Assessing Noise Amplitude in Remotely Sensed Images Using ... noise amplitude.pdf · Assessing Noise Amplitude in Remotely Sensed Images Using Bit-Plane and Scatterplot Approaches

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 8, AUGUST 2007 2665

Assessing Noise Amplitude in Remotely SensedImages Using Bit-Plane and Scatterplot Approaches

Alessandro Barducci, Member, IEEE, Donatella Guzzi, Paolo Marcoionni, and Ivan Pippi

Abstract—The problem of assessing the noise amplitude af-fecting remotely sensed hyperspectral images and the corre-sponding signal-to-noise ratio is discussed. An original algorithmfor noise estimation, which performs the analysis of image bit-planes in order to assess their randomness, is described. Dif-ferently from more traditional signal-to-noise estimators, whichneed a homogeneous area in the concerned image to isolatenoise contributions, this estimator is almost insensitive to scenetexture, a circumstance that allows the developed method tocarefully assess the noise amplitude of nearly any observed tar-gets. The developed algorithm has been compared with the well-known noise estimator scatterplot method, for which a novelimplementation based on the Hough transform is presented.Hyperspectral and multispectral data cubes collected by the fol-lowing aerospace imagers, MIVIS, VIRS-200, and MOMS-2P onPRIRODA, have been utilized for investigating the performanceof the two considered estimators. Outcomes from processing syn-thetic and natural images are presented and discussed alongthis paper.

Index Terms—Bit-plane analysis, Hough transform, hyperspec-tral remote sensing, image processing, noise amplitude, scatter-plot, signal-to-noise ratio (SNR).

I. INTRODUCTION

D IGITAL images are generally affected by different kindsof noise that can be distinguished in two main classes:

pattern and temporal noise [1]. Periodic noise and “stripe-noise” (“striping”) are two important examples of pattern noise.Striping, the amplitude of which may be as large as 20%–30%of the unperturbed signal [2], has partially deterministic nature,and gives rise to different spatial patterns in each sensor spectralchannel. Random noise (temporal noise) is instead generatedby a fully stochastic process and represents the most importantlimitation to the image signal-to-noise ratio (SNR). The mainsources of random noise are categorized as follows [3]–[5].

1) Photonic noise, sometime termed “shot noise,” resultsfrom the inherent statistical variation in the photon flux,thus it is distributed according to Poisson’s statistics.

2) Thermal noise is the main component of the “darkcurrent,” and arises from statistical variation in the num-ber of electrons thermally generated inside the photosen-sitive area.

Manuscript received February 14, 2006; revised February 7, 2007.The authors are with the Consiglio Nazionale delle Ricerche, Istituto

di Fisica Applicata “Nello Carrara,” 50019 Sesto Fiorentino, Italy (e-mail:[email protected]).

Digital Object Identifier 10.1109/TGRS.2007.897421

3) Read noise (e.g., reset and flicker noise) is inherent to theprocess of converting charge carriers into a voltage signal,and its analog-to-digital conversion.

4) Charge transfer errors take place during carrier transferbetween adjacent detector elements, as a result of chargetrapping by substrate bulk defects [6].

5) Round-off error has for a uniform signal the well-knownstandard deviation 1/(2

√3) Digital Number (DN).

An additional source of systematic disturbance is the “smear-ing effect” [7] (photoelectrons generated during the chargetransfer phase), which is specific of “frame-transfer” devices.

Noise amplitude estimation is a topic relevant to almost anyscience and technology domain, from signal processing to as-trophysical investigation. Careful noise estimation is necessaryfor assessing data quality and SNR, for computing Kalman,matched and Wiener’s filters, for signal classification or detec-tion driven by χ2 optimization, for covariance matrix estima-tion, and so on. Evidently, knowledge of noise characteristicsis relevant to many hyperspectral image classification algo-rithms, as well as to atmospheric correction of remote sensingdata procedures. Recent works unveiled the important effectsof noise for the unmixing of hyperspectral and multispectraldata [8], [9].

A variety of methods have been proposed to model andmeasure random noise [10]–[22] in digital images. Part ofthem relies on the assumption of an additive white noise,other employ homogeneous image areas for measuring thenoise variance [10] by means of statistical estimation. Theproblem with direct estimation methods is that they must besupervised, requiring a priori knowledge about homogeneousregions, and that their noise amplitude estimations are corruptedfrom residual scene texture frequently.

In years past, different approaches have been proposed inorder to overcome these troubles [11]–[21]. The “scatterplot”method [10], [11], [14], [25], [27] computes the noise am-plitude by means of a η − σ scatterplot obtained from localestimates of signal mean η and standard deviation σ using amoving window procedure. Scatterpoints accumulate aroundhorizontal or slant regression lines whose intercept with thescatterplot σ axis (η = 0) represents a robust estimation ofnoise amplitude. This algorithm shows reduced sensitivity toimage texture, although some troubles affecting the scatterplotanalysis are unresolved yet.

The geostatistical algorithm [12] achieves its noise esti-mation using the semivariogram [26] computed on an imagetransect selected in a homogeneous area. Because the semivari-ogram variance is the expectation of the quadratic difference of

0196-2892/$25.00 © 2007 IEEE

2666 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 8, AUGUST 2007

pixels apart of a certain lag-distance, it is strictly related to thesignal spatial covariance, and its limit to a null lag is a soundestimate of the noise variance. The geostatistical algorithmsuffers from the same drawbacks that affect the direct statisticalestimation over homogeneous regions: bias from residual idealsignal texture and needing of user supervision.

Rank et al. describe in [17] a procedure for noise estima-tion relying on image filtering and histogram analysis. Imagefiltering aims at removing texture, while preserving high spa-tial frequency contributions originated by noise. This methodrequires the noise to obey the normal statistics, be stationary,and independent of the ideal signal. The main shortcomingof this procedure is that its noise estimation is biased fromtargets edges revealed by the employed filters, while high-passfiltering may dim noise spectral density at moderate-to-mediumspatial frequencies. The same noise estimation approach hasbeen reported in other papers [16], the main difference beingrelated to the use of the Laplacian filter. In [28], the idealcomponent of the image signal is suppressed after local surfacefitting, then noise is assessed using direct statistical estimation.All these algorithms are built around an initial processingphase in which the causal component of the signal is rejected,without substantial modification of its casual part. For thisreason, they are sometimes referred to as “data masking”algorithms [17].

