Noise-enhanced Contrast Stretching of Dark Imagesin SVD-DWT domain

Rajlaxmi ChouhanIndian Institute of Technology KharagpurKharagpur, West Bengal, India 721 302

Email: rajlaxmi@ece.iitkgp.ernet.in

Rajib Kumar JhaIndian Institute of Technology Patna

Patna, Bihar, India 800 013Email: jharajib@gmail.com

Prabir Kumar BiswasIndian Institute of Technology KharagpurKharagpur, West Bengal, India 721 302

Email: pkb@ece.iitkgp.ernet.in

Abstract—A noise-enhanced contrast stretching algorithm forenhancement of dark images in SVD-DWT domain has been pre-sented in this paper. A dark or low-contrast image is consideredto be comprising a weak signal (information) and noise (dueto insufficient illumination). Since singular values of an imagerepresent luminance of independent image layers, the internalnoise may be considered to be inherent in the singular values ofthe image, as well that of the wavelet approximation coefficientsof a dark image. The singular values of the approximationcoefficients are iteratively processed using the analogy of adouble-well system exhibiting dynamic stochastic resonance (SR),where addition of noise is utilized to enhance system performance.Iteration is terminated with the criterion of target contrast andvisual qualities. Comparison with other SR-based and non-SRbased algorithms shows noteworthy performance of the algorithmin the SVD-DWT domain.

I. INTRODUCTION

OVER the past two decades, research in noise-enhancedimage processing has gained widespread momentum due

to its counterintuitive mechanism and its vast potential forapplication to image enhancement, detection, and analysis.Low or non-uniform illumination in image capturing scenescauses severe problems in object identification and informationprocessing, and such images, therefore, need to be enhancedbefore further analysis. The concept of ‘stochastic resonance’,where noise is utilized to amplify a weak subthreshold signal,has particularly been found to work remarkably in contrastenhancement of dark and low-contrast images [1], [2], [3], [4],[5], [6], [7], [8], [9], [10], [11], [12]. The broad framework ofeach of these works is to add controlled amounts of noise tothe image (values or coefficients) so as to increase its contrastand comprehension.

A large number of techniques have focussed on enhancementof graylevel images in the spatial domain, like adaptive his-togram equalization, gamma correction etc. [13]. Jobson et al.[14] have reported the retinex theory that requires filteringwith multiscale Gaussian kernels and post-processing stagesfor adjusting colors. They have also reported an extension ofthe previously designed single-scale center/surround retinex toa multiscale version that achieves simultaneous dynamic rangecompression, color consistency and lightness rendition [15]. Amodified high-pass filtering has been described in [16], wheresome specific spatial frequencies are selectively magnified byexaggerating the local visibility of an image followed by high-pass filter to adjust those critical frequencies. Since singularvalues of an image hold luminance information on an image,

nonlinear scaling of these values also leads to increase inoverall luminance of the image [9].

The above mentioned enhancement techniques are based onspatial-domain operations. However, DFT and DCT domainsprovide spectral separation, and due to this property it ispossible to enhance features by treating different frequencycomponents differently. A popular technique called alpha-rooting [17] produces an increase in contrast of the imagewhen the magnitude of each transform coefficient is raised to apower a, 0 < a < 1, and the sign or the phase of the coefficientis unchanged. Many algorithms reported in literature havebeen designed for both colored and grayscale images in blockDCT domain [18], [19], [10]. However, there are problemsof blocking artifacts when operating in block neighborhood.Wavelet domain is another promising platform for imageenhancement due to its multiresolutional characteristics. Asthe approximation band consists of illumination content, whilethe other subbands contain edge information, processing ofthis approximation band, may, therefore protect the edgesand details from degradation. Hybrid domains like SVD-DWThave been explored by [20], [21] where the broad mechanismadopted is the equalization of singular values (of image orapproximation band) by a coefficient found by ratio of largestsingular values of the image and an equalized image.

