An Adaptive Noise Filtering Algorithm for AVIRIS Data
with Implications for Classification Accuracy
Rhonda D. Phillips Student Member, IEEE, Christine E. Blinn Member, IEEE,
Layne T. Watson, Fellow, IEEE, and Randolph H. Wynne, Member, IEEE
Abstract—This paper describes a new algorithm used to adaptively filter a remote sensing dataset based on
signal-to-noise ratios (SNRs) once the maximum noise fraction (MNF) has been applied. This algorithm uses
Hermite splines to calculate the approximate area underneath the SNR curve as a function of band number, and
that area is used to place bands into “bins” with other bands having similar SNRs. A median filter with a
variable sized kernel is then applied to each band, with the same size kernel used for each band in a particular
bin. The proposed adaptive filters are applied to a hyperspectral image generated by the AVIRIS sensor, and
results are given for the identification of three different pine species located within the study area. The adaptive
filtering scheme improves image quality as shown by estimated SNRs, and classification accuracies improved
by more than 10% on the sample study area, indicating that the proposed methods improve the image quality,
thereby aiding in species discrimination.
I. Background
Hyperspectral images provide a powerful tool as the wave spectrum is finely discretized using hundreds of channels
on a scanner. The large dimensionality of a hyperspectral dataset often requires a data transformation such as principal
components analysis (PCA) or the singular value decomposition (SVD) to reduce the number of variables, or bands,
within an image prior to further processing. Furthermore, these images tend to be noisy as a result of the fine
discretization and other factors such as the method of acquisition (small aircraft) [1]. Green et al. first proposed the
maximum noise transform (alternately called the minimum noise transform, minimum noise fraction, or MNF) to align
a dataset in order of decreasing signal-to-noise ratio (SNR) using an eigenvalue decomposition similar to PCA [2]. Lee
et al. equivalently defined the MNF (or noise adjusted PCA) as two PCA transformations, and used the MNF to reduce
the noise level in an image [3]. The MNF can be used to reduce noise and the number of dimensions in an image.
Reduction in noise in imagery is essential to many remote sensing applications, as Landgrebe has documented the
relationship between noise in imagery and classification errors [4]. Furthermore, certain applications require a minimum
SNR, for example, estimating foliar biochemical concentrations [5].
Noise can be reduced using a variety of filters defined on the frequency or spatial domains [6]. While certain
frequency domain filters (using the Fourier transform) have been shown to be more effective than spatial filters with
respect to specific types of noise, a spatial filter such as a median filter can produce similar results and requires
significantly less computation [7]. An adaptive filter can alter the size of the filter kernel (spatial domain) or change
the frequencies filtered (frequency domain) depending on image characteristics and noise levels. Lennon et al. used an
adaptive median filter on data transformed to MNF coordinates [8], and Pok et al. vary the kernel size between three
and five depending on the detected noise in a particular window in a three-band image [9]. King et al. present an
adaptive frequency domain filter used on medical imagery [10].
The properties of the MNF are well suited to an adaptive filter, yet adaptive spatial filtering is not commonly used
on MNF transformed data, although the idea was proposed by [8]. Typical data processing using the MNF truncates
the data, resulting in loss of signal; uses all bands in the MNF coordinate system without noise removal, or applies a
spatial convolution with a uniform kernel size across all bands despite all bands having drastically different SNRs. This
paper introduces an algorithm for an adaptive median filter applied to data transformed using the MNF in which the
filter support size varies with noise. To demonstrate the effectiveness of this technique, a real dataset is filtered using
R.D. Phillips and L.T. Watson are with the Departments of Computer Science and Mathematics, Virginia Polytechnic Institute and State University,
Blacksburg, VA 24061.
C.E. Blinn and R.H. Wynne are with the Department of Forestry, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061.
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the algorithm presented in this paper, and the dataset is subsequently used to identify three different species of pines
within a forest, an application for which the high spectral resolution imagery is well-suited.