More recently [29], new noise estimators based on entropymodeling have been developed with application to medical 3-Dimage processing. The entropy rate of multispectral and hyper-spectral images has been also investigated in a previous paper[11], where scatterplot and bit-plane oriented noise estimationwas also discussed.

Bit-plane analysis is a novel approach to noise estimationin which the entire sequence of image bit-planes is exploitedin order to assess their randomness. In our previous works[10], [11], the algorithm has been outlined neglecting manymathematical details, nonetheless its performance was analyzedin depth. It resulted that the algorithm is almost insensitive tothe ideal image component, while having a modest accuracy inassessing the noise standard deviation. In this paper, we presentan updated version of the bit-plane noise estimation algorithm,which overcomes the low-accuracy drawback pointed out pre-viously. The improved algorithm is now capable of estimatingthe noise amplitude with good accuracy, remaining insensitiveto image texture. We compare the performance of this algorithmwith that provided by the scatterplot procedure, for which wepropose a new implementation.

This paper is organized as follows. Theoretical descriptionof the bit-plane algorithm is given in Section II. Section IIIdiscusses the main properties of the “scatterplot” method andthe solutions adopted by us in order to retrieve the noiseamplitude. Section IV contains the outcomes from applying thetwo algorithms to test images corrupted by additive white noise,and to hyperspectral data cubes collected by various aerospaceimagers such as the Multispectral Infrared/Visible ImagingSpectrometer (MIVIS), Visible/Infrared Spectrometer (VIRS-200), and Modular Optoelectronic Multispectral Scanner onPRIRODA (MOMS-2P). Conclusions and open problems areoutlined in Section V.

II. BIT-PLANE METHOD FOR NOISE

AMPLITUDE ESTIMATION

A. Overview of the Original Bit-Plane Algorithm

The bit-plane algorithm [10], [11] is almost insensitive toscene texture, and its noise estimates are independent of the ac-tual spectral properties of noise statistics (noise amplitude andinterband correlation are allowed to change with wavelength).The algorithm is based on the assumption that noise is additiveand spatially stationary, with vanishing spatial autocovariance(quasi-mean-ergodic noise). The basic idea is that bit-planeswith amplitude less than or comparable with noise standarddeviation will be dominated by noise, and they should look likea characteristics salt–pepper 0–1 distribution. This property hasin our belief a general validity as discussed in the following.

Let us suppose that the input signal u(t)(0 ≤ u(t) ≤ umax)has to be digitized with Nbit bits of accuracy, and that theanalog-to-digital-converter (ADC) operation can be given asNbit successive comparisons, starting from the most significantbit (MSB) down to the least significant one (LSB). Let tk anduk be the generic threshold and the residual input signal for thekth bit bk ∈ {0, 1}, 1 ≤ k ≤ Nbit. The bit bk is computed bygauging the value of the residual signal uk with respect to thethreshold tk, as shown in the following relationships.

t1 =umax

2tk =

tk−1

2=

umax

2k

b1 ={

1, u1(t) ≥ t10, otherwise

bk ={

1, uk(t) ≥ tk0, otherwise

u1(t) = u(t) uk(t) = uk−1(t) − bk−1tk−1. (1)

If the kth threshold tk is less than or comparable with thenoise standard deviation, the comparison between uk and tkhas a casual outcome (bk is random noise) and all the remainingLSBs would behave in the same manner.

A simulated noise-free image digitized with eight bit ofaccuracy, together with its bit-plane sequence, is shown inFig. 1: as shown, none of its eight bit-planes is corruptedby noise. Fig. 2 shows the effect of adding normal randomnoise (4.0 DN standard deviation) to the image of Fig. 1. It isclear that the noise corrupts the LSB-planes leaving unchangedthe MSBs, hence confirming the aforementioned hypothesis.In view of this behavior, the problem of noise estimation indigital images can be reduced to assessing the randomness of itsbit-planes.

Let g(x, y) be a Nr rows by Nc columns image digitized withNbit bits of accuracy, and let g(x, y, k) ∈ {0, 1} indicate its kthbit-plane LSB first (LSBF). The algorithm aims at estimatingthe bit-plane statistics of the following absolute differences∆i(x, y, k):

∆1(x, y, k) = |g(x + 1, y, k) − g(x, y, k)|∆2(x, y, k) = |g(x, y + 1, k) − g(x, y, k)|∆3(x, y, k) = |g(x + 1, y + 1, k) − g(x, y, k)|∆4(x, y, k) = |g(x − 1, y, k) − g(x, y, k)| . (2)

BARDUCCI et al.: ASSESSING NOISE AMPLITUDE IN REMOTELY SENSED IMAGES 2667

Fig. 1. Noise-free test image (1025 × 1025) coded at eight bits per sample,and its bit-planes sequence. The fractal-background of this image has beencreated by using the recursive diamond–square algorithm, as derived from thewell-known midpoint displacement procedure [23]. As can be seen, all the bitplanes are strongly spatially correlated.

Fig. 2. Test image (1025 × 1025) coded at eight bits per sample with super-imposed a Gaussian random noise having null mean, and standard deviation of4 DN. The fractal-background of this image has been created by using arecursive generation technique based on the diamond–square algorithm [23].In this case, the spatial correlation of the less significant bit-planes vanishes.

