Tone Reproduction: A Perspective from Luminance-Driven Perceptual Grouping

Tone Reproduction: A Perspective from Luminance-Driven Perceptual Grouping

HWANN-TZONG CHEN

Institute of Information Science, Academia Sinica, Nankang, Taipei 115, Taiwan;

Department of CSIE, National Taiwan University, Taipei 106, Taiwan

[email protected]

TYNG-LUH LIU

Institute of Information Science, Academia Sinica, Nankang, Taipei 115, Taiwan

[email protected]

CHIOU-SHANN FUH

Department of CSIE, National Taiwan University, Taipei 106, Taiwan

[email protected]

Received June 16, 2004; Revised May 24, 2005; Accepted June 29, 2005 First online version published in January, 2006

Abstract. We address the tone reproduction problem by integrating the local adaptation effect with the consistency in global contrast impression. Many previous works on tone reproduction have focused on investigating the local adaptation mechanism of human eyes to compress high-dynamic-range (HDR) luminance into a displayable range.

Nevertheless, while the realization of local adaptation properties is not theoretically defined, exaggerating such effects often leads to unnatural visual impression of global contrast. We propose to perceptually decompose the luminance into a small number of regions that sequentially encode the overall impression of an HDR image. A piecewise tone mapping can then be constructed to region-wise perform HDR compressions, using local mappings constrained by the estimated global perception. Indeed, in our approach, the region information is used not only to practically approximate the local properties of luminance, but more importantly to retain the global impression.

Besides, it is worth mentioning that the proposed algorithm is efficient, and mostly does not require excessive parameter fine-tuning. Our experimental results and comparisons indicate that the described framework gives a good balance in both preserving local details and maintaining global perceptual impression of HDR scenes.

1. Introduction

The attempt to reproduce the visual perception of the real world is at the heart of painting and photography.

Artists have long been endeavoring to develop skills in simulating actual reflected light within the limitation of the medium, since our world generally delivers a much wider range of luminance than pigments can reflect. Apart from artistic concern, recreating real-

scene impressions on limited media is also inevitable in many vision and graphics applications (see Tumblin and Rushmeier,1993for a more detailed discussion).

For example, the contrast of a today’s fine LCD monitor could be around 1,000 to 1, but we may need to use it to display an indoor scene comprising very bright windows that results in a contrast of 250,000 to 1.

By a tone reproduction problem, we are interested in establishing a method to satisfactorily reconstruct the

high dynamic range (HDR) radiance on a low dynamic range (LDR) image. The dynamic range of a digital image typically refers to the contrast ratio between its brightest and darkest parts, though a more compre- hensive definition should include issues such as noise sensitivity and possible nonlinear distortion in encod- ing the pixel values. In Ward (1991), Ward proposes a floating-point picture format to record HDR radiance in 32 bits per pixel, and designs a graphics rendering system to output images in the format. Debevec and Malik (1997) have shown that HDR radiance of real scenes may also be captured using regular digital cam- eras. They describe a technique to combine a series of pictures taken with different exposure settings into a single HDR image, called the radiance map. In this context, the focus of our work can be stated as solving the tone reproduction problem of radiance maps, that is, generating a displayable standard RGB image that preserves visually significant properties of the original HDR radiance map.

1.1. Related Work

Several previous works have been devoted to producing HDR images of real scenes. The method by Debevec and Malik (1997) requires multiple photos of a scene taken under different exposures. To simultaneously capture luminance under multiple exposures, Nayar and Mitsunaga (2000) develop a camera, including a mask with cells of different optical transparencies. The grid mask provides different exposures in neighboring pixels, which are averaged to output an HDR image.

Naturally, panoramic imaging also involves high dy- namic range. The process of collecting a broader range of luminance at each location can be coupled with the process of constructing a wider field of view by adding spatially varying optical filters to a camera (Aggarwal and Ahuja,2001; Schechner and Nayar,2003). Yet, on a different focus, Kang et al. (2003) propose to gener- ate HDR videos by varying the exposure of each frame and then by stitching consecutive frames.