II. Maximum Noise Fraction
Data transformations such as PCA and SVD transform an image to a new coordinate system without taking factors
such as noise into consideration [2]. PCA uses the eigenvectors (V ) resulting from an eigen decomposition:
Σ = V ΛV
−1
,
where Σ is the covariance matrix of the image, and Λ is a diagonal matrix containing the eigenvalues corresponding
to V , as a new coordinate basis for the image. This transformed image has the property that each successive band is
aligned along an axis of decreasing overall variance in the original image; that is, as the component number increases,
the variance within the component decreases. When PCA is used for data reduction, ideally these higher order bands
with decreasing variances are not necessary to represent the majority of the original image, and these components
can be removed, resulting in a data reduction. Unfortunately with datasets that are particularly noisy (the case with
hyperspectral data), the first few components are not sufficient to represent the image as they capture much of the noise
as well as the signal. The MNF is similar in spirit to PCA with the additional quality that it considers image noise
when selecting a new coordinate system. While the PCA aligns the axes along directions of the maximum variance in
the original image, the MNF aligns the axes along directions of the maximum SNR.
Theoretically, the MNF orders the data along the axes of maximum SNR using the eigen decomposition:
Σ
S
Σ
−1
N
= V ΛV
−1
where Σ
S
is the covariance matrix of the signal, Σ
N
is the covariance matrix of the noise, V is an (orthogonal) matrix
containing the eigenvectors of Σ
S
Σ
−1
N
, and Λ is a diagonal matrix containing eigenvalues that correspond to V . V
provides the basis for the transformed dataset. In practice, Σ
S
and Σ
N
are unknown and must be estimated from the
data [2]. Σ
S
is generally taken to be the covariance matrix of the image, and Σ
N
can be estimated using various
procedures [2][3]. The eigenvalues contained in Λ are the estimated variance of the signal (σ
S
) divided by the estimated
variance of the noise (σ
N
), and therefore the diagonal element λ
b
in Λ is an approximation for the SNR of band b in
the transformed image.
III. Adaptive Filter
The MNF is commonly used in remote sensing for data reduction and noise removal. The MNF of an image can be
truncated while still preserving most of the information within the image, which is especially useful in the hyperspectral
image processing domain as images contain hundreds of highly correlated bands and noise. The higher order bands that
are truncated commonly contain very low SNRs, and truncating the MNF can have the added effect of eliminating much
of the noise without losing much signal. Determining the precise location to truncate the MNF is problematic, and a
judgement call is often made by looking at a plot of the eigenvalues relative to the band number and determining where
this eigenvalue curve begins to approach an asymptote (λ = 1). In practice, this truncation is performed as a means
of reducing the overall noise within the image, but this method does not fully take advantage of the properties of the
MNF. If the truncation includes too many bands, too much noise is left in the image, and if the truncation includes too
few bands, useful signal may be excluded from the resulting image. A likely scenario would be that truncation includes
noise in the bands that are kept while discarding good signal with the higher order bands that are discarded.
Green et al. suggest that with low SNR bands, all values can be replaced with the mean of the band, and the MNF
image can be retransformed to the original subspace, resulting in a less noisy image [2]. This is an example of a rather
extreme mean filter. Another approach to reducing the noise in an image is to apply a small (typically a three by three
window) spatial filter such as a mean or median filter. However, applying a filter uniformly to all bands within the MNF
will not take advantage of the specific ordering of the bands. Bands with lower SNRs might benefit from a filter with a
larger window, while bands with high SNRs require little or no filtration. Bands with low SNRs have comparatively
low signal relative to noise, yet may have enough signal to warrant smoothing of the noise. A large filter will degrade
that signal, but will hopefully affect the noise more, resulting in a greater signal relative to noise.