It can easily be shown that the remaining four spatial di-rections not considered in (2) are redundant for statisticalcalculations extended to the entire bit-plane. Let p[q] be theprobability to find a bit g(x, y, k) in the upper (lower) state “1”[“0”]; the probabilities P∆i=0, P∆i=1 for the single difference∆i(x, y, k) to be 0 or 1 are written as

P∆i=0 = P (∆ = |0 − 0|) + P (∆ = |1 − 1|)= q2 + (1 − q)2

P∆i=1 = P (∆ = |1 − 0|) + P (∆ = |0 − 1|)= 2q(1 − q). (3)

These expressions mean that if the concerned bit-planeg(x, y, k) is a random field with equiprobable states, the twopossible values of the difference ∆i(x, y, k) are equally proba-

ble too, and the Bernoulli random variable ∆i(x, y, k) obeys thebinomial distribution with 1/2 mean. It can also be shown thatif the ADC introduces a systematic bias (i.e., the probability qof the failure event is perturbed by a small amount ε 1/2 sothat q = (1/2) + ε), the random variable ∆i(x, y, k) is almostunaffected, and the probabilities P∆i=0 and P∆i=1 are 1/2 inthe less of a second-order power of ε. When the concerned bit-plane is not noisy, g(x, y, k) autocorrelation will grow due toscene texture as in Fig. 1, and the average value of process∆i(x, y, k) will fall below 1/2. In conclusion, the differenceδ(k) = |1/2 − 〈∆i(x, y, k)〉x,y,i| between the expected 1/2 (fora noisy bit-plane) and the actual 〈∆i(x, y, k)〉x,y,i mean shouldrepresent a reliable randomness index for the kth bit-plane.

In order to use this index for bit-plane randomness assess-ment it is necessary to account for its statistical variability,using for instance the local central limit theorem [30]. Let n0

be the number of “0” (failure) occurrences in the kth bit-planefor the process ∆i(x, y, k) ∀x, y, i, the randomness index δ(k)can be written as

δ(k) =∣∣∣∣4qNrNc − n0

4NrNc

∣∣∣∣=

∣∣∣∣q − n0

4NrNc

∣∣∣∣=

∣∣∣∣12 − 〈(∆i(x, y, k))=0〉x,y,i

∣∣∣∣ . (4)

As long as the number of computed differences 4NrNc

is high, and assuming that differences ∆i(x, y, k) computedin different image locations (x, y) or directions (i) are in-dependent each other, the quantity δ(k) is a null mean ran-dom variable that obeys the normal statistics. According toMoivre–Laplace central limit theorem σδ, the expected stan-dard deviation of δ(k), is given by

σδ =

√q(1 − q)4NrNc

=(NrNc)−1/2

4. (5)

The original bit-plane algorithm computes the actual valueof variable δ(k) and compares it with the theoretical standarddeviation expected for a noisy bit-plane. The bit-plane is labeledas “noisy” if δ(k) is less than the threshold nσδ , n being auser-defined multiplier that sets the confidence of algorithmpredictions. Let us note that accepting as noisy the valuesδ(k) ≤ 3σδ leads to assess noisy bit-planes with roughly only1% probability of missed hits (noisy bit-planes classified astextured ones). Calculation of false alarm probability (texturedbit-planes classified as noisy ones) is really complex, as itsprediction requires strong assumptions concerning the texturedbit-planes difference ∆i(x, y, k) distribution. In our experience,the presence in the analyzed bit-plane of even a small texturedarea heavily affects the index δ(k), which becomes far abovethe selected threshold (e.g., 10σδ in our algorithm implemen-tation). Hence, the two possible cases of a noisy or a texturedbit-plane always originate different δ values, without significantsuperposition of the two corresponding statistics. This propertygives rise to an ideal case of perfect two-state classificationwith an almost diagonal confusion matrix. Based on this highly


confident bit-plane classification scheme, the original bit-planealgorithm estimated the digital noise standard deviation σ̃n as

σ̃n = σ0 × 2k(noise). (6)

k(noise) is the index of the more significant bit-plane beingrecognized as noisy, and σ0 an empirically retrieved constantfactor [10], [11]. The main drawback of the original bit-planeprocedure is its inability to provide careful estimations ofnoise amplitude, since noise estimate only changes with amultiplicative factor of two. This was a severe limitation tothe applicability of the algorithm, which on the other handallows unprecedented and reliable assessment of bit-planerandomness.

B. Bit-Plane New Formulation

Suppose we have an input noise level σn for which theoriginal algorithm gave a perfect estimate σ̃n = σn, and letus consider a case in which the input noise is augmented ofa small amount δσn. It is clear that the tiny input noise increaseis insufficient to change the bit-plane classification (the actualk(noise) index), hence the original algorithm erroneously es-timates exactly the same noise standard deviation. However,the little noise increase δσn will change the randomness indexδ(k) of bit-planes more significant than k(noise). Usually,nonrandom bit-planes have higher δ(k), which decreases withincreasing σn. This kind of general behavior has been verifiedafter processing several synthetic images affected by randomnoise of different amplitudes as shown in Section IV. Therefore,more accurate noise estimation can be obtained using the entireset of randomness indexes for various ks, rather than a simplelabeling output as in (6). The new bit-plane algorithm adoptsthe following empirically derived noise estimator:

σ̃n = σ0 ×

k<k(noise)+m∑k=k(noise)

(2k − 1) × (1 − 2δ(k))4

m. (7)

In this relationship, 2k − 1 is the digital signal amplitudeequivalent to bit-plane k, while the term 1 − 2δ(k) is a measureof randomness equals to 1 for a random bit-plane, and nullfor a textured one. σ0 is an empirical multiplier that in ouralgorithm was set to 0.8, and m is the number of bit-planeshigher than k(noise) involved in noise estimation (m = 4 in ourimplementation). Bit-planes less significant than k(noise) havenot been included in the average of (7), since all them bringa fixed contribution (δ(k) = 0 ∀k < k(noise)) and wouldmake the estimate less sensitive to the actual noise amplitude.

