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An adaptive dynamic range compression with local contrast enhancement algorithm for real-time color image enhancement

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Abstract

Dynamic range compression has become an important function used in modern digital video cameras to improve visual quality of color images suffered from low dynamic range and poor contrast defects. This study addresses real-time implementation of an adaptive dynamic range compression algorithm for color image/video enhancement. To achieve this purpose, we first propose a new image-dependent nonlinear intensity-transfer function to produce a satisfactory dynamic range compression result with less color artifacts. The proposed algorithm is then derived by combining the new adaptive nonlinear intensity-transfer function with an efficient local contrast enhancement algorithm. Moreover, an algorithmic acceleration method is also presented to accelerate the processing speed of the proposed color image enhancement method, achieving real-time performance in processing high-definition video signals. Experimental results validate the performance of the developed method in terms of quantitative evaluation, visual quality, computational efficiency, and power consumption.

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Acknowledgments

This study was supported by the National Science Council of Taiwan, ROC under Grant NSC 101-2221-E-032-022.

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Correspondence to Chi-Yi Tsai.

Appendix

Appendix

In this Appendix, we prove that the variability range of the parameter φ is [0, Φ], where Φ is the solution of the equation \( \varphi (1 + \varepsilon )^{(\varphi - 1)} - S = 0 \) with \( S \in (0,2] \). Moreover, we justify the choice \( \varphi \in [0,\hbox{Min} (1,S)] \subseteq [0,\Upphi ] \) made in our algorithm. Let us compute the variability range of φ. First of all, we note that the normalization factor f n (x,y) defined in (5b) must be a nonzero positive value for each pixel. Hence, when applying the SDRCLCE algorithm to the intensity-transfer function (10), it has to satisfy a necessary condition such that

$$ \gamma_{ \hbox{Max} } (x,y)T_{2} (L_{\text{in}}^{\hbox{Max} } ) + [1 - \gamma_{\hbox{Max} } (x,y)][\alpha T^{\prime}_{2} (L_{\text{in}}^{\hbox{Max} } )L_{\text{in}}^{\hbox{Max} } ] > 0, $$
(22)

which guarantees that the normalization factor \( f_{n}^{{T_{2} }} (x,y) \) defined in (11) is always nonzero positive. Suppose that \( L_{\text{in}}^{\hbox{Max} } = 1 \) (i.e., normalizing the input luminance value to range [0,1]), then we have \( T_{2} (L_{\text{in}}^{\hbox{Max} } ) = 1, \)

$$ \gamma_{\hbox{Max} } (x,y) = \frac{{L_{\text{in}}^{\hbox{Max} } + \varepsilon }}{{\bar{L}_{\text{in}} (x,y) + \varepsilon }} = \frac{1 + \varepsilon }{{\bar{L}_{\text{in}} (x,y) + \varepsilon }},\quad {\text{and}} $$
(23)
$$ T^{\prime}_{2} \, (L_{\text{in}}^{\hbox{Max} } ) = \frac{1}{2}\left\{ {[(1 - \varphi )z + \varphi ](1 + \varepsilon )^{[(1 - \varphi )z + \varphi - 1]} + (2 - z) - S(1 - z)} \right\}, $$
(24)

where \( T^{\prime}_{2} \, (L_{\text{in}}^{\hbox{Max} } ) \ge 0 \) for every \( \varphi \ge 0 \) under the conditions of \( z \in [0,1] \) and \( S \in (0,2]. \) Consequently, the necessary condition (22) becomes

$$ \begin{gathered} \frac{1 + \varepsilon }{{\bar{L}_{\text{in}} (x,y) + \varepsilon }} + \alpha \left[ {1 - \frac{1 + \varepsilon }{{\bar{L}_{\text{in}} (x,y) + \varepsilon }}} \right]T^{\prime}_{2} \, (L_{\text{in}}^{\hbox{Max} } ) \hfill \\ \quad = \frac{1 + \varepsilon }{{\bar{L}_{\text{in}} (x,y) + \varepsilon }} + \frac{{\alpha [\bar{L}_{\text{in}} (x,y) - 1]T^{\prime}_{2} \, (L_{\text{in}}^{\hbox{Max} } )}}{{\bar{L}_{\text{in}} (x,y) + \varepsilon }} > 0. \hfill \\ \end{gathered} $$
(25)

The above expression can further be simplified as

$$ 1 + \varepsilon + \alpha [\bar{L}_{\text{in}} (x,y) - 1]T^{\prime}_{2} \, (L_{\text{in}}^{\hbox{Max} } ) > 0 , $$
(26)

where the value of α can be −1 or 1. For the case of α = −1, the expression (26) becomes

$$ 1 + \varepsilon - [\bar{L}_{\text{in}} (x,y) - 1]T^{\prime}_{2} \, (L_{\text{in}}^{\hbox{Max} } ) = 1 + \varepsilon + [1 - \bar{L}_{\text{in}} (x,y)]T^{\prime}_{2} \, (L_{\text{in}}^{\hbox{Max} } ). $$
(27)

As \( T^{\prime}_{2} \, (L_{\text{in}}^{\hbox{Max} } ) \ge 0 \) and \( \bar{L}_{in} (x,y) \in [0,1] \) due to the condition \( L_{\text{in}}^{\hbox{Max} } = 1 \), it is clear that the condition (26) is always satisfied for every \( \varphi \ge 0 \) when α = −1.