Spatial median filters work by decreasing the variance within a small window (kernel) by assigning a pixel the
median value of the surrounding pixels. For example, using a 3 × 3 window, a median filter would assign a pixel the
median value of itself and its eight immediate neighbors (top, left, right, bottom, and four diagonal locations). As
geographic data is highly correlated, the variance of the signal within such a window should be small, and noise should
be random and not correlated within a neighborhood, making a large variance probable. With the assumption that the
2
1
2 3
4 5
x
1
2
3
4
5
fHxL
Fig. 1. Typical MNF eigenvalue curve shape.
1
2 3
4 5
x
1
2
3
4
5
fHxL
1
2 3
4 5
x
1
2
3
4
5
fHxL
Fig. 2. Dividing area under curve into bins.
variance of the noise is larger than that of the signal in these small windows, a spatial filter such as a median filter will
preserve most of the signal while eliminating much of the noise. A filter with a larger window has a more dramatic
smoothing effect over a filter with a small window, resulting in a larger SNR at the expense of the signal. A median
filter has the property of preserving original values unlike a mean filter.
The MNF is ordered such that for any two bands numbered m and n (assume n > m), SNR
m
≥ SNR
n
. Recall
that the eigenvalues associated with the MNF are estimates of SNR, meaning λ
m
≥ λ
n
. Based on the reasoning offered
above, the size K of the filter kernel for band m should not be greater than that for band n, K
m
≤ K
n
. Similarly, the
same size filter should be applied to bands with the most similar eigenvalues.
Consider the shape of the typical MNF eigenvalue curve, shown in Fig. 1. The first few bands with the largest
decrease in slope should be grouped together in smaller groups than the last bands with very similar small negative
slopes. In order to divide the bands into bins in this manner, the area underneath the eigenvalue curve can be divided
evenly into a number of bins corresponding to the number of different sized filters to be applied, as shown in Fig. 2.
The three colors represent three different bins and three different kernel sizes. A formal algorithm for the adaptive filter
(AF) is given below.
Algorithm AF(M, Λ, nb)
input/output:
M (image transformed to MNF coordinates,
filtered upon exit)
input:
Λ (B eigenvalues where B is number of bands in M )
nb (number of bins to use)
1 begin
2 approximate f (b) = λ with C(b) = λ
3
3 area :=
Z
B
1
C(b)db
4 area
per bin :=
area
nb
5 sare a := 0
6 for i := 1 step 1 until B − 1 do
7 begin
8 sare a := sarea +
Z
i+1
i
C(b)db
9 bin := ceiling
sarea
area per bin
10 kernel
i
:= 2 · (bin − 1) + 1
11 end
12 kernel
B
:= ker nel
B−1
13 for i := 1 step 1 until B do
14 apply kernel
i
× kernel
i
median filter
to band
i
in image M
15 end
Calculating the area under the curve will require a function to approximate the eigenvalues as a function of band
number, as indicated in line 2 of the above algorithm. A description of Hermite splines, which are recommended given
their suitability to this particular application, is included in Section 3.1. The assignment of bins occurs in line 9, and
warrants further explanation. Starting with the first band, the area under the curve is calculated as
Z
2
1
C(b)db.
The total area under the curve through band i is therefore
Z
i+1
1
C(b)db.
The results of previously calculated integrals are stored in sarea to prevent redundant calculations. Taking the ceiling
of the cumulative area under the curve divided by the area per bin results in bands one through B − 1 being placed in
bins one through nb, and band B is placed in the same bin as B − 1. Line 10 continues with the conversion of a bin
number to the size of a spatial filter kernel that corresponds to bin number. The bands in bin one should have no filter
(equivalent to a kernel of size one) applied, and the bands in bin two should have a 3 × 3 filter applied.
This approach is valid for convex eigenvalue curves that are similarly shaped to Fig. 1, which is usually the case.