III. SCATTERPLOT METHOD FOR NOISE

AMPLITUDE ESTIMATION

A. Scatterplot Algorithm Overview

In the well-known “scatterplot” approach [10], [11], [14],[25], [27], the spatially stationary noise affecting the imageg(x, y) is found as the σ-intercept of a regression line drawnon the scatterplot of local standard deviation σg(x, y) and localmean ηg(x, y). First- and second-order statistical moments are

computed on a window sliding the input image independentlyfor any available spectral channels. Homogeneous areas willproduce clusters aligned along a horizontal regression linewhose intercept yields the estimate of noise standard deviation.On the contrary, the presence of textured signal has the effectof spreading out and rotate the clusters in the scatterplot images(σg(x, y), ηg(x, y)). This last effect is evident for an ideal caseof perfectly periodic image texture p(x, y), where image signalg(x, y) is modeled as weighted superposition of noise n(x, y)and texture p(x, y), g(x, y) = A · p(x, y) + n(x, y), A being aluminance factor. When statistical moments are estimated bylocal spatial averages, we have

σ2g(x, y) =A2 · σ2

p(x, y) + σ2n

ηg(x, y) =A · ηp(x, y). (8)

σ2p being the local average of texture power, and ηp(x, y) its

not null local mean. The σg to ηg relationship is therefore

σ2g(x, y) =

σ2p(x, y)

η2p(x, y)

· η2g(x, y) + σ2

n. (9)

When the size of the moving window adopted for localstatistics estimation is large enough, all terms on the right-handside of (9) are constant with exception of η2

g that may changewith position, producing therefore slant or curved clusters.The standard scatterplot procedure consists of the followingsteps [14], [25]:

1) generation of the scatterplot image;2) the scatterplot image is Hough transformed [24], [27] in

order to assess the orientation (ρM , ϑM ) of the principalscatterplot clusters;

3) the previous scatterplot peak designs a regression lineγ in the form σ = αη + β whose intercept is the noiseestimation of this algorithm.

This method has been successfully employed for measuringadditive noise in digital images, as well as the multiplicativespeckle noise in synthetic aperture radar data [14]. In ourimplementation of this algorithm, we have introduced littlechanges in the last two steps, those connected with Houghtransform analysis and noise estimation. We noted in fact thatnot always the maximum (ρM , ϑM ) of the Hough transformoriginates the best regression line and the optimal noise esti-mation. In our belief, this phenomenon is related to the actualspread of the scatterplot main cluster, a characteristics that ismeasured by the width of the maximum peak in the Houghtransform. In order to mitigate this shortcoming, we have takenas best noise estimation the intercept of the straight line definedby the photocenter (ρC , ϑC) of the Hough transform highestpeak. Photocenter has been computed integrating the Houghtransform signal around its maximum up to a distance ∆, whichis one of the two free parameters of our implementation (theother being the size of the sliding window used for estimatingmoments of local statistics). Sometimes this noise estimationhappens to be negative, a circumstance that forced us to look foralternative estimations. The second highest peak (ρM2, ϑM2)in the Hough transform also provides a reliable noise estima-tion, while as last-choice estimation, we considered the stan-dard deviation of interception point between the straight lines


Fig. 3. Plot of retrieved standard deviation values. On the x-axis expected input noise is indicated, and the y-axis reports the noise standard deviation as estimatedwith different methods: (a) new bit-plane algorithm and (b) our implementation of the scatterplot method. Estimated noise has been fitted with a regression linein order to establish how noise amplitude estimates differ from true values. (c) Curve of randomness index δ(k) versus input noise standard deviation for threedifferent bit-planes. (d) Logarithm of the δ(k) inverse plotted versus the input noise standard deviation. (e) Plot of logarithm of the δ(k) inverse as a function ofthe ratio between input noise standard deviation and signal amplitude for the concerned bit-plane.

(ρM , ϑM ), and (ρM2, ϑM2) corresponding to the highest twomaxima in the Hough domain. To recap, we build up a prioritylist with four independent noise estimations, and we acceptas correct the first positive estimate on it. Finally, we assumethat the difference between the two possible noise estimatesobtained with peak photocenter (ρC , ϑC) and maximum coor-dinates (ρM , ϑM ) is a measure of noise estimation uncertainty.

IV. DATA PROCESSING

A. Test Images

In order to validate the performance of the new bit-planealgorithm and compare it with the outcomes of a state-of-art

noise estimation method (the scatterplot), synthetic and naturalimages were employed. Synthetic images have been generatedusing different textures, and then they were corrupted addingGaussian white noise of increasing standard deviation. Fig. 1shows an example of synthetic image not degraded by noiseyet. Standard deviation of the added random term ranged from0.2 DN until 30 DN, for eight-bit synthetic images of roughly200 different signal levels. Using synthetic images offers theoption to precisely control the noise amplitude held in theprocessed data, hence providing an ideal test-bed for inves-tigating algorithm performance. The starting ideal image isnoise-free (see Fig. 1), apart from a small noise contributionembedded in the adopted background, and neglecting the pixel


Fig. 4. Plot of estimated SNR of a synthetic image (Fig. 1) versus noiseamplitude as computed by two algorithms. As shown, the agreement betweenthe outcomes from the concerned methods improves as the noise amplitudebecomes dominant.

round-off error, characterized by a standard deviation of1/(2

√3) DN typical of a uniform probability density function.

Fig. 2 shows the same image of Fig. 1 after the superpositionof an additive white-noise term (normal distribution) havingstandard deviation equal to 4 DN. The results of the new bit-plane algorithm, together with those obtained by our imple-mentation of the scatterplot method, are shown in Figs. 3 and4. These figures show the noise standard deviation retrieved(y-axis) after processing the synthetic image of Fig. 1 degradedwith increasing levels of input noise (x-axis). The comparisonbetween the original and the new bit-plane noise estimationsis shown in Fig. 3(a), which depicts the significant estima-tion accuracy improvement achieved by the new algorithm.The new bit-plane algorithm performs at the same level asour implementation of the scatterplot method, when theseprocedures are applied to synthetic images. Fig. 3(b) showsthe noise estimations provided by the scatterplot algorithmfor the same input image data (Fig. 1). We point out thatwhile the new bit-plane method is completely autonomous,the scatterplot algorithm requires the setting of two input freeparameters, which significantly affect the actual noise esti-mation. Specifically, scatterplot noise estimations reported inFigs. 3(b) and 4 have been obtained (best fitted with respect toknown input noise amplitude) using a 5 × 5 sliding window,and averaging the main peak in the Hough domain over a30 × 30 patch. Due to this circumstance, the bit-plane pro-cedure would have wider and more stable application thanscatterplot estimate. Let us note that the new bit-plane algo-rithm provides on average a slightly better estimate than thescatterplot method, as can be deduced by the coefficients of theregression line in Fig. 3(a) and (b). These figures show a goodmatch for both the considered methods between the expectedand retrieved values, considering that the angular coefficient ofthe linear fitting is generally a few percent smaller than unit.The bit-plane method yields an offset value (∼0.44), whichtakes into account the round-off error and a partially randomtexture introduced with the cloudy background.