On the other hand, for the case of α = 1, the expression (26) becomes

$$ 1 + \varepsilon + [\bar{L}_{\text{in}} (x,y) - 1]T^{\prime}_{2} \, (L_{\text{in}}^{\hbox{Max} } ) \ge 1 + \varepsilon - T^{\prime}_{2} \, (L_{\text{in}}^{\hbox{Max} } ). $$
(28)

This implies that the condition (26) can be satisfied if we have

$$ 1 + \varepsilon > T^{\prime}_{2} \, (L_{\text{in}}^{\hbox{Max} } ). $$
(29)

Substituting (24) into (29) yields

$$ 2(1 + \varepsilon ) > [(1 - \varphi )z + \varphi ](1 + \varepsilon )^{[(1 - \varphi )z + \varphi - 1]} + (2 - z) - S(1 - z) ,\;{\text{or}}\;{\text{equivalently,}} $$
$$ 2(1 + \varepsilon ) - (2 - z) + S(1 - z) = S + 2\varepsilon + z(1 - S) > [(1 - \varphi )z + \varphi ](1 + \varepsilon )^{[(1 - \varphi )z + \varphi - 1]} . $$
(30)

Now, we have to compute the upper bound of the parameter φ. Let \( \eta = 1 + \varepsilon \) and \( x = (1 - \varphi )z + \varphi = z + (1 - z)\varphi \) with \( z \in [0,1] \) and \( \varphi \ge 0 \). We note that the function \( f(x): = x\eta^{(x - 1)} \) is strictly convex since its second derivative \( f^{\prime\prime}(x) = \eta^{(x - 1)} (2 + x\ln \eta )\ln \eta \) is positive for all \( x \ge 0 \). Then, from the definition of strictly convex functions, the right-hand side of the inequality (30) also satisfies the following inequality

$$ \left. {f(z + (1 - z)\varphi )} \right|_{\eta = 1 + \varepsilon } < z\left. {f(1)} \right|_{\eta = 1 + \varepsilon } + (1 - z)\left. {f(\varphi )} \right|_{\eta = 1 + \varepsilon } = z + (1 - z)\varphi (1 + \varepsilon )^{(\varphi - 1)} . $$
(31)

Next, the left-hand side of (30) can be rewritten as

$$ S + 2\varepsilon + z(1 - S) = z + (1 - z)S + 2\varepsilon . $$
(32)

Observing (30), (31) and (32) finds that if the positive constant S satisfies the condition

$$ S \ge \varphi (1 + \varepsilon )^{(\varphi - 1)} , $$
(33)

then the following relationship is guaranteed

$$ z + (1 - z)S + 2\varepsilon > z + (1 - z)\varphi (1 + \varepsilon )^{(\varphi - 1)} > \left. {f(z + (1 - z)\varphi )} \right|_{\eta = 1 + \varepsilon } . $$

Expression (33) shows that the upper bound of φ is the root of the equation

$$ \varphi (1 + \varepsilon )^{(\varphi - 1)} - S = 0. $$
(34)

For instance, if \( S = 2 \) and \( \varepsilon = {1 \mathord{\left/ {\vphantom {1 {255}}} \right. \kern-0pt} {255}} \), then the root of (34) is 1.9922 (searched by using “fzero” command in Matlab), and the range of φ can be determined as \( \varphi \in [0,1.9922] \). However, this method is too complicated to use in practice as it requires employing a one-dimensional minimization method [39] in the proposed algorithm, increasing the complexity of the implementation of the algorithm. Instead, a simpler way is that if we set \( \varphi \le 1 \), then we have \( 1 \ge \varphi \ge \varphi (1 + \varepsilon )^{(\varphi - 1)} \). Let Φ denote the root of the Eq. (34). In fact, we have \( S \le \Upphi \) for all \( S \in (0,1) \), and \( \Upphi \le S \) for all \( S \in [1,2] \). Based on the above observations, the upper bound of φ thus can be assigned as S for the case of \( S < 1 \) and assigned as 1 for the case of \( S \ge 1 \). This means that the range \( [0,\hbox{Min} (1,S)] \) is guaranteed to be a subset of range \( [0,\Upphi ] \) under the condition of \( \varphi \le 1 \) and \( S \in (0,2] \). By doing so, the process of one-dimensional minimization can be omitted while the necessary condition (22) is always satisfied. This concludes the choice of variability range \( \varphi \in [0,\hbox{Min} (1,S)] \subseteq [0,\Upphi ] \) used in our method.

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Tsai, CY., Huang, CH. An adaptive dynamic range compression with local contrast enhancement algorithm for real-time color image enhancement. J Real-Time Image Proc 10, 255–272 (2015). https://doi.org/10.1007/s11554-012-0299-9

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