To cite this version: Ba Chien Thai, Anissa Mokraoui, Basarab Matei. HDR Image Tone Mapping Approach based on Near Optimal Separable Adaptive Lifting Scheme. International Conference on Signal Processing: Algorithms, Architectures, Arrangements, and Applications, Sep 2018, Poznan, Poland. �10.23919/SPA.2018.8563293�. �hal-02428256�

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HDR Image Tone Mapping Approach based on Near Optimal Separable Adaptive Lifting Scheme Ba chien Thai∗ , Anissa Mokraoui∗ and Basarab Matei† ∗ L2TI, † LIPN,

Institut Galil´ee, Universit´e Paris 13 Sorbonne Paris Cit´e 99, Avenue Jean-Baptiste Cl´ement, 93430 Villetaneuse, France {bachien.thai, anissa.mokraoui}@univ-paris13.fr [email protected]

Abstract—This paper proposes a Tone Mapping (TM) approach converting a High Dynamic Range (HDR) image into a Low Dynamic Range (LDR) image while preserving as much information of the HDR image as possible to ensure a good LDR image visual quality. This approach is based on a separable near optimal lifting scheme using an adaptive powerful prediction step. The latter relies on a linear weighted combination depending on the neighboring coefficients extracting then the relevant finest details in the HDR image at each resolution level. Moreover the approximation and detail coefficients are modified according to the entropy of each subband. The pixel’s distribution of the coarse reconstructed LDR image is then adjusted according to a perceptual quantizer with respect to the human visual system using a piecewise linear function. Simulation results provide good results, both in terms of visual quality and TMQI metric, compared to existing competitive TM approaches.

I. I NTRODUCTION The objective of a High Dynamic Range (HDR) image Tone Mapping (TM) approach is to find a trade-off between the relevant information (e.g. details, contrast, brightness...) to be preserved or discarded in the image ensuring a good visual quality of the displayed image on Low Dynamic Range (LDR) devices that would be appreciated by observers. A state of the art on HDR image TM approaches is fairly complete in [1], [2] and [3] where a classification is provided. Among the developed TM strategies, this paper quickly reviews those that caught our attention because of their performance. In [6], the TM approach reduces the HDR contrast while preserving the image details. This work uses an edge-preserving bilateral filter to decompose the HDR image into two layers: a base layer encoding large-scale variations and a detail one. Contrast is then reduced only in the first layer while the details are kept unchanged. An adaptive logarithmic mapping method of luminance values is presented in [7]. It concerns the adjustment of the logarithmic basis depending on the radiance of the pixels. In [8], a subband architecture related on an oversampled Haar pyramid representation is proposed. Subband coefficients are re-scaled according to a gain control function reducing the high frequency magnitudes and boosting low ones. In [9], a TM optimization approach using a histogram adjustment between linear mapping and the equalized histogram mapping is developed. A modification of this approach is made where revisited histogram equalization approaches are discussed in [10]. The latter considers both

histogram equalization and human sensitivity to the light function. In [11], a second generation of wavelets based on the edge content of the image avoiding having pixels from both sides of an edge is proposed. In [12], a separable non-linear multiresolution approach based on essentially non-oscillatory interpolation strategy has been investigated. These approaches take into account the singularities (e.g. edge points) in their mathematical models preserving then the structural information of the HDR images. In [13], a non-separable non-linear multiresolution approach is proposed. The results provided in [11], [12] and [13] show that the decomposition of the HDR image on different resolution levels would seem to be a good strategy. However, the choice of the decomposition filters is extremely important and decisive in the extractive power of the approximation and detail information. To do so, this paper proposes a separable near optimal lifting scheme. The ”Predict” operation is performed by a new adaptive prediction operation to extract the finest detail coefficients to highlight the sharp transitions in the given HDR image. The proposed HDR image TM approach has two main goals namely the preservation of the details and the adjustment of the contrast in accordance with the LDR display devices. It is composed of four stages. The first one judiciously decomposes the HDR image into different resolution levels (section II-A). The second one concerns the weighting strategy of the detail and approximation coefficients (section II-B). The third one reconstructs the coarse LDR image (section II-C). Finally the fourth one adjusts the contrast according to a perceptual linear quantizer (section II-D). Section III discusses the simulation results. Section IV concludes the paper. II. P ROPOSED HDR IMAGE TONE MAPPING APPROACH This section concerns the first, second, third and fourth stages of the proposed HDR image TM mapping approach (i.e. decomposition, weighting, reconstruction and contrast adjustment stages). Before describing the approach, introduce some notations. The original HDR image, at the finest resolution level J, is assumed to be of size N J × M J . The index j refers to the resolution level (with j = 0, ..., J −1). Denote lHDR the HDR image luminance. In the rest of this paper, the HDR image luminance is considered in the logarithm domain since it is