The dependence of the randomness index δ(k) [see (4)] onthe standard deviation of the input noise is shown in Fig. 3(c),(d), and (e). Fig. 3(c) shows that the (δ(k), σn) curve has a

Fig. 5. Images of San Rossore (Pisa, Italy) test site. (a) At-sensor radianceimage (755 × 2000) acquired by MIVIS on June 21, 2000 in the sixth spectralchannel (540 nm). (b) At-sensor radiance image (512 × 2000) acquired byVIRS-200 on June 21, 2000 in the eighth spectral channel (553.75 nm).

quasi-Gaussian shape, an appearance only partially confirmedby successive Fig. 3(d). Here, the logarithm of the δ(k) inverseis plotted versus the input noise standard deviation, revealing analmost rational (polynomial) link between these two quantities.In any case, high sensitivity of δ(k) value on the actual inputnoise level is confirmed for any bit-planes (k). Fig. 3(e) reportsthe − log(δ(k)) value as a function of the σn to (2k − 1) ratio,showing that different bit-planes originate similar curves. Thisproves that the δ(k) depends on the ratio between input noisestandard deviation σn and the signal amplitude (2k − 1) forthe concerned bit-plane, confirming the optimality of the newbit-plane noise estimator of (7). Fig. 4 shows the SNR plotsversus the imposed noise amplitude as computed by using thetwo methods. It is clear from this figure that the agreement be-tween the two outcomes becomes closer as the noise amplitudeaugments.

B. Remotely Sensed Data

As shown in Fig. 5(a) and (b), we considered imagesacquired by means of two airborne hyperspectral sensors:the MIVIS and the VIRS-200. An additional image col-lected by the spaceborne MOMS-2P was also processed, andthe related results (omitted) confirmed those obtained af-ter processing the MIVIS and VIRS images. Configurationfacts concerning the above imaging spectrometers are detailedin Table I.

Fig. 6 shows the entire sequence of bit-planes extracted fromone spectral channel of the MIVIS acquisition of Fig. 5(a).As shown noise affects the LSB-planes, while scene texture


TABLE ISPECTRAL CHANNELS AND WAVELENGTHS FOR THE AIRBORNE SPECTROMETERS MIVIS AND VIRS-200 AND THE SATELLITE SENSOR MOMS-2P

(information) dominates the MSB-planes. Images gathered bythe MOMS, and the VIRS-200 are not shown here, since theyexhibits exactly the same distribution of noise on their bit-planesequence. This behavior holds true for raw data, but theprocessing used for calculating remote sensing products mightaffect it. Multiplying an image by a floating-point constant(not a whole number) augments the number of available bit-planes, mixing texture and noise contributions in the sup-plementary LSBs. Fortunately, this kind of data processingcannot change the data SNR, and the MSB-planes maintainthe natural distribution of noise effects: noise mainly affectsthe LSBs of the top Nbit(raw) bit-planes composing theprocessed data.

In Fig. 7(a), the scatterplot computed from the VIRS-200image acquired in the fourth spectral channel is displayed. Asshown, the cluster is spread out and slanted due to the scenetexture (signal component having nonzero variance). Fig. 7(b)shows the Hough transform of the image of Fig. 7(a). Thetransform is arranged so that the slope ϑ is aligned alongthe horizontal axis, while the distance ρ from the referenceorigin (located in the upper left corner of the scatterplot por-trait) is reported on the vertical direction. In Fig. 8(a) and(b), the SNR spectrum as computed by the two consideredalgorithms is shown for the two sensors as far considered.The exceptionally low SNR revealed by the new bit-planealgorithm in the third MIVIS spectrometer [see Fig. 8(b)]appears reasonable when compared with the correspondingimages.

We point out that noise estimations originated by the two dis-cussed algorithms when processing natural images fairly agreeeach other, with the new bit-plane method originating better

results. This property has been confirmed after processing anadditional MIVIS image of the Tuscany coast (Italy) gatheredon December 25, 2005. This MIVIS image contains a largesea region and is completed with a dark signal image, whichgave us the opportunity to estimate the noise level with a directmethod. Fig. 9 shows the MIVIS image, together with its dark-signal counterpart [Fig. 9(b)]. The size of the sliding windowof the scatterplot was set to 3 × 3, with photocenter averagingbounded to an 11 × 11 region. Let us note that the scatterplotmethod has been unable to get meaningful noise estimationin a limited number of spectral channels, mainly located inthe thermal infrared (TIR) spectral interval. These wrong es-timates (negative standard deviation) have been dropped fromsubsequent analysis. We also point out that at the epoch ofimage acquisition the MIVIS was needing maintenance, sohardware revision and radiometric calibration were executedimmediately after this flight. Due to this circumstance, thenoise level revealed in this campaign is suboptimal with respectto standard MIVIS performance. We have also dropped datafrom the second and third spectrometer that showed an unre-alistic spectral pattern of noise revealed by all the estimationalgorithms. Fig. 10 shows the noise estimates in the visibleand near-infrared spectral ranges for the Tuscany coast sceneand the dark-signal acquisition. As shown, the direct methodproduces different estimates for the two datasets. Reasons forhaving different noise standard deviations between the twoacquisitions are as follows:

1) the lack of photonic noise in the dark-signal;2) possible detector temperature drift between the two

acquisitions;


Fig. 6. Sequence of bit-planes of the MIVIS image of Fig. 5(a). Let us notethat bit-planes #06 and #07 are noisy, and hold less information than lowerrank bit-planes #02 until #05. This kind of inversion in the sequence of noisybit-planes is related with the radiometric calibration of data, a task which isaccomplished subtracting the dark-signal contribution and scaling the resultby a radiometric calibration factor (gain). The combined effects of these twooperations is to add extra LSB-planes with unpredictable noise and texturecontributions, which may show this kind of inversion. However, the MSB-planes maintain the original sequence of noisy bit-planes, where noise onlyaffects the LSB-planes. This point is commented in Section IV-B.

