experiment, the lower accuracy is most probably caused by imprecise estimation of luminance perceived as white.
5.3.4 Net Lightness
Given the decomposed frameworks and estimated local anchors we compute the net lightness of the pixels by merging the frameworks. We process each framework indi-vidually. We sum the original luminance values of the HDR image normalized by the locally estimated anchor value and proportionally to the probability map:
L(x,y) =30%·
∑
i
(Y−Wi)·Pi(x,y) +70%·(Y−W0), (5.4) where L denotes the final lightness value, Y the original luminance of the HDR im-age, Withe local anchor of framework i, W0 the anchor in the global framework (all these values are in the log10space), and Piis the probability map. The 30% and 70%
coefficients for local and global anchor influence respectively are arbitrarily suggested in [Gilchrist et al. 1999] and can be modified if necessary. In Figure5.3we illustrate how the net lightness has been computed for the sample HDR image. A comparison of the net lightness result to the original HDR image in Figure5.1illustrates an improved perception of image contents in the processed image.
5.4 Model Analysis
The main focus of this chapter is the computational model of the lightness perception theory applied to the tone mapping of HDR images. A thorough verification of the pre-sented model would require a psychophysical experiment which is beyond the scope of this thesis. Instead, we test our computational model by simulating two experiments related to the perception of lightness. The first one analyzes the accuracy of the decom-position into frameworks for natural scenes and the second experiment is a simulation of the Gelb illusion using various lightness mapping algorithms for HDR images (tone mapping operators).
5.4.1 Frameworks within Multi-Illuminant Scenes
According to the anchoring theory of lightness perception, successfully identified frame-works should define the areas in which the lightness is perceived homogeneously [Gilchrist et al. 1999]. The evidence for such lightness perception can be obtained through a distribution of probe disks of constant known luminance value across an im-age. The disks should have the same lightness within a framework independently of the ratio of their luminance to the background luminance on which they are placed.
Such an experiment has recently been presented by Gilchrist and Radonjic [Gilchrist and Radonjic 2005]. In Figure5.4we provide an HDR reproduction of this experiment using an image similar to the original material. We decompose the HDR image into frameworks using our computational model and place several probe disks of constant
(a) probe discs in natural scene (b) probe discs (c) identified frameworks
Figure 5.4: The probe discs of constant luminance (middle) are inserted to a multi-illuminant natural scene (left). The perceived brightness of the probes changes in the context the scene, but is constant within the two identified frameworks (right) and is independent of the local background luminance. The contrast ratios of the probes with the background range in the shadow framework from 1:2 to 1:9 (disc is an increment) and in the light framework from 9:1 to 2:1 (disc is a decrement).
luminance value in various areas of the image. The contrast ratios between the probe disks and the background in both shadow and light frameworks range from 1:2 to 1:9.
The lightness of the probes is perceived consistently within the area of the frameworks independently of the background. These leads to the same conclusions as in the orig-inal experiment: lightness is determined by the frameworks and the influence of local contrasts is minimal. Our contribution here is not to confirm the theory, but to pro-vide an automated method for obtaining an appropriate decomposition. In this sense, the reproduction of the experiment serves as the evidence that frameworks areas are accurately identified using our computational model.
5.4.2 Anchoring in the Gelb Illusion
The Gelb Effect is a well known illusion which provides a good example of lightness constancy failure [Gilchrist et al. 1999]. In the illusion, one observes perceptual dark-ening of a surface despite its constant reflectance and constant illumination. The failure is caused by the appearance of new brighter surfaces in the scene. The illusion can be reproduced in a darkroom with low ambient light using several patches of gray paper with a different reflectance. A single beam of light should first illuminate only the dark-est paper, which will appear to be white. Placing a bit brighter paper beside the existing one causes perceptual darkening of the darkest paper which has initially appeared to be white. Each time a brighter paper is added to the scene, it becomes white and all others immediately become darker. This perceptual illusion can by definition be attributed to the anchoring in general and to the highest luminance rule in particular. It can neither be explained with the contrast theories because the papers do not have to be placed adjacent to each other [Gilchrist et al. 1999], nor with intrinsic image models because the illumination does not change. Furthermore, if the scene occupies only a part of the
5.4. MODEL ANALYSIS 65 visual field, like a tone mapped image observed on a display, the perceptual darkening will not appear because other visible surfaces may serve as a white reference. There-fore it needs to be reproduced during tone mapping to preserve the appearance of the original scene.