well adapted to the human visual system. It is denoted I J and defined as follows I J = {I J (xn , ym ) = log10 (lHDR (xn , ym )) for 1 ≤ n ≤ N J and 1 ≤ m ≤ M J where I J (xn , ym ) is the HDR logarithm luminance value of the pixel located at position (xn , ym ) on the image. A. First stage: HDR image decomposition

E = arg minkIˆj (xn , y2k−1 ) − I j (xn , y2k−1 )k22 uj−1 i

for 1 ≤ k ≤ M j /2.

(6)

This results in solving the following equation:

The proposed HDR image decomposition is performed according to the forward process of a separable near optimal cell-average lifting scheme. This strategy is motivated by the fact that the relevant details are accurately predicted since the coefficients of the filters are adapted locally to the image to be processed. The decomposition consists to go from the finest resolution level J to the coarse resolution level 0. At a given resolution level j − 1 (with 0 ≤ j ≤ J − 1), the algorithm deals with the approximation coefficients denoted I j (xn , yk ) (with 1 ≤ n ≤ N j and 1 ≤ k ≤ M j ) computed at resolution level j. For a given n belonging to [1, N j ], the algorithm starts with splitting the horizontal 1D-signal, i.e. I j (xn , yk ) for 1 ≤ k ≤ M j , into a set of odd and even indexes as follows: {I j (xn , yk ) with 1 ≤ k ≤ M j } j

Error (MSE) between Iˆj (xn , y2k−1 ) and I j (xn , y2k−1 ) is minimized:

j

Γj−1 · uj−1 = rj−1 ,

(7)

where uj−1 is the weight vector. rj−1 is the crosscorrelation vector rj−1 = (rj−1 (0), rj−1 (1), rj−1 (2))t where rj−1 (i) represents the cross-correlation function between V j−1 (xn , yk+i−1 ) and I j (xn , y2k−1 ) for 1 ≤ k ≤ M j /2. Γj−1 is the autocorrelation matrix defined as: j−1 R (0) Rj−1 (1) Rj−1 (2) Rj−1 (1) , Γj−1 = Rj−1 (−1) Rj−1 (0) (8) j−1 j−1 R (−2) R (−1) Rj−1 (0) where Rj−1 (i) is the autocorrelation of V j−1 (xn , yk ). These weights, associated to the row (xn ), are then deduced so that the partial derivatives of the MSE (given by equation (i.e. i = 0, 1, 2) are equal to zero: (6)) with respect to uj−1 i

:= {I (xn , y2k−1 ), I (xn , y2k ) with 1 ≤ k ≤ M j /2 }. (1)

uj−1 (xn ) = (Γj−1 )−1 · rj−1 .

This process is then repeated for all n. Based on this split, the approximation coefficient located at position (xn , yk ), denoted V j−1 (xn , yk ), is computed on a Cell-Average (CA) scheme. For a given n, it is expressed as follows: I j (xn , y2k−1 ) + I j (xn , y2k ) V j−1 (xn , yk ) = 2 for 1 ≤ k ≤ M j /2. (2)

The weight vectors are then stored to be used in the adaptive lifting scheme backward process to reconstruct the decomposed 1D-signal. For sake of convenience, the approximation and details coefficients are organized as follows:

This process is then repeated for all n. The detail coefficient, denoted dj−1 (xn , y2k−1 ), is computed at odd indexes and is provided below for a given n: dj−1 (xn , y2k−1 ) = Iˆj (xn , y2k−1 ) − I j (xn , y2k−1 ) for 1 ≤ k ≤ M j /2,

(3)

where Iˆj (xn , y2k−1 ) is the predicted logarithm luminance value at odd position (xn , y2k−1 ) and resolution level j. This predicted value is expressed as a linear weighted combination of the neighboring approximation coefficients at resolution level j − 1: P2 Iˆj (xn , y2k−1 ) = i=0 uj−1 (xn ) · V j−1 (xn , yk+i−1 ) i for 1 ≤ k ≤ M j /2,

(4)

where the weights uj−1 (xn ) must preserve the initial 1Di signal average which results in satisfying this condition: P2 j−1 (xn ) = 1. (5) i=0 ui In what follows, the weights are rather denoted in a vector j−1 j−1 t form uj−1 = (uj−1 0 (xn ), u1 (xn ), u2 (xn )) to lighten the writing. These weights are deduced so that the Mean Squared

(9)

{W j (xk , ym ) for 1 ≤ k ≤ N j and ∀m ∈ [1, M j /2]} := {V j−1 (xn , yk ) for 1 ≤ k ≤ M j /2 and ∀n ∈ [1, N j ]}, {U j (xk , ym ) for 1 ≤ k ≤ N j , ∀m ∈ [1, M j /2]} := {dj−1 (xn , y2k−1 ) for 1 ≤ k ≤ M j /2, ∀n ∈ [1, N j ]}. (10) The split, approximation and detail operations are applied on W j (xk , ym ) and U j (xk , ym ) for a given m ∈ [1, M j /2] (i.e. on the vertical direction). Note that the approximation ˆ j (x2k−1 , ym ) (respectively step requires the prediction of W j ˆ U (x2k−1 , ym )) based on a set of weights vij−1 (respectively wij−1 ). These weights need to be stored for the backward process lifting scheme to reconstruct the decomposed image. Finally, the approximation resolution level I j (xn , ym ) j−1 j−1 is divided into 4 blocks I j := (I j−1 , dj−1 HL , dLH , dHH ). The decomposition is thus iterated on I j−1 until j = 0. The finest HDR image I J is then represented by 3J + 1 resolution levels I J := J−1 J−1 J−1 (I 0 , d0HL , d0LH , d0HH , ..., dHL , dLH , dHH ). B. Second stage: Weighting strategy of the coefficients To reduce the dynamic range of the HDR image, the TM approach proposes to weight the approximation and detail coefficients in an appropriate way before performing the adaptive lifting scheme backward process described in section II-C.

Denote Nl the number of resolution levels equal to J; Ea the entropy of the approximation coefficients at the coarsest resolution level (i.e. j = 0); and Edj the entropy of the detail coefficients at resolution level j (i.e. djHL , djLH and djHH ) getting therefore Nl + 1 entropies (Ea , Ed0 , Ed1 , ..., Edj ,..., EdNl −1 ). From these entropies, positive weights smaller than one are deduced as follows: PNl −1 i Ed i=0 αa = PNl −1 i for j = 0 Ea + i=0 E PNl −1 d i (11) = j Ed αdj = Ea + Pi=0,6 for j = 0, ..., N − 1 l Nl −1 i Ea +

i=0

Ed

αa (respectively αdj ) is the weight associated to the approximation (respectively detail) coefficients at resolution level j = 0 (respectively j). The coefficients of the four coarsest resolution levels are first modified according to: 0 0 I 00 = αa × I 0 , d00 HL = αd × dHL , 0 0 d00 LH = αd × dLH ,

0 0 d00 HH = αd × dHH .