3) the necessity to subtract the dark-signal average from anydaylight acquisition in order to compute radiometricallycalibrated remote sensing products;

4) a change in high-frequency electromagnetic interferencesaffecting the sensor.

Notwithstanding this, the direct estimate applied to the di-urnal image seems to be biased by residual sea texture, partic-ularly in the 500- to 700-nm interval. The scatterplot insteadappears underestimate the true noise level for both the consid-ered datasets, a circumstance confirmed by the noise retrievalin the TIR shown in Fig. 10(b). We remark that the new bit-plane algorithm gives rise to noise estimation better than thosegenerated by the scatterplot. This conclusion is supported byexperimental evidence shown in Figs. 3(a) and (b), 4, 8(b), andFigs. 10(a) and (b). Our experiment indicates a tendency of thebit plane procedure to overestimate the noise in dark scenes of amultiplicative factor between 2 and 4 (see Fig. 10). We interpretthis phenomenon as due to the lack of the ideal component ofthe signal, which let the noise extend its effects to higher bit-

Fig. 7. (a) Scatterplot (256 × 256) computed from the VIRS-200 imageacquired in the fourth spectral channel (located at 443.75 nm). Image localmean value ηg(x, y) (horizontal axis) and local standard deviation σg(x, y)(vertical axis) were normalized in the range from 0 to 255. This circumstancerestricts the number of states in the scatterplot, allowing the extraction from itof statistically significant estimates. The image is displayed using an invertedlookup table: higher values of the transform correspond to darker imageregions. As shown, the cluster displayed is spread out due to the scene texture(signal component having nonzero variance). (b) Hough transform of the imageshown in (a). The image is displayed using an inverted lookup table: highervalues of the transform correspond to darker image regions. Each columncorresponds to a fixed value of the slope ϑ (0 to 1000), while image rowsindicate different distance ρ (0 to 181) of the regression straight line from thereference origin.

planes. To explain this effect we consider a dataset g(x, y, k)LSBF where only one bit-plane k0 is nontrivial (g(x, y, k) = 0∀k �= k0), and the additive constant A =

∑k=k0+ik=k0

2k − 1, ibeing an arbitrary factor giving the number of not-null bitsin A. It is easy to show that adding the constant A to g(x, y, k)originates a new image p(x, y, k) where the original bit-planeg(x, y, k0) is shifted to bit k0 + i + 1

p(x, y, k) = g(x, y, k) + A =

g(x, y, k0), k0≤k≤k0+ig(x, y, k0), k=k0+i+10, otherwise.

(10)

The upper line on the right-hand side of (10) indicates thenegated (complementary) bit-plane. It is worth noting that theadded constant in this elementary example is able to shiftthe original bit-plane to the higher digits. As long as the bit-plane g(x, y, k0) is a random distribution, the image p(x, y, k)also contains noisy bit-planes ∀k|k0 ≤ k ≤ k0 + i + 1. In adifferent wording, adding such a constant to a pure noise imageapparently amplifies the noise amplitude, whose effect reaches


Fig. 8. SNR versus wavelength for hyperspectral images collected by(a) VIRS-200 and (b) MIVIS. SNR estimates obtained by the new bit-planealgorithm have been compared with outcomes from our implementation of thescatterplot procedure.

Fig. 9. (a) At-sensor radiance image (755 × 2000) acquired by MIVIS onDecember 15, 2005 in the sixth spectral channel (540 nm). (b) Dark-signalimage (755 × 402) collected with the MIVIS foreoptics occluded. The imageis displayed in the first spectral channel (440 nm).

the more significant bit-planes. This phenomenon explainswhy the bit-plane algorithm overestimates the noise standarddeviation in dark-signal acquisitions. Understanding why thesame behavior would not take place for daylight standardacquisitions is a bit more complex. We outline this property

Fig. 10. Plot of noise standard deviation versus wavelength for two MIVISspectrometers (visible and TIR) as computed from the hyperspectral imageof Fig. 10(a). The new bit-plane has been compared with direct statisticalestimation and with our implementation of the scatterplot algorithm. (a) Noiseamplitude in the visible and near-infrared range (from 400–900 nm). (b) Noiseamplitude in the TIR interval (from 8000–12 500 nm).

with a simple example using the decimal numerical system.Let us consider the additive constant A = 1099 and a signalaffected by random noise having a standard deviation σ = 2.When the causal part of the signal is absent the hundreds digitin the summation A + σ is casual, assuming the value zeroor one depending on the current noise realization. However,if a causal texture is present, with values for instance in therange from 10 up to 100, the noise will no longer be able torandomly change the hundreds digit, remaining bounded to thelower digit.

We have verified that adding a constant to a random im-age changes the noise amplitude estimated by the new bit-plane algorithm, while adding the same constant to a naturalimage having not null texture does not change the estimatednoise. We conclude that the new bit-plane algorithm is notsuitable for noise estimation in dark-signal images, a task thatshould be performed by means of a direct statistical estimationprocedure.

V. CONCLUSION

The problem of noise estimation in presence of additive,spatially stationary, and weakly correlated input random noisehas been reexamined. An improved version of the bit-planealgorithm has been described, and its performance has been


compared with that of the original procedure showing a strongimprovement of noise estimation accuracy. Comparison ofthe new algorithm with our implementation of the scatterplotmethod has shown good agreement and similar levels of esti-mation accuracy, although the new bit-plane algorithm got onaverage the more reliable estimate.

Investigation of performance of the two algorithms, as farconsidered, has been carried out utilizing both synthetic andnatural images. To this purpose some hyperspectral imagesacquired by three different aerospace sensors have been se-lected. Both noise estimation methods have been comparedwith the outcome of direct statistical estimation over uniformregions (i.e., sea). It resulted that the bit-plane estimate closelyapproaches the direct one, while the scatterplot method of-ten underestimate the noise and is sometime unable to getmeaningful values. Evidence exists that the new bit-plane al-gorithm significantly overestimates the input noise amplitudeof dark scenes, due to the absence of any image deterministicpattern.

