The high spectral redundancy of hyper/ultraspectral Earth-observation
satellite imaging raises three challenges: (a) to design
accurate noise estimation methods, (b) to denoise images with
very high signal-to-noise ratio (SNR), and (c) to secure
unbiased denoising. We solve (a) by a new noise estimation,
(b) by a novel Bayesian algorithm exploiting spectral
redundancy and spectral clustering, and (c) by accurate
measurements of the interchannel correlation after denoising. We
demonstrate the effectiveness of our method on two ultraspectral Earth
imagers, IASI and IASI-NG, one flying and the other in project, and
sketch the major resolution gain of future instruments entailed by
such unbiased denoising.
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MSNRs Obtained When Varying the Hyperparameters of the
Algorithm for HSI #2a
Optimal
Case #1
Case #2
Case #3
63.44
62.39
62.08
62.84
Optimal is the default case of IASI-NG
with , case #1 with
, case #2 with
, and case #3 with
. Deviations from the optimal
hyperparameters cause a slight decrease of the
performance.
Table 2.
Mean MSNR for the Seven Simulated IASI-NG USIs for VMB3D, DBBD,
Chen’s method, PCA Band-by-Band, and TDLa
The column labeled Noisy corresponds to
the MSNR when no denoising is applied. The fact that TDL
shows a MSNR lower than the noise itself must be
understood as this method not being adapted to these kinds
of high-SNR frequency-dependent noise HSIs, despite being
one of the best methods available.
Table 3.
for the Seven Simulated IASI-NG
USIs for VBM3D, DBBD, Chen, PCA Band-by-Band, and TDLa
Noisy refers to the case when no denoising
is applied. The evaluation of TDL shows again that certain
methods (including VBM3D) are not adapted to high-SNR
frequency-dependent HSIs, even if they can be considered
the best methods for low and medium SNR HSIs.
Table 4.
(a) STD Reduction Factor after Denoising with DBBD,
(b) the Associated Surface Reduction for the IASI-NG
Simulated USIs, (c) Mean and (d) STD of the
Pearson Autocorrelation of (the Removed Signal) Excluding
the Diagonala
USI #
(a)
(b)
(c)
(d)
#0
7.492
56.124
0.021
#1
7.331
53.740
0.021
#2
6.663
44.400
0.021
#3
6.966
48.530
0.021
#4
6.406
41.037
0.021
#5
6.778
45.943
0.021
#6
7.125
50.767
0.021
The noise is simulated according to the tabulated NEDT of
the sensor and by applying Eq. (1) to obtain its
realistic frequency-dependent STD.
Table 5.
Mean MSNR for the Seven Simulated IASI-NG USIs for VMB3D, DBBD,
Chen’s Method, PCA Band-by-Band, and TDL in the Case of
Low SNR and a Reduced Number of Channelsa
We multiplied by 10 the STD of the noise and took one out
of 70 wavenumbers (a total of 242). Now the difference
between Chen and DBBD is not significant. The method by
Chen should be preferred in this scenario, since it
performed better in a majority of USIs. TDL shows
competitive results, as expected.
Table 6.
Statistics of the Histograms of the Pearson Cross-Correlations
Coefficients along All Frequencies of the Removed Signal in
Real USIs #7 … #11a
Mean
STD
USI #
(a)
(b)
(c)
(d)
#7
#8
#9
#10
#11
Mean
(a) means in DBBD, (b) means in PCA-bands and
Chen, (c) STDs in DBBD, and (d) STDs in
PCA-bands and Chen. For the real IASI data (8461 channels)
Chen’s method has values very similar to PCA-bands
because with a reduced number of channels its thresholding
simply set to zero the less significant PCs (as PCA does).
The autocorrelation is better in DBBD. Note, however, that
when the number of available channels is larger (as in the
IASI-NG case, see Table 5, and also Tables 2 and 3), Chen clearly
outperforms PCA. The goal of this experiment is to show
that indeed these methods are valid to denoise
ultraspectral images.
Tables (8)
Algorithm 1.
Estimate the Noise Level Function of an HSI
1: HSI noise estimation
Input: acquired noisy HSI
Input: window size
Output: noise estimation
2: ▹ Pearson
cross-correlation coeffs.