(12)

The approximation subband, denoted I 01 , is then reconstructed (see section II-C) and the number of levels is reduced to Nl − 1. The Nl entropies (associated to 3Nl − 2 subbands) are calculated again to update the weights αa , αd1 (equation (11) with Nl = Nl − 1). After that, these weights are applied on the coefficients I 01 , d1HL , d1HL , d1HL to build I 02 as explained above. This process is iterated until Nl = 0 to reconstruct the J , called coarse coarse tone mapped HDR image denoted IeLDR LDR image. C. Third stage: Reconstruction of the coarse LDR image The reconstruction stage is carried out inversely to the decomposition stage. Assume that the adaptive lifting scheme backward algorithm processed all resolution levels until j − 1. The next step consists to recover the approximation coefficients I 0j of size N j × M j using the four weighted blocks 0j−1 0j−1 I 0j−1 , d0j−1 LH , dHL and dHH . The algorithm first deals with j−1 j−1 the coefficients in a vertical direction using I 0 and d0 LH 0j−1 (denoted d ). At a given m, the approximation coefficient at odd position, denoted W 0j (x2k−1 , ym ), is deduced: ˆ 0j (x2k−1 , ym ) − d0j−1 (xk , ym ) W 0j (x2k−1 , ym ) = W for 1 ≤ k ≤ N j /2,

(13)

ˆ 0j (x2k−1 , ym ) is the predicted coefficient, at odd where W position, deduced from the weighted combination vij−1 (with i = 0, 1, 2) of the neighboring approximation coefficients I 0j−1 (x2k−1 , ym ) as: ˆ 0j (x2k−1 , ym ) = P2 v j−1 (ym ) · I 0j−1 (xk+i−1 , ym ) W i=0 i for 1 ≤ k ≤ N j /2. (14) The weights vij−1 (ym ) are those computed and stored in the decomposition process. The approximation coefficient W 0j (x2k , ym ), at even position, is deduced in a CA scheme: W 0j (x2k , ym ) = 2I 0j−1 (xk , ym ) − W 0j (x2k−1 , ym ) for 1 ≤ k ≤ N j /2.

(15)

The odd and even approximation coefficients are then merged to built 1D-signal: {W 0j (xk , ym ) for 1 ≤ k ≤ N j } := {W 0j−1 (x2k−1 , ym ), W 0j−1 (x2k , ym ) for 1 ≤ k ≤ N j /2}. (16) This process is repeated for all m to reconstruct W 0j of size j−1 j−1 N j × M j /2. These same steps are applied on d0 HL and d0 HH to generate the new block U 0j of size N j × M j /2. As in the decomposition strategy, W 0j and U 0j are renamed V 0j−1 and d0j−1 . The same steps, as described above, are then performed but according to a horizontal direction. The reconstruction is iterated to finally build the image called coarse LDR image J which is denoted IeLDR . D. Fourth stage: Piecewise linear perceptual quantizer This stage proposes to adjust locally the distribution of the J coarse LDR image logarithm luminance IeLDR according to the HVS to enhance the contrast using a piecewise linear function. This strategy is inspired from [19] which has been developed for compression purpose. However, modifications are made mainly to avoid the problem of empty bins of equal size. J To do so, the IeLDR values are first sorted and classified into equal B bins defined by cutting points denoted ciuLDR (with 1 ≤ i ≤ B). A non-uniform histogram equalization is also performed. cinuLDR (with 1 ≤ i ≤ B) cutting points, defining the bounds of the non-uniform consecutive B bins, are deduced. The lower bound (cutting point) of each bin is then adjusted as follows: i e lLDR (1) = ciuLDR + β(cinuLDR − ciuLDR ),

(17)

where β is a positive parameter smaller than 1. J values are classified into non-uniform Therefore the IeLDR B bins as follows: = {e lLDR (k) for k = 1, ..., N J × M J } = IeJ LDR

1 1 i i {[e lLDR (1), ..., e lLDR (K1 )], ..., [e lLDR (1), ..., e lLDR (Ki )], ..., (18) B B (KB )]}, (1), ..., e l [e l LDR

LDR

depending on the quantization level set, where Ki is the number of values in the i − th bin (i.e. PB1 ≤ i ≤ B; Ki > 0) and satisfying the following relation i=1 Ki = N J × M J . The ”s-shape” TM perceptual curve, as discussed in [4] and [5], is modelled by a piecewise linear curve on each bin (see Fig. 1). Consider the i − th bin, defined by i i [e lLDR (1), ..., e lLDR (Ki )], the coarse LDR values are then modeled as follows: iei i ˆli LDR (k) = a lLDR (k) + b with k ∈ [1, Ki ],

