A MAXIMUM NOISE FRACTION TRANSFORM BASED ON A SENSOR NOISE
MODEL FOR HYPERSPECTRAL DATA
12Naoto Yokoya1 and Akira Iwasaki2
Graduate Student, Department of Aeronautics and Astronautics, The University of Tokyo,
4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan; Tel: + 81-3-5452-5319;
E-mail: yokoya@sal.rcast.u-tokyo.ac.jp

Professor, Research Center for Advanced Science and Technology, The University of Tokyo,
4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan; Tel: + 81-3-5452-5285;
Email: aiwasaki@sal.rcast.u-tokyo.ac.jp

KEY WORDS: feature extraction, MNF transform, noise estimation, hyperspectral data, spectral
distortion
ABSTRACT: The maximum noise fraction (MNF) transform, which produces the improvedorder of components by signal to noise ratio (SNR), has been commonly used for spectral feature
extraction from hyperspectral remote sensing data before image classification. When
hyperspectral data contains a spectral distortion, also known as a “smile” property, the firstcomponent of the MNF, which should have high image quality, suffers from noisy brightness
gradient pattern which thus reduces classification accuracy. This is probably because the classic
noise estimation of the MNF is different from the real noise model. The noise estimation is themost important procedure because the noise covariance matrix determines the characteristics ofthe MNF transform. An improved noise estimation method from a single image based on a noisemodel of a charge coupled device (CCD) sensor is introduced to enhance the feature extraction
performance of the MNF. This method is applied to both airborne and spaceborne hyperspectral
data, acquired from the airborne visible infrared/imaging spectrometer (AVIRIS) and theEO-1/Hyperion, respectively. The experiment for the Hyperion data demonstrates that theproposed MNF is resistant to the spectral distortion of hyperspectral data. Furthermore, the image
classification experiment for the AVIRIS Indian pines data using the MNF as a preprocessing step
to extract spectral features shows that the proposed method extracts higher SNR components inlower MNF components than the existing feature extraction methods.
1. INTRODUCTION
Hyperspectral data contains hundreds of spectral bands collecting spectrograms continuously witha pushbroom imaging system. At each pixel, a continuous spectral profile can be obtained, whichenables discrimination among land-cover classes that are spectrally similar (Kruse et al., 2003).
Before classifying these ground surface objects, spectral feature extraction is commonly
performed, such as the principal component analysis (PCA) and the maximum noise fraction(MNF) transform. PCA generates linear combinations of hyperspectral pixel radiance which aremutually uncorrelated and have maximum variance. Since hyperspectral data always contains
noise, PCA extracts noisy images in low order components due to no consideration of image noise.
MNF transform was developed to produce new components ordered by image quality maximizing
the signal to noise ratio (SNR) rather than maximizing variance (Green et al., 1988). It has beencommonly used for spectral dimensionality reduction of hyperspectral data (Underwood et al.,
2003; Mundt et al., 2005).
Hyperspectral data contains spectral distortion also known as a “smile” property. Smile property
is a wavelength shift in spectral domain expressed as a function of the cross-track pixel number.
Past researches showed that MNF transform suffers from smile property (Goodenough et al.,2003; Dadon et al., 2010). When MNF transform is applied to hyperspectral data with smile
distortion, brightness gradients in the cross-track direction appear in the first component. These
artifacts never represent spectral features of ground surface objects and thus degrade classification
results. Enhancement of smile distortion as a brightness gradient in MNF occurs because noise
estimation in MNF is different from the real noise model. In this work, we propose a new MNF
transform improving noise estimation based on a sensor noise model. Resistance to smile property
and performance of spectral dimensionality reduction before classification are evaluated applying
the proposed method to airborne and spaceborne hyperspectral data.
2. MNF BASED ON A SENSOR NOISE MODEL
2.1 Classic MNF
A hyperspectral profile x at each pixel can be expressed as x = s + n, where s and n represent thesignal and noise part of data, respectively. When both components are assumed to be normally
distributed with respective covariance matrices Ȉs and Ȉn, to have zero mean, and to be
uncorrelated, the covariance of observed data is given by Ȉ = Ȉs + Ȉn. Let us seek a linear
combination y A t x to maximize the SNR. A is the MNF transform matrix defined by A = (a1,a2…an), where ai is the i th order transform vector. The SNR of the new component is given byat ¦ x aSNR
1.
at ¦ n aSetting its vector derivative with respect to a equal to zero, we get the generalized eigenvalue
problem given by¦n a O ¦x a .
Noise estimation is the most important procedure that determines the performance of MNF
transform. The classic MNF transform for remote sensing data analysis supposes that an observed
target at a pixel is almost the same at neighboring pixels, and thus estimates noise as follows:
1n i , j x i , j  (x i 1, j  x i , j 1 ) .
2Here, i and j denote the cross-track and along-track positions, respectively. Many past studies
show the effectiveness of this classic MNF transform for spectral dimensionality reduction of
hyperspectral data (Underwood et al., 2003; Mundt et al., 2005). However, as mentioned in theintroduction, when there is smile distortion at some level, brightness smile gradient appears in thefirst MNF component, as shown in the lower images of Figure 2. This phenomenon degrades theperformance of MNF transform for feature extraction and reduces classification accuracy.