The new bit-plane algorithm takes advantage of its totalautonomy, not needing any input parameter to be specified bythe user. For these reasons, the new bit-plane method appearsas one of the most promising methods for autonomous noiseestimation. The revised bit-plane algorithm can be used forboth single-band, as well as hyperspectral images, and allowsthe user to get reliable noise estimations independently ofthe spectral distribution of noise (e.g., algorithm accuracy isnot degraded from spectral correlation of noise). Due to itsindependence from image texture and to its accuracy the newbit-plane algorithm can be of help for applications that requirescareful noise estimation or covariance matrix determination;i.e., procedures for virtual dimensionality assessment, dimen-sionality reduction, classification, unmixing, data compression,and feature extraction [31], [32].

ACKNOWLEDGMENT

The authors would like to thank Dr. S. Baronti (Istitutodi Fisica Applicata “Nello Carrara”-Consiglio Nazionale delleRicerche) for his critical reading of the manuscript draft, whichbrought them many invaluable suggestions. The authors wouldalso like to thank the three anonymous reviewers for theircompetence and constructive criticism.

REFERENCES

[1] J. Nieke, M. Solbrig, and A. Neumann, “Noise contributions forimaging spectrometers,” Appl. Opt., vol. 38, no. 24, pp. 5191–5194,Aug. 1999.

[2] A. Barducci and I. Pippi, “Analysis and rejection of systematic distur-bances in hyperspectral remotely sensed images of the Earth,” Appl. Opt.,vol. 40, no. 9, pp. 1464–1477, Mar. 2001.

[3] K. Watson, “Processing remote sensing images using the 2-D FFT—Noisereduction and other applications,”Geophysics, vol. 58, no. 6, pp. 835–852,Jun. 1993.

[4] R. G. Sellar and G. D. Boreman, “Comparison of relative signal-to-noiseratios of different classes of imaging spectrometer,” Appl. Opt., vol. 44,no. 9, pp. 1614–1624, Mar. 2005.

[5] M. D. Nelson, J. F. Johnson, and T. S. Lomhein, “General noise processesin hybrid infrared focal planes arrays,” Opt. Eng., vol. 30, no. 11,pp. 1682–1699, Nov. 1991.

[6] A. Mohsen, T. McGill, and C. Mead, “Charge transfer in charge-coupleddevices,” in Proc. IEEE Int. Solid-State Circuits Conf. Dig. Tech. Papers,1972, vol. XV, pp. 248–249.

[7] W. Ruyten, “Smear correction for frame transfer charge-coupled-device cameras,” Opt. Lett., vol. 24, no. 13, pp. 878–880,Jul. 1999.

[8] J. M. P. Nascimiento and J. M. B. Dias, “Vertex component analysis: Afast algorithm to unmix hyperspectral data,” IEEE Trans. Geosci. RemoteSens., vol. 43, no. 4, pp. 898–910, Apr. 2005.

[9] A. Barducci and A. Mecocci, “Theoretical and experimental assess-ment of noise effects on least-squares spectral unmixing of hyper-spectral images,” Opt. Eng., vol. 44, no. 8, pp. 087 008.1–087 008.17,Aug. 2005.

[10] B. Aiazzi, L. Alparone, A. Barducci, S. Baronti, and I. Pippi, “Estimatingnoise and information of multispectral imagery,” Opt. Eng., vol. 41, no. 3,pp. 656–668, Mar. 2002.

[11] B. Aiazzi, L. Alparone, A. Barducci, S. Baronti, and I. Pippi,“Information-theoretic assessment of sampled hyperspectral imagers,”IEEE Trans. Geosci. Remote Sens., vol. 39, no. 7, pp. 1447–1458,Jul. 2001.

[12] P. J. Curran and J. L. Dungan, “Estimation of signal-to-noise: A newprocedure applied to AVIRIS data,” IEEE Trans. Geosci. Remote Sens.,vol. 27, no. 5, pp. 620–628, Sep. 1989.

[13] S. I. Olsen, “Estimation of noise in images: An evaluation,” CVGIP,Graph. Models Image Process., vol. 55, no. 4, pp. 319–323,Jul. 1993.

[14] J.-S. Lee and K. Hoppel, “Noise modelling and estimation of remotelysensed images,” in Proc. Int. 12th Can. Symp. IGARSS, Jul. 10–14, 1989,vol. 2, pp. 1005–1008.

[15] K.-S. Chuang and H. K. Huang, “Assessment of noise in digital imageusing the join-count statistics and the Moran test,” Phys. Med. Biol.,vol. 37, no. 2, pp. 357–369, Feb. 1992.

[16] B. R. Corner, R. M. Narayanan, and S. E. Reichenbach, “Noise estimationin remote sensing imagery using data masking,” Int. J. Remote Sens.,vol. 24, no. 4, pp. 689–702, Feb. 2003.

[17] K. Rank, M. Lendl, and R. Unbehauen, “Estimation of image noise vari-ance,” Proc. Inst. Elect. Eng.—Vision, Image Signal Process., vol. 146,no. 2, pp. 80–84, Apr. 1999.

[18] A. Amer and E. Dubois, “Fast and reliable structure-oriented video noiseestimation,” IEEE Trans. Circuits Syst. Video Technol., vol. 15, no. 1,pp. 113–118, Jan. 2005.

[19] R. Bracho and A. C. Sanderson, “Segmentation of images based on in-tensity gradient information,” in Proc. IEEE Comput. Soc. Conf. Comput.Vis. Pattern Recog., San Francisco, CA, 1985, pp. 341–347.

[20] J. Immerkaer, “Fast noise variance estimation,” Comput. Vis. Image Un-derst., vol. 64, no. 2, pp. 300–302, Sep. 1996.

[21] P. Meer, J.-M. Jolion, and A. Rosenfeld, “A fast parallel algorithm forblind estimation of noise variance,” IEEE Trans. Pattern Anal. Mach.Intell., vol. 12, no. 2, pp. 216–223, Feb. 1990.

[22] J. P. Véran and J. R. Wright, “Compression software for astronomicalimages,” in Proc. Astron. Data Anal. Softw. and Syst. III, ASP Conf.Series, D. R. Crabtree, R. J. Hanisch, and J. Barnes, Eds., 1994, vol. 61,p. 519.