In this paper, we have presented our investigation of noise-induced contrast enhancement using dynamic stochastic res-onance (DSR) in the hybrid SVD-DWT domain. Our earlierwork in application of DSR in SVD, DCT, DWT, and intensitydomains has displayed remarkable enhancement in contrastof dark images, and this paper presents an extension of thework in SVD domain to a hybrid domain of singular values ofwavelet coefficients (SV-DWT). The unique feature of our ap-proach is the use of internal noise, instead of externally addednoise to induce resonance, and the selection of parameters bySignal-to-Noise Ratio (SNR) maximization.

II. STOCHASTIC RESONANCE AND ITS CONNOTATION INIMAGE ENHANCEMENT

Stochastic resonance is a phenomenon in which a system(embedded in a noisy environment) acquires an enhancedsensitivity towards a small external periodic forcing when thenoise intensity reaches some finite level [22]. The underlyingmechanism of stochastic resonance may be understood usingthe double-well model proposed by Benzi et al. [23], [24] asfollows: If a particle is placed in a double-well potential systemhaving two stable states (or wells), and a weak force is exerted

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on the system, the double-well would oscillate asymmetrically.The particle may also oscillate but may not have sufficientforce to transit into another well. If a random noise fluctuationis applied on the particle, it still may not be able to makethe transition. However, at some optimum amount of noiseintensity, the particle may make the transition to the other welldue to constructive cooperation between the weak force andthe noise. This periodic inter-well transition occurs when noiseis tuned to the signal using the Kramer’s equation [25]. If theposition of the particle is considered as output at any instant,the periodic nature of the weak subthreshold signal is amplifiedand exhibited at the output.

The dynamics of the motion of this particle is governed byLangevin equation of motion [25] as follows:

dx(t)

dt= −dU(x)

dx+B sin(ωt) +

√Dξ(t) (1)

where B and ω are respectively, the amplitude and frequencyof the weak periodic signal.

Here, U(x) is a bistable quartic potential given by:

U(x) = −ax2

2+ b

x4

4(2)

Here, a and b are positive bistable double-well parameters.The double-well system is stable at xm = ±

√a/b separated

by a barrier of height ∆U = a2/4b when the ξ(t) is zero.

In context of a dark or low-contrast image, the intensity valuesconstitute a weak signal due to their low excursion about themean. The same logic applies to the wavelet approximationband that is a coarse representation of the image itself. Thesame idea extends down to the singular values of the image orits approximation band, as they represent luminance weights ofeach image layer. Therefore, if stochastic resonance is inducedin the singular values of the approximation band (henceforthto be called as SV-DWT in this paper), each of the SV-DWTvalues may be non-linearly scaled up. Readers are advised torefer to [9], [12] for a exhaustive mathematical formulation ofthe same in SVD and intensity domains respectively.

Eq 1, when discretized for this bistable double-well (as de-scribed in [9], [12]), produces the following iterative equation:

x(n+ 1) = x(n) + ∆t[ax(n)− bx3(n) + Input

](3)

Note that Input = B sin(ωt)+√Dξ(t) denotes the sequence

of input signal + noise. We assume that the noise is dueto degradation arisig from insufficient illumination, and thesignal is the image information. Therefore, both signal andnoise are inseparably and inherently present in the image, andany transformed coefficients or values. Hence, it is safe toassume that here, Input can be the SV-DWT values. ∆t is thesampling time for discretization, and a and b are the double-well parameters as described before. x(0) is a zero vector formathematical convenience.

By differentiation of the SNR equation for dynamic stochasticresonance w.r.t a, the optimum value of a for maximum SNRis found to be a=2σ0

2 (as derived in [9]). Another conditionis needed to ensure that the maximum allowable force onthe bistable well maintains its stability, i.e. the periodic inputsignal is less than or equal to maximum restoring force or

gradient of potential function. This is ensured by the conditionb < 4a3/27.