Concerning displaying HDR images with good per- ceptual fidelity, tone reproduction addresses visibil- ity and impression by finding an appropriate map- ping to compress the contrast into a displayable range.

For the convenience of our discussion, we categorize tone-mapping functions of related work into global mappings and local mappings, according to whether spatial information is incorporated in their respective formulations.

1.1.1. Global Tone Mapping. Global tone mappings are spatially uniform functions. They uniformly com- press the luminance values of pixels into a displayable range, regardless of pixels’ spatial or local properties.

The great advantage of using a global mapping is its efficiency Tumblin and Rushmeier,1993. Since global mappings are often conveniently designed as mono- tonically increasing, these functions therefore preserve the order of pixel luminance values. Such a property gives another advantage of using global mappings—

for avoiding halo artifacts, which are undesirable in displaying HDR images. Halos are due to reversed contrasts in a neighborhood, i.e., an area originally darker than its neighbors becomes brighter after tone mapping, or vice versa. However, strictly and glob- ally preserving the order of luminance is not always necessary (though sufficient) for avoiding halos, and may not be preferable in some circumstances. Every so often we may want to emphasize local contrasts to display more details, even at the risk of re-arranging the global order of luminance, as long as no perceivable ar- tifacts are produced. Ward et al. (1997) describe a more sophisticated approach to globally adjust contrast with respect to luminance histograms. Nevertheless, their method still smooths out image details in areas of flat histograms.

1.1.2. Local Tone Mapping. To better preserve im- portant visual features, such as image details, of high contrast scenes, several tone reproduction methods have exploited local (or spatially nonuniform) map- pings, as human visual system operates more likely this way. Particularly, human visual cells are organized in a center-surround manner so that we can see a broad range of luminance values by discriminating locally (Palmer,1999). Simulating the local adaptation effect is a common tactic to keep image details for tone map- ping. Its basic idea is to attenuate high contrast by computing locally-averaged luminance values for ef- fectively adjusting the tone-mapping parameters, e.g., the scale factor (Chiu et al.,1993; Tumblin et al.,1999), or the luminance gain control (Pattanaik et al.,1998).

Since the term adaptation luminance has already been used in global tone-mapping methods (Ferwerda et al., 1996; Tumblin and Rushmeier,1993; Ward,1994), the locally-averaged luminance in Ashikhmin (2002) and Yee and Pattanaik (2003) is referred to as local adap- tation luminance to emphasize its spatially-varying property. A related approach to non-uniformly reduc- ing pixels’ luminance values relies on extracting the

illumination from an HDR image (Kimmel et al.,2003;

Tumblin et al.,1999). Since the estimated illumination usually has a wider dynamic range, a displayable im- age can be obtained by reducing the dynamic range via compressing the illumination, and then by recombin- ing the compressed one with the rest of the original HDR image.

When displaying synthesized HDR images, the problem of extracting the layers of illumination and surface properties is inherently solved because images in graphics are usually rendered from illumination and reflectance (e.g., Tumblin et al.,1999). In dealing with images of natural scenes, Land’s Retinex theory (Land and McCann, 1971) can be used to compute the illu- mination, and has been implemented with multi-scale Gaussian filters (Jobson,1997). There are other tone- mapping methods also using multi-scale Gaussian fil- ters to derive the locally-averaged luminance for sim- ulating the local adaptation effect (Ashikhmin,2002;

Pattanaik et al., 1998), or the photographic process (Reinhard et al.,2002). The concern here is that since the global order of luminance is no longer guaranteed by local tone mappings, strategies to avoid halo arti- facts must be explicitly formulated. To gain insight into this issue, we look at a scenario of applying a large- scale Gaussian filter to smooth two neighboring and monotone areas that occupy significantly different lu- minance levels. Then, tone mapping with the resulting over-smoothed adaptation luminance would exagger- ate the contrast near the boundary of the two areas, and consequently yield halos. The example suggests that choosing an appropriate scale of the smoothing operator is critical for computing the locally-averaged luminance and avoiding halos. Ashikhmin (2002) and Reinhard et al. (2002) adopt similar schemes to deter- mine the scales of Gaussian filters. At each pixel lo- cation, they gradually increase the scale of a Gaussian filter until that large contrasts are encountered.