2.2 Noise Estimation Based on a Sensor Noise Model
We adopt a noise estimation method based on a sensor noise model that estimates the noise
dependent on radiance (Liu et al., 2008). A charge coupled device (CCD) imaging sensor iscommonly used for the two-dimensional detector array of satellite hyperspectral sensors. TheCCD imaging sensor produces five primary noise sources, namely, fixed pattern noise, darkcurrent noise, shot noise, amplifier noise and quantization noise (Healy & Kondepudy, 1994).
They can be classified into two noises, dependent and independent on radiance. We focus on
estimating the noise dependent on radiance.
First, we estimate the noise level function (NLF), which is a continuous function describing thenoise level as a function of image brightness. The NLF f NL is defined as the standard deviation
with respect to image intensity. The problem is how to measure the NLF only from observedimages. We estimate the NLFs in all bands by the block based noise level estimation as follows. In
all bands, each image component is divided into blocks of size NxN. Then, for all blocks, the mean(ȝ) and standard deviation (ı) are calculated. Finally, the lower envelope, which is the calculated

Figure 1. Noise level function.
NLF estimation, is fit to the sample set (ȝ,ı), as shown in Figure 1. We adopt the 95 percentile
method, which approximates the lower envelope as the value over which 95 percent ofobservations can be found. After the NLF estimation, the signal part s(z) of the original value x(z)is estimated by bilateral filtering (Tomasi & Manduchi, 1998) with the consideration of NLF,
where z is a position (i, j) for a certain band. Bilateral filtering is given bys(z )k (z ) 1 ³f

f³ x(z) g (ȟ, z) p( x(ȟ), x(z))dȟ ,f fwith the normalizationk (z )ff³ ³ g (ȟ, z) p( x(ȟ), x(z))dȟ .
f fHere, g(ȟ,z) and p(x(ȟ),x(z)) denote the geometric closeness and the photometric similarity,respectively, between the neighborhood center z and a nearby point ȟ. For the closeness functiong(ȟ,z) and the similarity function p(x(ȟ),x(z)), we use Gaussian functions defined as follows:§ 1 § d ([ , z ) · 2 ·§ 1 § G ( x([ ), x( z )) · 2 ·¨¸¸
¸ ¸,g ([ , z ) exp¨  ¨, p([ , z ) exp¨¨  ¨¸¨¸¨¸ ¸2© Vg ¹2©
Vp
¹ ¹
©
¹
©
where d (ȟ, z ) ȟ  z , G ( x(ȟ ), x(z )) x(ȟ )  x(z ) , and ıg and ıp denote the geometric andphotometric spread parameters, respectively. In this work, we set ıg as 1, and ıp as an output of theNLF. Finally, the noise part n(z) of the original value x(z) is given by n(z ) x(z )  s (z ) .
3. RESULTS AND DISCUSSIONSIn this section, the proposed MNF transform is evaluated by the following two points, theresistance to smile property and the spectral dimensionality reduction performance before
classification.
3.1 Resistance for Smile Property
The classic and proposed MNF transforms were applied to the EO-1/Hyperion/VNIR band 8-57
taken over various eight scenes. Comparisons of the MNF 1st component between the classic andproposed MNF transforms are shown in Figure 2. In the classic MNF, brightness gradient in thecross-track direction caused by smile property appeared in the first component for all imageries.
On the other hand, this artifact did not appear in the proposed MNF. This means that the new MNF
transform is little affected by smile property, which contributes to improved classification
mapping for hyperspectral data. Originally, the information of smile distortion should be small inthe entire hyperspectral data information. Therefore, the new MNF shows better performance thanthe classic counterpart as the feature extraction methods which should produce higher SNR
images in lower order components.

Figure 2. Comparison of the MNF 1st component between the classic (lower images) and proposed(upper images) MNF transforms
3.2 Spectral Dimensionality Reduction
The spectral dimensionality reduction performance of the proposed method was examined byhyperspectral data classification experiment comparing the method with PCA, classic MNF, andspectral MNF. Spectral MNF (Liu et al., 2009) is a MNF transform that improves noise estimation
using a neighboring band image correlation for a certain band. We used the AVIRIS data takenover Indian pines (USA) in 1992. This data set was chosen because its ground truth is available.
The data consists of 145x145 pixels with 220 bands. Its true color image and the class map areshown in Figure 3. First, the spectral dimensionality of the data was reduced by feature extraction.
Then, this image was classified using spectral angle mapper (SAM) commonly used as classifier.
All ground truth samples were used for test data and 20% of the data were randomly selected fortraining data. Average spectra of training data were set as the reference spectra. This experiment
was iterated 20 times. Figure 4 shows the comparison between four feature extraction methods
regarding the relationship between classification accuracies and feature component numbers used
for classification. Mapping results, with feature component number b set to 4, 10 and 20, areshown in Figure 5. Classification accuracy of the proposed new MNF outperforms the others
when feature component number is less than twelve. This means that this method extracts higher
quality features of hyperspectral data in lower order components than the others. When featurecomponent number is over twelve, classification accuracy remains at the same level and isreversed by the classic and spectral MNFs. This is probably because noisy images also appear inlower order components. In the noise estimation of the new MNF, there are some parameters,
which affect classification accuracy. Determination of these parameters is a future task.

Figure 3. The true color and class map for the AVIRIS Indian Pine data.

Figure 4. Comparison of relationship between classification accuracies and feature component
numbers used for classification for different four feature extraction methods.

Figure 5. Classification mapping results. Percentage values below images are classification
accuracies.

4. CONCLUSION
In this work, we proposed the new MNF transform based on the sensor noise model. Noiseestimation procedure was improved by estimating image noise dependent on measured radianceusing the NLF. The effectiveness of this method was proved from two points of view. One is theresistance for smile property, which makes the brightness gradient found in the first component of
the classic MNF. When applying the proposed method to the EO-1/Hyperion data, the firstcomponent of the new MNF is not affected by smile property. The other is the performance ofspectral dimensionality reduction as a preprocessing of hyperspectral classification mapping.
From the classification experiment using the AVIRIS data, it turned out that the proposed method
extracts higher SNR components in lower MNF components than the existing feature extraction
methods.
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