The properties of the MNF dictate that the eigenvalue function is strictly decreasing, but in the event that the eigenvalue
function is not convex, dividing up the area under the curve of the derivative of the function will group the most similar
eigenvalues and their corresponding bands together. Consider finding the area under the curve of the derivative:
Z
b
a
f
′
(x)dx = f (b) − f(a)
according to the fundamental theorem of calculus. This calculation requires no approximation of the function as the
actual functions’ values can be used. Because the eigenvalue function is monotonically decreasing, the area underneath
the curve will be negative, and therefore the area will be negated to produce a positive result necessary for bin
determination. The above algorithm is modified to produce the following adaptive filter with derivative (AFD) algorithm
using the area underneath the curve of the derivative to determine the location of bins.
Algorithm AFD(M, Λ, nb)
input/output:
M (image transformed to MNF coordinates,
filtered upon exit)
input:
Λ (B eigenvalues where B is number of bands in M )
nb (number of bins to use)
4
1 begin
2 area := Λ(1) − Λ(B)
3 area pe r bin :=
area
nb
4 sare a := 0
5 for i := 1 step 1 until B − 1 do
6 begin
7 sare a := sarea + Λ(i) − Λ(i + 1)
8 bin := ceiling
sarea
area per bin
9 kernel
i
:= 2 · (bin − 1) + 1
10 end
11 kernel
B
:= ker nel
B−1
12 for i := 1 step 1 until B do
13 apply kernel
i
× kernel
i
median filter
to band
i
in image M
14 end
Finally, either of the above variations on the adaptive filtering algorithm may be used on a particular range of
bands. For example, if prior knowledge or analysis of the MNF transformed dataset indicates that there is no usable
signal beyond a specific band, the MNF image can still be truncated and filtered adaptively. The value of B would be
changed from the total number of bands in the image to the number of bands desired after truncation. This is different
from simply truncating the MNF because the bands that are kept would be filtered to decrease the noise, and the number
of bands kept could be larger to ensure that very little signal is lost in the truncation.
A. Hermite Splines
The filtering algorithm requires a function that approximates the eigenvalue curve generated by the MNF. Cubic
splines are piecewise cubic polynomials that produce a visually appealing curve and interpolate a given set of points. In
particular, Hermite cubic splines have only one continuous derivative (standard cubic splines have two) and produce a
monotone cubic spline curve interpolating a monotonic function, rendering this type of spline ideal for interpolating the
monotonic SNR curve. The Hermite cubic spline C(x) is composed of 2n basis functions, c
i
(x), ˆc
i
(x), i = 1, . . ., n,
where n is the number of interpolation points. The function
C(x) =
n
X
i=1
y
i
c
i
(x) + d
i
ˆc
i
(x)
interpolates the points (x
i
, y
i
), i = 1, . . . , n if
c
i
(x
i
) = 1, c
i
(x
j
) = 0, j 6= i,
ˆc
i
(x
j
) = 0, for all i, j.
Furthermore,
c
′
j
(x
i
) = 0, for all i, j,
ˆc
′
i
(x
i
) = 1, ˆc
′
i
(x
j
) = 0, j 6= i,
making
C
′
(x
i
) = d
i
.
Only x
i
, y
i
, and d
i
, i = 1, . . ., n are required to define a Hermite cubic spline, and the d
i
are chosen to make C(x)
monotone (theoretically always possible for monotone data y
i
). Refer to [11] for a more detailed description of Hermite
cubic splines including definitions of the basis functions c
i
(x
i
), ˆc
i
(x
i
).
The derivative and the definite integral of Hermite cubic splines can be easily obtained as the cubic polynomials
(and basis functions) are easily differentiated or integrated analytically. Included in [11] is a set of subroutines designed
to define, evaluate, and integrate Hermite cubic splines, PCHEZ, PCHEV, and PCHQA, respectively. PCHEZ defines
continuous derivatives, d
i
, that result in a visually appealing function, PCHEV evaluates the function and the derivative at
a set of points, and PCHQA returns the definite integral of the function between two points, a and b.
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