.
19. A system according to claim 17 wherein said means for removing noise comprises:
means for removing noise from said data in a spectral domain.
20. A system according to claim 17 wherein said means for correcting errors comprises filters.