Dynamic range expansion based on image statistics

Abstract

As the dynamic range of displays keeps increasing, there is a need for reverse tone mapping methods, which aim at expanding the dynamic range of legacy low dynamic range images for viewing on higher dynamic range displays. While a number of strategies have been proposed, most of them are designed for well-exposed input images and are not optimal when dealing with ill-exposed (under- or over-exposed) content. Further, this type of content is more prone to artifacts which may arise when using local methods. In this work, we build on an existing, automatic, global reverse tone mapping operator based on a gamma expansion. We improve this method by providing a new way for automatic parameter calculation from the image statistics. We show that this method yields better results across the whole range of exposures.

Appendices

Appendix: A F-tests for assessing the appropriateness of adding new predictors to a model

An F-test is typically performed to decide whether or not a certain null hypothesis can be rejected. To do this, a test statistic (the F-statistic) is needed which under the null hypothesis follows an F-distribution. In our case, the null hypothesis is that, given two models, A and B, with a number of predictors p _A and p _B (p _A >p _B ), the two models fit equally well the data. The F-statistic is then given by:

$$ F_{p_{A} - p_{B}, n - p_{A}} = \frac{(SS_{B} - SS_{A})/(p_{A} - p_{B}) }{SS_{A} / (n - p_{A})}, $$

(6)

where S S _i , i={A,B}, is the sum of squared residuals of model i, and n is the number of data values [26]. It must be noted that in Equation 6, and throughout the document, p _i as a measure of the number of terms in the regression includes the constant term (i.e. the intercept).

For the particular case of creating model A by adding one variable to a model B that has p terms, and expressing the formula in terms of R ², the F-statistic becomes:

$$ F_{1, n-p-1} = \frac{{R^{2}_{A}} - {R^{2}_{B}}}{(1 - {R^{2}_{A}}) / (n-p-1)} $$

(7)

As it is well known, given a value for F in an F-test, the p-value is the probability of obtaining a value as extreme as the F obtained, assuming that the null hypothesis is true. As a consequence, the null hypothesis is typically rejected if the p-value is lower than the significance level α (which, in this work, has the usual value of α=0.05).

Appendix B: Goodness of fit in multilinear regressions

This appendix includes the description of a series of metrics which are typically used in regression analysis to measure the accuracy of the fitting of a certain model.

For a multilinear regression, RMSE is computed as shown in Equation 8, where Y _i are the observed data (i.e. the given γ values) and $\hat {Y}_{i}$ the data predicted by the model.

$$ RMSE = \sqrt{\sum\limits_{i=1}^{n}(Y_{i} - \hat{Y}_{i})^{2}/(n-p)}, $$

(8)

where, n is the data size and p the number of terms in the regression. Please recall that in this formulation the intercept is included in p. This metric provides an intuition on the error we would incur in when using a certain regression to estimate the value of a variable.

The overall F-statistic is simply an F-test in which the null hypothesis is that the data can be explained by a constant (which would be the mean of the observed data), versus the hypothesis that the data can be explained by the selected model. Therefore, a high F-statistic and, specially, a low associated p-value indicate that the hypothesis that our model explains the data (vs. the hypothesis that a constant explains them) is clearly correct.

Typically used to assess how well the values predicted by a model will adjust to the real values, in the case of linear regressions R ² is simply the square of the correlation coefficient between the observed and the predicted data.

However, in the case of multilinear regression, the R ² value will always increase as new variables are added to the model. For this reason sometimes the adjusted R ² is used, which corrects for the number of explanatory variables in the model. As a result, the adjusted R ² value will only increase if the new term improves the regression more than would be expected by chance. The adjusted R ² value is usually denoted by $\tilde {R}^{2}$ and computed as follows:

$$ \tilde{R}^{2} = 1 - (1 - R^{2})\frac{n - 1}{n - p} $$

(9)

where, again, n is the data size and p the number of terms in the regression. Please recall that in this formulation the intercept is included in p. It is well-known that the higher the R ² and the adjusted R ² values, the higher the correlation between the values predicted by the model and the values actually observed.