Contrast enhancement and noiseelimination using singular valuedecomposition for stereo imaging

Changwoo HaGwanggil JeonJechang Jeong

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Contrast enhancementand noise eliminationusing singular valuedecomposition for stereoimaging

Changwoo Ha,a Gwanggil Jeon,b and Jechang Jeonga

aHanyang University, Department of Electronics andComputer Engineering, 222 Wangsimni-ro, Seongdong-gu,Seoul 133-791, KoreabUniversity of Incheon, Department of Embedded SystemsEngineering, 12-1 Songdo-dong, Yeonsu-gu, Incheon406-772, KoreaE-mail: jjeong@hanyang.ac.kr

Abstract. A new stereoscopic satellite image enhancementmethod based on singular value decomposition (SVD) ispresented. The proposed stereoscopic image enhancementtechnique consists of four steps; SVD based on image ana-lysis, image enhancement, noise reduction, and tone consis-tency. The goal of this work is to enhance low-contrast stereosatellite images with noise elimination, and obtain identicalluminance levels from slightly different luminance in stereoimages. Experimental results show the proposed methodimproves contrast performance while maintaining the lumi-nance continuity. It also exhibits less noise than conventionalmethods. © 2012 Society of Photo-Optical Instrumentation Engineers(SPIE). [DOI: 10.1117/1.OE.51.9.090504]

Subject terms: stereo satellite image; singular value decomposition;contrast enhancement; midway image equalization.

Paper 121040L received Jul. 18, 2012; revised manuscript receivedAug. 19, 2012; accepted for publication Aug. 27, 2012; publishedonline Sep. 18, 2012.

1 IntroductionStereoscopic satellite image techniques have garnered muchattention since innovative observation satellites were put intouse. The view made possible by a stereo satellite has greatlyimproved our ability to analyze the phenomenon of severespace shift.1 However, stereo satellite images frequentlysuffer from low contrast and inconsistent illuminationproblems because existing satellite images are obtainedfrom several space weather satellites or other observationaldevices. Additionally, satellites often do not have the latestoptics because their launches predate the development ofsuch equipment, and they remain in orbit for an average often years. Consequently, post-processing techniques, such asimage enhancement, are the last options for improving theperceptual quality of the content. A pair of images is usedto maintain the tone consistency between each image, whileenables a better three-dimensional effect for interpretability.They are generally used in the gray level and transformdomains.

In this research, we specifically focus on the singularvalue decomposition (SVD) transform. The SVD is an

important factorization technique in linear algebra for rectan-gular matrices and has many useful applications in signalprocessing and statistics; yet, two prominent issues existwhen processing images based on SVD. The first importantissue is image denoising.2–5 The basic concept underlyingnoise reduction is the preservation of important small singu-lar values and the removal of unimportant small singularvalue including noisy components. The second issue isimage contrast enhancement.6–8 The singular value matrixdescribes the illumination information of the image, andcan be changed to reasonable values for contrast enhance-ment. SVD-based applications such as noise reduction andcontrast enhancement have been developed independently;however, such applications to stereo images have not yetbeen proposed.

In this letter, we propose a new approach for adjustablestereoscopic satellite image enhancement based on SVD thatis robust to noise. The rest of this letter is organized asfollows: The SVD algorithm is described briefly in Sec. 2.In Sec. 3, we present the proposed algorithm. Experimentalresults are provided in Sec. 4. Finally, conclusions are givenin Sec. 5.

2 Singular Value DecompositionThe SVD of a rectangular matrix A has many importantproperties and useful applications. Without loss of generality,for everym × n (m ≥ n) matrix A, the SVD can be written as

A ¼ UΣVT ¼Xni¼0

~uiαi~vTi ; (1)

where U¼ð~u1;~u2; : : : ;~umÞ∈Rm×n and V¼ð~v1;~v2; :::;~vnÞ∈Rn×n are two-column orthogonal matrices, and VT denotesthe transpose of V. Moreover, Σ is a diagonal matrix withelements αi, i ¼ 1; 2; : : : ; n. The diagonal elements can bearranged in a nonincreasing order and are called the singularvalues of A:

Σ ¼

264α1

. ..

αn

375¼ diagfα1; α2; : : : ; αng: (2)

The Σ matrix represents the intensity information of agiven image. Since α1 ≥ α2 ≥ : : :≥ αn, the first terms ofthis series will have the largest influence on the total sum,with the last few terms representing insignificant noiseeffects. Thus, we can approximate the matrix A by summingonly a few terms of the series. Note that we can correct for alow-contrast image by using a new α with singular valuesobtained from the adjustable enhanced image.