III. MECHANISM OF CONTRAST STRETCHING BYNOISE-ENHANCED ITERATIONS

From an investigative point of view, the effect of DSR it-erations has been observed on singular values (SVs) of allfour wavelet subbands. The effect of iterative processing onSV-DWT distribution has been shown in Fig. 1. It can beobserved that each of the distributions shifts and gets skewedtowards the larger end. This would mean that the count ofsmaller SVs, that was higher before DSR, has now decreased,while count of larger SVs which were very few has nowincreased. Also, the rate of distribution flattening is muchmore for the approximation coefficients. Since singular valuesdenote weights of image luminance, as the number of largerSVs increase, this can be attributed to increase the overallluminance of the image in question.

Fig. 2 shows zoomed-in regions of image (Fig. 4(b)) obtainedafter 8 iterations on SVs of only the approximation subband,and on that of all subbands. The increase in variance of the SV-Detail coefficients (as shown in Fig. 1) shows that this wouldlead to an increase in high-frequency content. This is visible inFig. 2 as the boundary of objects appears relatively smootherwhen processing is done only on SV-Approximation, than thatobtained for all subbands. With this reason, along with the factthat approximation band contains luminance information, ouralgorithm processes only SV-Approximation.

IV. ALGORITHM FOR NOISE-ENHANCED CONTRASTSTRETCHING

The highlight of the algorithm is the use of internal noiseof a dark image. Non-linear scaling of singular values ofcoarse information shall be performed to produce an increasein overall illumination and contrast of the image.

A. Performance Characterization

The performance of the algorithm is gauged in terms ofmetrics of contrast enhancement and visual quality. Metricof contrast enhancement (F ) is based on global varianceand mean of original and enhanced images [6]. Therefore,a descriptor called image quality index, Q, has been usedsuch that Q = σ2/µ where σ and µ are, respectively, thestandard deviation and mean of the image intensity values.Relative contrast enhancement factor, F , is computed as theratio of values of quality indices post-enhancement, (QB), andpre-enhancement, (QA). For evaluation of perceptual quality,we have used a no-reference metric of image quality, PQM ,that takes into account visible blocking and blurring artifactsreported [26]. According to [19], PQM should be close to 10for good perceptual quality (digression away from 10 on eitherdirection is an indication on decrease in perceptual quality).The code available at their website [26] has been used tocompute PQM . To observe the colorfulness of the enhancedimage, we have used a no-reference metric called colorfulnessmetric, (CM ) [27]. The color enhancement factor (CEF ) hasbeen defined as ratio of colorfulness of output to input image.For good color and contrast enhancement, respective valuesCEF and F should be greater than 1.

(a) Input Distribution (b) n=20 (c) n=40 (d) n=60 (e) n=80

(f) Input Image (g) n=20 (h) n=40 (i) n=60 (j) n=80

Fig. 1: (a) shows distribution of coefficients of a dark input image. (b)- (e) show the probability density function (pdf) of SV-DWTvalues (approximation, horizontal, vertical, diagonal details) of a dark low contrast image after 20, 40, 60 and 80 iterationsrespectively. (f ) shows the input dark low contrast image. (g)-(j) show output image after 20, 40, 60 and 80 iterations respectively(for ∆t = 0.001)

(a) Processing on SVs of all subbands (b) Processing on SVs of only approximation band

Fig. 2: Zoomed-in portions of enhanced output of Flowers after 8 iterations

B. Proposed algorithm

The basic steps of our noise-enhanced contrast stretchingalgorithm are as follows:Step 1 - Color model conversion, followed by DWTdecomposition, and computation of singular values ofapproximation band: The input image is projected into HSVcolor space to process only luminance vector. The value vectoris decomposed into approximation (LL) and detail (HL, LH ,HH) coefficients using 1-level discrete wavelet transform(here, the Haar wavelet). Singular value decomposition of the

approximation (LL) band is done using

LL = U SVLL V T (4)

where U and V are left and right singular matrices.

Step 2 - Computing SR parameters: Assume ∆t = 0.01,a = k × 2σ0

2, b = m × (4a3)/27, where σ0 is the standarddeviation of SVLL.

Here, k is a factor denoting image region dullness (given byinverse of (variance × dynamic range)) and m is a factor

much less than 1 (so that b < 4a3/27).