Besides multi-scale Gaussians, anisotropic diffu- sion is another popular choice as the smoothing op- erator for computing the locally-averaged luminance.

Tumblin and Turk (1999) establish the low curva- ture image simplifier (LCIS) to blur an HDR image without smearing the edges. Anisotropic diffusion can also be implemented in the form of bilateral filtering (Tomasi and Manduchi, 1998). Specifically, a bilat- eral filter includes two weights (such as Gaussians):

one for the spatial domain, and the other for the range domain to explain intensity differences between each pixel and its neighbors. Nearby areas with dissimilar

luminance values thus yield small weights, which pre- vent from blurring the edges. DiCarlo and Wandell (2000) have applied this concept to HDR tone map- ping. More recently, Durand and Dorsey (2002) de- velop a fast bilateral filtering to efficiently compute the locally-averaged luminance. Though seemingly differ- ent from the methods mentioned above, the gradient- domain approach of Fattal et al. (2002) is also a local tone-mapping. Their formulation emphasizes attenu- ating large gradients, and recovers the luminance by solving a Poisson equation.

Since estimating local adaptation luminance relates to the issues of how to separate dissimilar areas and how to determine an adequate region for averaging, it is worthwhile to address these problems from a seg- mentation viewpoint (Krawczyk et al.,2004; Schlick, 1994; Yee and Pattanaik, 2003). Schlick (1994) has proposed to divide the picture into zones of similar intensity values and then compute the average of each zone. He notes that although image segmentation is itself a challenging one, the segmentation problem in tone reproduction is a simplified case—on a gray-level image with few regions. Schlick further suggests to use gradient or histogram thresholding techniques to partition the image, but concludes without elaborating further details in Schlick (1994) that the improvement on anti-aliasing around high contrast is imperceptible.

Yee and Pattanaik (2003) develop a multi-layer parti- tioning and grouping algorithm to compute the local adaptation luminance. Their method computes layers under different intensity resolutions, i.e., to quantize intensities by different bin widths, and then for each intensity resolution, to collect pixels of the same bin into a same group. Each pixel’s value in the local adaptation luminance can be obtained by averaging the values of all pixels from the groups in different layers with which the underlying pixel is associated. Though not directly related to the problem of HDR tone repro- duction, the anchoring model proposed by Gilchrist et al. (1999) also considers the connection between the decomposition of an image and the perception of lightness. They propose to decompose a complex image into multiple local frameworks (in terms of the Gestalt grouping principles, e.g., see Palmer, 1999, p. 257). The perceived reflectance is estimated by combining the local and global anchoring properties according to the decomposed frameworks. For prac- tically implementing the model of Gilchrist et al., the main difficulties to be overcome include finding the right grouping factors to reasonably explain lightness

perception, and establishing a computational approach to suitably decompose an image for the anchoring.

1.2. Our Approach

We aim to develop an efficient tone-mapping opera- tor to compress the HDR luminance, accounting for the revealing of image details and the preserving of visual impressions. Often these two effects are hard to be dealt with in a compatible way, but rather as two competitive factors in formulating a tone-mapping function. More precisely, since only a limited dynamic range is allowed to display an image, achieving a good trade-off between local and global contrast is not always straightforward. This difficulty does reflect in our survey that most recent approaches have fo- cused mainly on local adaptation properties of tone reproduction (Ashikhmin,2002; Durand and Dorsey, 2002; Fattal et al.,2002; Reinhard et al.,2002; Yee and Pattanaik,2003), but somewhat overlooked an equally important issue about maintaining the consistency in global contrasts. Motivated by this aspect of consid- eration, we propose to perceptually decompose the luminance into a small number of regions that en- code the overall impression based on the sequential order of luminance levels, and thus term the process as perceptual grouping for deriving a sparse decom- position of an HDR image. Accordingly we gain the advantage of concentrating on designing a local tone mapping for each region, without the burden of worry- ing about breaking the overall impressions carried by the original luminance. Furthermore, it becomes fea- sible to estimate the local adaptation luminance with the perceptual-based region information. Having es- tablished the piecewise tone mapping, we can conve- niently perform region-wise HDR compressions us- ing local mappings constrained by the desirable global property, and produce a displayable LDR image of good quality.