3: ▹ First
dimension: frequencies
4: # Raw noise estimation
5: fordo
6: ▹
Exclude trivial diagonal maxs
7: ▹ Get most
correlated frequency
8:
9: ▹ Mean
equalization
10: ▹ Raw noise
estimation
11: end for
12: # Final filtered estimation
13: fordo ▹ Non-overlapping
window
14:
15: ifthen ▹ Take care of
array boundary
16:
17: end if
18: ▹
Assign min to window
19: end for
20: return
Algorithm 2.
DBBD Denoising of an HSI
1: # Input/output:
Input: acquired noisy HSI, of
size
Input: STDs of the noise
(wavenumber dep.)
Input: number of spectral
clusters
Input: number of similar
pixels
Input: number of PCA PCs
kept
Output: denoised HSI
2: ▹
Placeholder
3: Clusterize in spectral groups
▹ K-means
for cosine distance
4: for each cluster
do
5: Noise normalization: divide each
coordinate of each sample in by its STD taken from
6: ▹
Matrix of eigenvectors of the PCA of
, in decreasing abs.
eigenvalue order
MSNRs Obtained When Varying the Hyperparameters of the
Algorithm for HSI #2a
Optimal
Case #1
Case #2
Case #3
63.44
62.39
62.08
62.84
Optimal is the default case of IASI-NG
with , case #1 with
, case #2 with
, and case #3 with
. Deviations from the optimal
hyperparameters cause a slight decrease of the
performance.
Table 2.
Mean MSNR for the Seven Simulated IASI-NG USIs for VMB3D, DBBD,
Chen’s method, PCA Band-by-Band, and TDLa
The column labeled Noisy corresponds to
the MSNR when no denoising is applied. The fact that TDL
shows a MSNR lower than the noise itself must be
understood as this method not being adapted to these kinds
of high-SNR frequency-dependent noise HSIs, despite being
one of the best methods available.
Table 3.
for the Seven Simulated IASI-NG
USIs for VBM3D, DBBD, Chen, PCA Band-by-Band, and TDLa
Noisy refers to the case when no denoising
is applied. The evaluation of TDL shows again that certain
methods (including VBM3D) are not adapted to high-SNR
frequency-dependent HSIs, even if they can be considered
the best methods for low and medium SNR HSIs.
Table 4.
(a) STD Reduction Factor after Denoising with DBBD,
(b) the Associated Surface Reduction for the IASI-NG
Simulated USIs, (c) Mean and (d) STD of the
Pearson Autocorrelation of (the Removed Signal) Excluding
the Diagonala
USI #
(a)
(b)
(c)
(d)
#0
7.492
56.124
0.021
#1
7.331
53.740
0.021
#2
6.663
44.400
0.021
#3
6.966
48.530
0.021
#4
6.406
41.037
0.021
#5
6.778
45.943
0.021
#6
7.125
50.767
0.021
The noise is simulated according to the tabulated NEDT of
the sensor and by applying Eq. (1) to obtain its
realistic frequency-dependent STD.
Table 5.
Mean MSNR for the Seven Simulated IASI-NG USIs for VMB3D, DBBD,
Chen’s Method, PCA Band-by-Band, and TDL in the Case of
Low SNR and a Reduced Number of Channelsa
We multiplied by 10 the STD of the noise and took one out
of 70 wavenumbers (a total of 242). Now the difference
between Chen and DBBD is not significant. The method by
Chen should be preferred in this scenario, since it
performed better in a majority of USIs. TDL shows
competitive results, as expected.
Table 6.
Statistics of the Histograms of the Pearson Cross-Correlations
Coefficients along All Frequencies of the Removed Signal in
Real USIs #7 … #11a
Mean
STD
USI #
(a)
(b)
(c)
(d)
#7
#8
#9
#10
#11
Mean
(a) means in DBBD, (b) means in PCA-bands and
Chen, (c) STDs in DBBD, and (d) STDs in
PCA-bands and Chen. For the real IASI data (8461 channels)
Chen’s method has values very similar to PCA-bands
because with a reduced number of channels its thresholding
simply set to zero the less significant PCs (as PCA does).
The autocorrelation is better in DBBD. Note, however, that
when the number of available channels is larger (as in the
IASI-NG case, see Table 5, and also Tables 2 and 3), Chen clearly
outperforms PCA. The goal of this experiment is to show
that indeed these methods are valid to denoise
ultraspectral images.