(19)

where ai (with ai 6= 0) and bi are two unknown parameters depending on the i − th bin. This equation, after some mathematical manipulations, can be rewritten as follows: i ˆli (k) − ˆlLDR (1) ei i e + lLDR (1), (20) lLDR (k) = LDR ai where the unknown parameter ai is deduced so that the MSE between the coarse LDR value and its quantized version, i i denoted lLDR (k) (i.e. when ˆlLDR (k) has been supported a

ceiling process according to a scalar quantization), is minimized in the i − th bin: i i lLDR (k)k22 . arg min klLDR (k) − e

(21)

ai

This equation is simplified as follows: 2 i i PKi lLDR (k)−ˆ lLDR (k) arg min · pi , i k=1 a

(22)

ai

i where pi is the probability of the e lLDR (k) value in the i − th Ki P bin given by pi = B K . i i=1 Extending equation (22) to all bins involves the computation of a Global MSE (GMSE) deduced as follows: 2 i i PB PKi lLDR (k)−ˆ lLDR (k) GM SE = i=1 k=1 · pi . (23) ai i i The variance of (lLDR (k) − ˆlLDR (k)) on each bin is assumed to be equal and is denoted ξ. Equation (23) is then simplified and becomes: PB (24) GM SE = i=1 (apii)2 · ξ.

Denote lLDRmax (respectively lLDRmin ) the maximum (respectively minimum) LDR luminance value. Introduce δ i as the difference between coarse LDR luminance in two consecutive bins: i+1 i δ i = ˜lLDR (1) − ˜lLDR (1). (25) A constraint related to the limit sum of the projected heights equal to the entire LDR range results in: PB i i (26) i=1 a · δ = lLDRmax − lLDRmin . Therefore the optimization problem is written as follows: arg min ai

B X pi ·ξ, i )2 (a i=1

s.t

B X

ai ·δ i = lLDRmax −lLDRmin .

i=1

(27) This problem is solved analytically using the Lagrangian function. After some mathematical manipulations, the slope ai is deduced: (lLDRmax − lLDRmin ) · (pi )1/3 a = . PB i 1/3 i=1 δ · (pi ) i

(28)

Hence the unknown parameter bi is calculated (i.e. bi = i ei ˆli LDR (1)−a × lLDR (1)) and LDR mapped values are deduced according to equation (19). The global piecewise linear curve is continuous and strictly monotonic increasing according to the positive slopes (i.e ai > 0, or angles 0◦ < atan(ai ) < 90◦ ). III. S IMULATION RESULTS This section provides the performance of the proposed tone mapped HDR image. The tone mapped image quality is measured with the TMQI (Tone-Mapped image Quality Index) metric [18]. Simulations have been conducted under Matlab environnement using the HDR Toolbox ( [1]) with 274 test HDR images. For lack of space, we only present the results obtained with 8 HDR images (”Anturium”, ”Bottle Small”, ”Office”, ”Oxford Church”, ”Memorial”, ”Light”,