In the following sections, we shall describe first how we derive the sparse decomposition of an HDR image based on a luminance-driven grouping process. We also discuss how the region information is embedded into the computation of the local adaptation luminance.

We then explain how we region-wise construct a lo- cal tone-mapping function, and modify the mapping with the global property encoded in the decomposi- tion. Finally, we conclude with several experimental results and comparisons to illustrate the characteristics of HDR compressions yielded by our method.

2. A Sparse Decomposition for HDR Images The important link between visual grouping and light- ness perception in human visual system has been em- pirically studied in, e.g., Adelson (2000) and Gilchrist et al. (1999). Still an attempt to devise overall corre- sponding computational models that explain the theo- ries of lightness perception and the responses of human visual system would be extremely challenging, though worthwhile. For the problem of HDR tone reproduc- tion, our formulation is not to build a complicated algo- rithm emulating the machinery/process of human vi- sual system, but mainly to establish an efficient method yielding satisfactory results based on useful properties of the proposed perceptual grouping.

To reduce the dynamic range by tone mapping, we decompose an HDR image into a small number of re- gions through a grouping process on luminance, using a perceptual similarity measure for image retrieval de- veloped by Rubner and Tomasi (2001) and Rubner et al.

(2000). We regard such a result of perceptual grouping on luminance as a sparse decomposition for an HDR image. Besides giving preference to fewer regions, the eventual decomposition would comprise regions that are brighter inside and darker near the boundary. As we will elaborate later, these two properties are essen- tial to our approach that locally and globally considers the reduction of dynamic range.

A number of works have explored the idea to com- pute the local adaptation luminance by referencing im- age segmentation (e.g., Krawczyk et al.,2004; Schlick, 1994; Yee and Pattanaik, 2003). In particular, while Schlick (1994) and Krawczyk et al. (2004) take ad- vantage of existing segmentation techniques, Yee and Pattanaik (2003) specifically develop a fast segmen- tation algorithm for computing local adaptation lumi- nance. They describe a multi-layer grouping process (quantized with multiple bin widths in the intensity do- main) to obtain the locally-averaged luminance value of each pixel. Nevertheless, all these aforementioned approaches simply focus on how to estimate local adap- tation luminance. None of them seeks to use the re- gion or segmentation information for modulating tone- mapping functions.

In our formulation the region information is used not only to compute local adaptation luminance values, but also to establish more comprehensive tone-mapping functions for maintaining global contrasts. We shall in- troduce a new perceptual grouping algorithm that can fast construct the sparse decomposition for an HDR

Figure 1. Garage. (a) The corresponding log-luminance ˜L. Note that values of ˜L are normalized to [0, 1] for displaying. (b) The adaptive block partition of ˜L, where blocks of the smaller size are located in those shaded areas on top of the Canny edges. (c) A sparse decomposition of 8 regions for ˜L. (d) The displayablegarageimage derived by our method. (e) A close-up of the block partition inside the red rectangle.

(f) The corresponding regions in close-up. (g) The resulting displayable image patch in close-up.

image. The advantages of using the proposed decom- position to encode the overall perceptual impression will be thoroughly discussed in Section 3. Suffice it to say now that working with an adequate number of re- gions, we can perform the HDR compression without incurring excessive overheads in assembling the results across different regions. Furthermore, with the region information, we can devise tone-mapping functions ac- counting for local and global contrast adjustments. On deriving such a sparse/compact decomposition, it takes

two steps: adaptive block partitioning and perceptual grouping, detailed in what follows.