Step 3 - SR processing of singular values of approximationband: Initialize a zero vector, S, such that S(0)=0. Usingthe bistable dynamic SR parameters tune the SVLL valuesaccording to Eq. 3 as follows:

S(n+ 1) = S(n) + ∆t[aS(n)− bS3(n) + SVLL

](5)

Singular value recomposition is done as before to compute thetuned LL band. Inverse DWT is computed using this tuned LLand original detail bands, and performance metrics, F , PQM ,and CEF are computed for each backprojected image afterevery iteration. Iteration is terminated at count, n0, such thatthe sum of F (n)+CEF (n) becomes maximum in the nearestpossible vicinity of PQM = 10, (say PQM ± 1.5) for thisn=n0.

V. EXPERIMENTAL RESULTS AND DISCUSSION

The enhanced images on some test images obtained afteriterative processing on SV-DWT are shown in Fig. 3-Fig. 5.Fig. 4(c) and 5(a) are naturally dark, while Fig. 3(a) and4(a) have been synthetically made dark for testing purposes.Fig. 3(c) was obtained from the Internet [28] in its currentform.

Table I shows the values of performance metrics for theproposed technique in comparison with that of several othercontrast enhancement techniques. A Mean Opinion Score(MOS) of subjective evaluation of each image by five subjectshas also been included. Since we are already operating in HSVdomain, the empirical values of color enhancement have beentabulated only for observation, since this enhancement is aresult of mapping of the enhanced value vector (in HSV).

Comparative analysis with non-SR-based techniques, like con-trast limited adaptive histogram equalization (CLAHE) [13],gamma correction (Gamma), single-scale retinex (Retinex)[14], multi-scale retinex (MSR) [15], and modified high-passfiltering (MHPF) [16], alpha rooting (AR) [17], multicontrastenhancement (MCE) [18], MCE with dynamic-range compres-sion (MCE-DRC) [29], and color enhancement using scaling(CES) [19] has been performed. A comparison with SVD-DWT based enhancement [21] has also been tabulated. Sincethe proposed technique is an automatic algorithm, a compar-ison has been made with outputs of ‘Auto Contrast’ controlof Adobe Photoshop CS2 (Photoshop). Among SR-basedtechniques, comparison with singular-value-based DSR (SVD-DSR) [9], DCT-DSR [10], DWT-DSR [11], and suprathresholdSR-based technique (SSR) [8] has been done.

A common issue while addressing dark images is the presenceof noise that arises on any attempts of contrast enhancement.Though empirically the contrast metrics may go higher andhigher, but perceptual quality goes on degrading. Note inFig. 5(c), that though maximum F+CEF occur at aroundn=26, the iteration is terminated at n=20 to preserve visualquality through PQM . In this way, a trade-off has to bemaintained between contrast and visual quality. Incrementalrefinement may be varied by changing the value of samplingstep, ∆t.

Empirical values of performance metrics show that the SVD-DWT-DSR technique performs comparable to conventional al-gorithms, and better than many enhancement algorithms whichare unable to address dark images. However, it performs infe-rior only to some of the DSR-based techniques in frequencydomains, and to Photoshop. With respect to MOS ratings, ourSVD-DWT-DSR technique ranks in top four amongst all thecompared techniques for most of the test images. For a visualcomparison among various techniques, readers are advised torefer to [9], [11], [12].

The computation time of the algorithm is guided by twofactors: the iteration count and complexity of transformation.Let the size of an image be M×N , and optimal iteration countby n0. Wavelet decomposition (O(MNlgN)) followed bysingular value decomposition (O(MN2)) is performed once.For each iteration from 1 to n0, the computation for DSR stepis O(N) (since singular values of approximation coefficientsare being processed). The inverse DWT (O(MNlgN)) andsingular value recombination (O(MN2 + M2N)) is alsoper iteration. Therefore, the combined complexity may beestimated as O(MNlgN +MN2) + n0 ·O(N +MNlgN +MN2 +M2N). An average 512× 512 dark grayscale imagetakes about 20 seconds to reach optimal output on an Intel(R)Core(TM)2 Duo Processor working on 2.53 GHz with 2 GBof RAM.