In addition, histogram equalization (HE) is a widely usedimage contrast enhancement method,9 but it tends to exces-sively enhance the contrast in some images and generateadditional noise. Hence, we introduce a method based onSVD that controls the level of contrast enhancement whilereducing the noise.

3 Proposed MethodThe main concept underlying our approach is to decomposestereoscopic satellite images through SVD, and then calculatethe adjustable image enhancement ratio by controlling the0091-3286/2012/$25.00 © 2012 SPIE

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contrast level. The noise components can be subsequentlyeliminated by the threshold bound through the rank approx-imation. Finally, each enhanced stereo satellite images can beequalized by midway image equalization (MIE). The overallstructure of the proposed method is shown in Fig. 1.

The adjustable singular value with noise reduction can beseen as a solution to a three-criterion optimization problem.The object functions are to find the adjustable singular valueαAI that is close to αAH in the first two terms and to find theoptimal singular value eαk that has a small bin values for noisereduction in the last term. Here, αAI and αAH represent theaverage image between the original images and the averagehistogram equalized images, respectively. Then, the adjusta-ble singular value with a decrease in noise can be formulatedas a weighted sum of the first two objectives with an addi-tional term for noise cancelling such that

eαk ¼ arg minαk

fkαk − αAIkþ λkαk − αAHkþ kDεkαkkg; (3)

where k denotes the view (left or right), and αk, αAI, and αAHsignify the singular values of the kth image, the averageimage between the original images, and the average histo-gram equalized images, respectively. The diagonal matrixDεkðik; ikÞ ¼ 1 for ik ¼ ½0; εk� and the remaining diagonalelements are zero.

An analytical solution of Eq. (3) can be obtained when thesquared sum of the Euclidean norm is used,

eαk ¼ arg minαk

fkαk − αAIk22 þ λkαk − αAHk22 þ kDεkαkk22g;

(4)

which results in a quadratic optimization problem.

eαk ¼ arg minαk

8>><>>:

ðαk − αAIÞTðαk − αAIÞþλðαk − αAHÞTðαk − αAHÞ

þðαkÞTDεkðαkÞ

9>>=>>;: (5)

The solution to this minimization problem of Eq. (5) is

eαk ¼ fð1þ λÞI þDεkg−1ðαAI − λαAHÞ: (6)

The controllable singular value eαk is a weighted averageof αAI and αAH with noise reduction. In the first and the sec-ond terms in Eq. (3), the parameter λ varies over ½0;∞Þ, andthe solution of Eq. (3) gives the optimal trade-off curvebetween the first and second objectives. If λ is zero, eαk isthe singular value that includes the average values of thetwo images, and as λ approaches infinity, eαk approachesthe maximum singular value of the histogram equalizedimage. Consequently, many enhancement levels can beeasily tuned by changing the parameter λ. In fact, the adjus-table contrast enhancement ratio is defined as

Γk ¼supðeαkÞsupðαkÞ

; (7)

where supðαkÞ and supðeαkÞ are maximum singular values ofαk and eαk, respectively.

To reduce the noise, the last term in Eq. (3) introduces alow rank matrix approximation for a noise estimator thateffectively removes the high noise frequency elements.The threshold is

εk ¼ γkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimknkσk

p; (8)

wherem and n are the numbers of rows and columns, respec-tively. The σk ¼ medianðjYHH

k jÞ∕0.6745 is the global noisevariance that can be estimated from the detailed waveletcoefficients (YHH

k ), and γk is an empirical constant that deter-mines the filter strength.

Finally, the enhanced image Ak with noise reduction iscomposed by

Ak ¼ ðUkÞðΣ̄kÞðVkÞT; Σ̄k ¼ Γkαk; (9)

where Σ̄k represents new singular values, which is obtainedfrom Eq. (7) and singular values of input image αk.

We can obtain nearly identical equalized stereo imagesfrom Eq. (9); however, they do not have the exact same his-tograms because U and V contain important information forthe recognition of each image.10 Thus, after employing theproposed stereo satellite equalized method, we apply a tunecontinuity between the equalized images. More specifically,we use MIE, which creates intermediate histogram betweenthe left and right histograms while remaining coherent withtheir previous gray level dynamics.11 The intermediatemidway cumulative histogram (CH) can be constructed asfollows

HMIE ¼�H−1

L þH−1R

2

�−1; (10)

whereHL,HR, andHMIE are two CHs of the original imagesand a midway CH, respectively.