2.1. Adaptive Block Partitioning

The luminance values of an HDR image can be com- puted from its R, G, and B channels by

L(x, y) = 0.2126R(x, y) + 0.7152G(x, y)

+ 0.0722B(x, y), (1)

Figure 2. An illustration of a possible transformation from one set of weighted features to another. The EMD between these two feature sets is the minimum amount of “work” (with respect tosi− s^j and the weights) needed to transform{(s1, w1), (s2, w2), (s3, w3)} to {(s1^, w1^), (s2^, w2^), (s3^, w^3)}, e.g., moving the whole of w1from s1

to s₁^, moving parts ofw2from s2to s₁^, s2^, and s₃^, and movingw3

from s3to s₂^ and s^₃. Note that in our algorithm we have

iwⁱ =

iw^i= 1, and the transformation is symmetric.

where the sRGB color space and a gamma of 2.2 are assumed (ITU, 1990; Stokes et al., 1996). Previous experience on studying perception has suggested that human visual system senses the contrast of light based on intensity ratio rather than intensity difference (e.g., the Weber’s law discussed in Palmer 1999, p. 672).

Following this observation, we consider the decompo- sition of an HDR image by examining its luminance property in the logarithmic domain. More precisely, the luminance L is transformed into log-luminance by

L(x˜ , y) = log L(x, y). (2)

A modification to the scene’s key value, as is used in Reinhard et al. (2002), has been performed on the

(original) log-luminance. Its effect is equivalent to shifting the log-luminance values for providing a pre- ferred middle-gray tone, a technique commonly used by photographers. Notice that the adjustment will not affect the result of the perceptual grouping process we are about to discuss, because the difference in log-luminance values between any two pixels remains the same. Hence, we will assume ˜L denotes the log- luminance after the key value adjustment hereafter.

While partitioning on a pixel level is both time- consuming and sensitive to noise, we also find working with blocks of uniform size often leads to unsatisfac- tory segmentation results. Though the situation could be improved by using small-size blocks, such a tactic again has the drawback of inefficiency. We thus design an adaptive scheme to partition the image with blocks of two different sizes. The smaller blocks are placed in the areas of strong log-luminance gradients; the larger ones are in the areas of less log-luminance variation.

Precisely we use Canny edge detector (Canny,1986) to obtain the edge information, and then divide ˜L into blocks of larger size b
× b
. For those image blocks containing Canny edges, they are further split into blocks of smaller size bs × bs. A complete example to illustrate these steps is given in Fig.1 with block sizes b
= 8 and bs = 2. Note that the two specified sizes are used in all our experiments reported through- out this work too.

2.2. Luminance-Driven Perceptual Grouping Among the many possible ways to group the blocks, we are interested in finding a sparseness one, driven

Figure 3. Intermediate stages of displayinggarage: ˜V and V. Both images are normalized to fit into the range between 0 and 1 for displaying.

Generally, a local adaptation luminance takes a significant portion of the dynamic range but leaves out the details.

Figure 4. Comparison of the global and local mappings. The result of local mapping provides more details of thegarageimage. However, in both (a) and (b), the top-right and bottom-right areas (inside the garage) are still too dark.

Figure 5. (a) The mappingϕ^γ(x)= (x/(1 + x))^γwith differentγ values. (b) and (c) The results ofgaragebyϕ^0.3(L) andψ(L, V ; ρ = 0.3, γ = 0.3), cf. Figs.4(a) and4(b). Although the details in the bottom-right area are visible now, the dynamic range of the rest area is also squeezed.