(a) scene setup for Gelb illusion (b) decomposed frameworks (c) illumination layer
Figure 5.5: Photograph of the scene in which the Gelb effect can be observed (left).
The middle image shows the decomposition into frameworks obtained from our model – red, green, and blue define the distinct frameworks, yellow marks the shared influence of the red and the green frameworks. The right image shows the illumination layer obtained with the bilateral filtering which is used in the intrinsic images model.
We have performed a study of this experiment to validate the results of our algorithm.
For comparison, we chose two other methods whose principle goal includes the re-production of appearance of the original image. The photographic tone rere-production algorithm presented by Reinhard et al.[Reinhard et al. 2002] is based on a sigmoid function and follows the rule of anchoring to middle-gray. The fast bilateral filtering presented by Durand and Dorsey[Durand and Dorsey 2002] is inspired by the theory of intrinsic images. We will refer to the first one as the middle-gray anchoring and to the latter as the intrinsic images model. In the study, we used four HDR captures of exactly the same scene setup, showing from one to four patches with progressively increasing maximum reflectance. The relative reflectance of the patches was respectively equal to 39%,56%,72%,and 100% with the reference to the brightest one. The area, in which we showed the patches, was illuminated from the top and in our conditions the Gelb il-lusion was reproduced. A photograph of the setup with all the patches visible is shown in Figure5.5.
The results of tone mapping of the four HDR images are shown in Figure5.6 and the respective reproductions of lightness of the patches are plotted in Figure5.7. All tone mapping methods reveal the objects placed outside of the main illumination that are not visible in a standard photograph in Figure5.5. The intrinsic images model maps the lightness of the patches in each of the images to an approximately constant value and maintains the overall brightness of the scene background constant. This is in accordance with the lightness constancy rule, but contrary to what was observed in the real setup. The middle-gray anchoring reproduces the perceptual darkening of the patches, however the brightest one is mapped to white only when all four patches are visible. Further, each brighter patch causes the darkening of scene background which was originally not observed. The lightness perception model presented in this chapter reproduces both the Gelb illusion on the patches and holds the lightness constancy of the objects in the scene background.
middle−gray anchoring
1 patch 2 patches 3 patches 4 patches
lightness percep− tion modelintrinsic images model
Figure 5.6: Simulation of the Gelb Effect by three tone mapping methods. The map-ping of lightness by each of the tone mapmap-ping is plot in Figure 5.7. The intrinsic images model refers to [Durand and Dorsey 2002] operator, the middle-gray anchoring to [Reinhard et al. 2002], and lightness perception model to the operator presented in this chapter.
Analysis
The decomposition of the scene into frameworks (shown in Figure5.5) in the lightness perception model permits the processing of patches and the rest of the scene separately.
The estimation of local anchors using the highest luminance rule estimates the appear-ance of patches in accordappear-ance with the observations in the original conditions. The net lightness calculation with the influence of a global anchor maintains the brightness relation between the frameworks.
The lightness constancy of the intrinsic images model can be explained as follows. In the illumination layer (shown in Figure5.5), obtained by processing the original HDR image with the bilateral filter, the brightness of each patch is approximately equal while the actual differences are in the reflectance layer. The tone mapping reduces the dy-namic range of the illumination layer and overlays the unmodified reflectance layer.
Since the intensities in the illumination layer do not significantly change between the four images, the lightness mapping is constant. Therefore, neither the average lumi-nance rule nor the highest lumilumi-nance rule applied to the illumination layer could repro-duce the Gelb illusion. The application of the highest luminance rule to the reflectance layer or to the final tone mapping result could reproduce the darkening of the patches, however it would also cause the undesired darkening of other image parts.
The middle-gray anchoring reproduces the darkening of the patches because the addi-tion of a new brighter patch causes change in the average luminance of the scene. When the average luminance increases in a new image, the patches, which have constant lu-minance, are mapped to darker gray shades. Unfortunately, such a global connection causes the overall darkening of the scene which is not expected, and the brightest patch is mapped to white only when all four patches are present in the scene.