Fig. 1 Flowchart of proposed method. HE ¼; histograph equaliza-tion SVD ¼ singular value decomposition; MIE ¼ midway imageequalization.

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4 Experimental ResultsThis letter presents a novel method for enhancing stereo-scopic satellite image. In this experiment, the enhancementparameter λ was set to 9, and γ was set to 0.08. These valueswere obtained empirically. The experimental results areshown in Fig. 2. Figure 2(a) and 2(b) are the original leftand right images, respectively. Figure 2(c) and 2(d) wereobtained by HE, and Fig. 2(e) and 2(f) were obtained by theproposed method. Figure 2(g) is the histogram of the leftimage in each method, and 2(h) is the CH of the original leftand right images, the HE images, the proposed methodleft and right images, and proposed method with MIE image.

Figure 2(c) and 2(d) shows the equalized results by HE;noise and over-enhancement is clearly evident. On the otherhand, Fig. 2(e) and 2(f) presents images that appear morenatural looking than the others. Figure 2(g) shows that thehistogram of the proposed algorithm has good extensionwhile maintaining the original histogram shape. Figure 2(h)shows that the CH of the proposed method has simultaneoussimilar tendencies between HE and the original CH.

5 ConclusionsAn important issue in stereoscopic satellite image enhance-ment based on SVD is how to control the enhancement inorder to obtain a natural-looking image. This goal is toreduce the noise resulting from its enhancement and lumi-nance continuity for a natural looking, three-dimensionaleffect. Hence, we used an adjustable enhancement, as wellas noise estimation techniques developed via three-criterionoptimization. The proposed method effectively enhanced thesatellite images. The experimental results verified that theproposed algorithm not only reduced noise, but also

enhanced the stereo image contrast. In the future work, wewill study on speedup of SVD and the optimization by adap-tive parameter.

AcknowledgmentsThis research was supported by the MKE (Ministry ofKnowledge Economy), Korea, under the ITRC (InformationTechnology Research Center) support program supervisedby the NIPA (National IT Industry Promotion Agency(NIPA-2012-C1090-1200-0010).

References

1. R. A. Howard et al., “Sun earth connection coronal and heliosphericinvestigation (SECCHI),” Adv. Space Res. 29(12), 2017–2026 (2002).

2. K. Konstantinides and K. Yao, “Statistical analysis of effective singularvalues in matrix rand determination,” IEEE Trans. Acoustics, Speech,Signal Process. 36(5), 757–763 (1988).

3. D. Kalman, “A singularly valuable decomposition: the SVD of amatrix,” College Math J. 27(1), 2–23 (1996).

4. Z. Hou, “Adaptive singular value decomposition in wavelet domain forimage denoising,” Pattern Recogn. 36(8), 1747–1763 (2003).

5. Y. He et al., “Adaptive denoising by singular value decomposition,”IEEE Sign. Process. Lett. 18(4), 215–218 (2011).

6. H. Demirel, G. Anbarjafari, and M. N. S. Jahromi, “Image equalizationbased on singular value decomposition,” in Proc. ISCIS 23rd Int. Symp.,pp. 1–5, IEEE, Istanbul, Turkey (2008).

7. H. Demirel, C. Ozcinar, and G. Anbarjafari, “Satellite image contrastenhancement using discrete wavelet transform and singular valuedecomposition,” IEEE Geosci. Remote Sens. Lett. 7(2), 333–337 (2010).

8. W. Lin, J. Wang, and S. Lin, “Contrast enhancement of x-ray imagebased on singular value selection,” Opt. Eng. 49(4), 047005 (2010).

9. R. C. Gonzalez and R. E. Wood, Digital Image Processing, 3rd ed.,Prentice Hall, Upper Saddle River, NJ (2008).

10. Y. Tian et al., “Do singular values contain adequate information for facerecognition?” Pattern Recogn. 36(3), 649–655 (2003).

11. J. Delon, “Midway image equalization,” J. Math. Imag. Vis. 21(2),119–134 (2004).

Fig. 2 Comparison of stereoscopic satellite images: (a) Original (left); (b) Original (right); (c) histogram equalization (HE, left); (d) HE (right);(e) Proposed (left); (f) Proposed (right); (g) Histograms of left (a), middle (c), and right (e); (h) Cumulative histograms (CH).

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