VI. CONCLUSION

A noise-enhanced contrast stretching algorithm in SVD-DWTdomain was discussed in this paper. The algorithm is iterativeand aims to perform non-linear scaling of singular values ofapproximation coefficients of a dark image. Since approxima-tion subband emulates the actual image properties, the singularvalues of this band appropriately represent the luminance ofthe image. An equation, derived from establishing resonancein a double-well system by motion dynamics of a particleunder force, has been used here to induce resonance in the SV-DWT values. With the termination criteria of optimal contrastwhile ascertaining good perceptual quality, a trade-off betweenperceptual quality measure and contrast metric is observed.Subjective scores have been used to present an estimate ofactual visual quality. The algorithm is found to give noteworthyperformance in the hybrid domain in comparison with otherenhancement techniques.

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(a) Low-contrast image, Grass (b) SV-DWT-DSR enhanced image,n=10

(c) Low-contrast image, Sat-Map (d) SV-DWT-DSR enhanced image,n=3

Fig. 3: Enhanced images for dynamic SR processing on SV-DWT values

(a) Low-contrast image, Flowers (b) SV-DWT-DSR enhanced image,n=8

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Fig. 4: Enhanced images for dynamic SR processing on SV-DWT values

(a) Low-contrast image,Chair

(b) SV-DWT-DSR en-hanced image, n=20

(c) Output characteristic for Chair

Fig. 5: Enhanced images for dynamic SR processing on SV-DWT values, and output characteristics.

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TABLE I: Comparative performance of the proposed technique with various existing techniques using three performance metricsF [6], PQM [26], CEF [19], and Mean Opinion Score (MOS)1 for Fig. 4(a) and Fig. 3(a).

Flower (Fig. 4(a)) Grass (Fig. 3(a))

Method PQM F CEF n MOS PQM F CEF n MOS

SVD-DWT-DSR 8.96 2.29 2.31 8 3 6.5 4.48 4.52 10 3.4

Intensity-DSR [12] 10.57 5.18 4.52 5 2.4 8.9 4.61 4.62 6 3

SVD-DSR [9] 9.71 3.08 3.09 18 3 8.89 5.06 4.49 18 3

DWT-DSR [11] 10 3 3.1 50 3.2 8.72 5.28 5.76 95 3.4

DCT-DSR [10] 9.81 2.99 2.72 17 3.8 8.84 2.58 2.57 15 2.8

SSR2[8] 9.61 2.47 3.71 1.4 8.86 2.5 6 1.4

CLAHE [13] 10.49 2.18 1.26 2 8.84 1.98 2.73 2.2

Gamma 10.95 1.22 1.48 2.4 7.92 5.91 5 2.6

Photoshop 10.45 6.7 2.7 4.4 9.12 4.69 4.75 3.4

MHPF [16] 11.55 0.6 0.84 0 9 5.02 7.22 1.8

Retinex [14] 12.37 0.09 0.27 0 7.96 4.78 8.34 1.4

MSR [15] 11.67 0.37 0.72 0.8 7.18 1.68 2.77 1.6

AR [17] 12.06 0.96 1.00 1.2 9.57 0.93 0.99 0.8

MCE [18] 12.2 1.2 0.3 0.8 8.78 1.17 0.96 0

MCE-DRC [29] 11.9 0.7 0.3 1.6 9.27 0.96 0.99 1

CES [19] 11.3 1.5 0.35 2.2 8.32 1.13 1.58 0.2

SVD-DWT [21] 10.37 2.67 3.57 2.8 6.84 5.53 6.72 2.41 Mean opinion score was given by five subjects with the following code: 0 - Very poor, 1 - Poor, 2 - Average, 3 - Good, 4 - Very good, 5 - Excellent.

The MOS (five subjects) for input images was found to be - Flower: 1.2, Grass: 1.4.2 Optimal SSR output at standard deviation, σ0, of added noise: σ0=24 for Flower, σ0=22 for Grass

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