[23] G. S. P. Miller, “The definition and rendering of terrain maps,” in Proc.SIGGRAPH Conf., 1986, vol. 20, pp. 39–48. No. 4.

[24] A. Barducci and I. Pippi, “Object recognition by edge analysis: A casestudy,” Opt. Eng., vol. 38, no. 2, pp. 284–294, Feb. 1999.

[25] B. Aiazzi, L. Alparone, and S. Baronti, “Reliably estimating the specklenoise from SAR data,” in Proc. IEEE Int. Geosci. and Remote Sens.Symp., 1999, pp. 1546–1548.

[26] P. J. Curran, “The semivariogram in remote sensing: An introduction,”Remote Sens. Environ., vol. 24, no. 3, pp. 493–507, 1988.

[27] W. K. Pratt, Digital Image Processing. New York: Wiley, 1978.[28] P. J. Besl and R. C. Jain, “Segmentation through variable-order surface

fitting,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 10, no. 2, pp. 167–192, Mar. 1988.

[29] M. Liévin, F. Luthon, and E. Keeve, “Entropic estimation of noise forMedical Volume Restoration,” in Proc. ICPR, Quebec City, QC, Canada,Aug. 11–15, 2002, pp. 871–874.

[30] B. V. Gnedenko, Teoria Della Probabilità. Roma, Italy: Edizioni Riuniti,1979.

[31] C.-I Chang and Q. Du, “Estimation of number of spectrally distinct signalsources in hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens.,vol. 42, no. 3, pp. 608–619, Mar. 2004.

[32] C.-I Chang and Q. Du, “Interference and noise-adjusted principal com-ponents analysis,” IEEE Trans. Geosci. Remote Sens., vol. 37, no. 5,pp. 2387–2396, Sep. 1999.


Alessandro Barducci (M’96) received the Laureadegree in physics from the University of Florence,Florence, Italy, in 1989.

From 1990 to 1992, he was Postgraduate Fel-low at the Research Institute on ElectromagneticWaves “IROE-CNR.” From April 1993 to April1994, he was Researcher at the Centro di EccellenzaOptronica, and from April to September 1994, hewas Fellow of the Département d’Astrophysique,Université de Nice-Sophia Antipolis, France. Since1994, he has been Consultant for high-tech industries

and the Istituto di Fisica Applicata “Nello Carrara”-Consiglio Nazionale delleRicerche (CNR), Florence (former Istituto Ricerca Onde Eletttromagnetiche(IROE)-CNR); since 1997, he has also been Assistant Professor on the En-gineering Faculty of the University of Siena, Siena, Italy. His main researchinterests include hyperspectral remote sensing, inverse modeling of remotelysensed data, hyperspectral interferometric imagers, atmospheric corrections,sensor characterization, spectral unmixing, digital image processing, and bidi-rectional reflectance distribution functions.

Prof. Barducci is a member of the IEEE Society for Geoscience and RemoteSensing, of the International Society for Optical Engineering, and of the SocietàItaliana di Fisica (Italian Physical Society).

Donatella Guzzi was born in Florence, Italy, onMay 2, 1963. She received the Laurea degree inphysics from the University of Florence, Firenze,Italy, in 1990.

Since 1991, she has been working for four yearsat the Istituto Ricerca Onde Eletttromagnetiche(IROE)-Consiglio Nazionale delle Ricerche (CNR)in the field of fiber optic sensors for environmentalmonitoring. Since 1995, she has been woking on thescattering of light by particulates in the atmosphereand in the analysis of lidar data at IROE-CNR with

the LIDAR group of the institute. Her activities have regarded analysis of thescattering properties of the atmospheric aerosols and clouds and their charac-terization. Since January 2001, she has been working and collaborating withthe group “Aerospace high-resolution optical sensor” at CNR-Istituto di FisicaApplicata “Nello Carrara,” where her main activities are the following: theimplementation and development of algorithms for atmospheric correction ofremote sensed data, the study of the propagation of radiation in the atmosphere,the development and calibration of aerospace high-resolution optical sensors,and the validation of remote sensed data by means of in-field measurements.Currently, she is working with “Dipartimento di Colture Arboree—BolognaUniversity,” Bologna, Italy, in the integration of remote sensed data withmodeling and in situ canopy reflectance measurements for the carbon balanceestimates in vegetation.

Paolo Marcoionni was born in Prato, Italy, in 1973.He received the Laurea degree in physics from theUniversity of Florence, Florence, Italy, in 1999, andthe Ph.D. degree in earth science from the Universityof Parma, Parma, Italy, in 2006.

Since 2006, he has been with Integrated ColorLine srl, Italy, where he is involved in the develop-ment of robots for industrial automation and qualitycontrol spectrophotometric systems. He collaborateswith the Istituto di Fisica Applicata “Nello Carrara,”Florence, where he participates in several research

projects devoted to high-resolution remote sensing by aerospace imaging spec-trometers. His research interests include hyperspectral remote sensing, inversemodeling of remotely sensed data, digital image processing, high-resolutioninterferometric imaging, and sensor characterization.

Ivan Pippi was born in Florence, Italy, in 1949. Hereceived the Diploma in electronics from TechnicalHigh School, Florence, in 1968.

From 1969 to 1970, he was with the Departmentof Physics of the University of Florence. Since 1970,he has been with Consiglio Nazionale delle Ricerchefirst dealing with astrophysics research. Then, since1976, he has been dealing with remote sensing tech-niques. His research interest in remote sensing wasfirst focused on laser-radar development for mete-orological studies and Earth observation. Then, he

started studying the applications to the environment monitoring of aerospaceoptical sensors operating in the visible and infrared wavelengths. He has beenparticipating in developing and characterizing several imaging spectrometersand interferometers, and in their data calibration and validation through remotesensing campaigns performed on equipped test sites. Since 1986, he hasbeen the leader of the research group on “high resolution aerospace opticalsensors” at the Istituto di Fisica Applicata “Nello Carrara,” managing severalnational and international research projects mainly supported by the Italian andEuropean Space Agencies.

Documents

Assessing Noise Amplitude in Remotely Sensed Images Using ... noise amplitude.pdf · Assessing Noise Amplitude in Remotely Sensed Images Using Bit-Plane and Scatterplot Approaches