”WardFlowers” and ”StreetLamp”) with different dynamic range (or contrast ratio) from 8 f-stops to 19 f-stops. The proposed approach is compared to: (i) NONSEP ENOCA [13] ; (ii) SEP ENO-CA [12] with parameters α1 = 0.3, α2 = 0.7; (iii) Li TMO [8] with Haar multiscale; (iv) Fattal [11] using RBW method with parameters α = 0.8, β = 0.3, γ = 0.8; (v) Duan [9] using β = 0.5; (vi) TMOs in HDR Toolbox: Drago [7] , Reinhard [14] , Ward [15] , Durand [6] , Schlick [17] with the default parameters as given in the HDR Toolbox. The different parameters are chosen so as to give the best results in terms of TMQI metric in all methods. Table I provides the TMQI metrics. The proposed TM approach namely ”Proposed LJ” is deployed with B = 256, β = 0.25, lLDRmax = 255, lLDRmin = 0 and J = 1, ..., 5. Our approach is competitive to those developed in the literature. More the number of resolution levels increase, more the performance increase. Fig. 2 compares the visual quality of the ”Church” tone mapped image using ”Duan” method and our approach. The stained glass window at the church background presents a better contrast and details with our approach although the TQMI are identical. Fig. 3 compares the ”Memorial” tone mapped image using ”Duan” and ”Fattal” methods and our approach. The details on tills (see Fig. 3) and rosette (see Fig. 4) are better rendered by our approach. Fig. 5 compares the visual quality of the ”WardFlowers” tone mapped image using ”Fattal” and our approach. Some details, on flowers and rocks, are lost on ”Fattal” tone mapped image compared to our approach. Moreover, our tone mapped image if of better contrast. A similar result is provided by Fig. 6 where the HDR ”StreetLamp” image has been mapped using ”SEP ENO” method and our method. The brightness of our tone mapped is better. The performance of our approach is confirmed on more than 274 test HDR images where the details and contrast are better represented than other competitive methods. IV. C ONCLUSION This paper proposed a new HDR image TM approach able at the same time to extract the relevant details and enhance the contrast of LDR images. This is essentially related to : (i) the forward process of the near optimal local adaptive cell average lifting scheme where the filter coefficients are locally adapted to the content; (ii) the weighting operation depending on the information of each subband; (iii) the adjustment of the coarse LDR image luminance distribution according to the perceptual piecewise linear function. Simulation results confirm the relevance of the proposed approach both in terms of the TMQI metric and the visual quality of the displayed image.

Fig. 3: ”Memorial” HDR test image (18.38 f-stops) - Left image: proposed (5 levels, TMQI=0.951); Middle image: ”Duan” (TMQI=0.935); Right image: ”Fattal” (TMQI=0.927). Fig. 1: Piecewise linear curve modelization (”s-shape” curve).

Fig. 4: LDR luminance ”Rosette” zoom - Left image: proposed (5 levels, TMQI=0.951); Middle image: ”Duan” (TMQI=0.935); Right image: ”Fattal” (TMQI=0.927).

Fig. 2: ”Oxford Church” HDR test image (15.46 f-stops) Up image: proposed (5 levels, TMQI=0.985); Down image: ”Duan” (TMQI=0.986).

Fig. 5: ”WardFlowers” HDR test image (14.01 f-stops) - Up image: proposed (5 levels, TMQI=0.930); Down image: ”Fattal” (TMQI=0.875).

R EFERENCES

Fig. 6: ”StreetLamp” HDR test image (13.83 f-stops) - Up image: proposed (5 levels, TMQI=0.911); Down image: ”SEP ENO” (TMQI=0.855).

TABLE I: Tone Mapped Image Quality Index (TMQI) TMOs DR f-stops Drago [7] Reinhard [14] Ward [15] Durand [6] Tumblin [16] Schlick [17] Duan [9] Fattal [11] Li [8] SEP ENO [12] NONSEP [13] Proposed L1 Proposed L2 Proposed L3 Proposed L4 Proposed L5

Anturium 8.73 0.874 0.778 0.806 0.811 0.715 0.770 0.964 0.889 0.964 0.896 0.938 0.946 0.965 0.978 0.980 0.982

Bottle 16.03 0.801 0.807 0.783 0.892 0.713 0.835 0.916 0.928 0.954 0.934 0.873 0.863 0.882 0.904 0.921 0.933

Office 16.29 0.800 0.826 0.775 0.825 0.735 0.926 0.955 0.943 0.854 0.943 0.935 0.934 0.938 0.945 0.946 0.948

Church 15.46 0.814 0.789 0.817 0.929 0.675 0.970 0.986 0.889 0.877 0.895 0.820 0.954 0.970 0.981 0.984 0.985

Memorial 18.38 0.800 0.791 0.795 0.814 0.759 0.787 0.935 0.927 0.834 0.932 0.832 0.918 0.929 0.941 0.949 0.951

Light 17.46 0.800 0.794 0.789 0.800 0.750 0.780 0.969 0.971 0.888 0.970 0.932 0.954 0.963 0.966 0.969 0.969

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