Figure 6. The setting for estimatingρ in Rk. LetDkdenote the set of ×  grid points that are inside the boundary ∂Rk, and for all points x∈ Dkwe have miny∈∂Rkx − y2 ≥ . Later on, we will assign pixels n∈ Dksome preliminaryρnvalues from which theρ values for all pixels inRkwill be derived.

by examining the log-luminance values. For that, we look at two important matters: (1) how to define an appropriate distance function to measure the degree of similarity between two regions, and (2) how to pro- ceed with a reasonable grouping process to derive the decomposition. Our choice to seek a sparse decompo- sition for segmenting an HDR image is mostly due to computational concern: We would like to region-wise perform tone mappings, and then integrate them into the final solution. It is thus useful to partition the image into a few (perceptually) significant regions, provided that the decomposition reasonably encodes the overall impression.

2.2.1. Perceptual Distance. We apply the earth mover’s distance (EMD) to evaluate the perceptual similarity between two image regions (Rubner and Tomasi, 2001; Rubner et al., 2000). Used mostly in image retrieval, EMD has been shown to be an ef- fective measure to perceptually correlate two images.

The physiological experiments in Allred et al. (2003, 2004) further support that the image similarity mea- sured by EMD pertains to the neural responses elicited by the same set of images. The EMD between two sets of weighted features is computed as the minimum amount of “work” (pertaining to the weights and the Euclidean distances between features) that is required to transform one feature set to the other (see Fig.2).

In our formulation, we generate features by equally di- viding the dynamic range of a region, which could be an image block or a set of neighboring blocks,

into three bins. The mean value si and the number hi of the pixels in each bin are then calculated. It fol- lows that the weighted-feature set p of each region contains three pairs{(s1, w1), (s2, w2), (s3, w3)} where wi = hi/
3

j=1hj, and they accordingly represent the bright, the middle-gray, and the dark parts of that re- gion. Clearly, the weighted-feature description is more general than the histogram representation in that the feature siis not necessary the bin center. Details about the computation and the analysis of EMD can be found in Rubner et al. (2000). For our formulation, the per- ceptual distance between two regions R1 andR2 is defined by

D(R1, R2)= EMD (p1, p2), (3)

where piis the weighted-feature set of regionRi. 2.2.2. Luminance-Driven Grouping. While analytic arguments on how to optimally decompose an image are difficult to establish, we prefer a compact/sparse representation Barlow(1961)to decompose an image into a few regions (see Fig.1(c)). Specifically, we adopt a greedy approach to grow a new region as large as possible, starting each time from the location of the brightest log-luminance value in the unvisited areas.

That is, the algorithm follows a brightest-block-first rule to determine the seeds and to merge image blocks.

Since the feature set of each block includes three clus- ters, namely, the bright, the middle-gray, and the dark parts, the brightest block can be simply identified as the one with the largest s₁ value. Unlike ours, other grouping algorithms for HDR luminance such as Yee and Pattanaik (2003) and Krawczyk et al. (2004) do not specify the order of accomplishing the process.

Owing to the brightest-block-first rule, each region ob- tained by our algorithm tends to be brighter in the inner area and darker in the peripheral. This property will be further exploited in regulating tone-mapping functions and smoothly piecing together adjacent regions.

Our grouping process proceeds as follows. All im- age blocks are initially marked as unvisited. Later on as the grouping process iterates, the number of unvisited blocks decreases. At iteration k, we pick the brightest and unvisited block, say, block B, and start to grow the regionRkfrom it. These steps of region grouping are summarized in Algorithm 1. Upon termination, the process will yield a decomposition that each derived region consists of connected blocks of similar luminance distributions. Indeed, our algorithm works

by balancing the local and global similarity in a region.

Similar blocks are pulled into the same region if the EMD between two neighboring blocks is smaller than δ. On the other hand, a region will stop growing when all blocks right beside the region boundary are not close enough to the region withinθ. (See Algorithm 1 for details.) The typical values of EMD thresholdθ are from 1.5 to 2.0, and those of δ are between 0.5 and 1.0.

2.3. Local Adaptation Luminance

Human visual system attains the HDR perception by locally adapting to different levels of luminance to en- sure a proper dynamic range can be recreated by the re- sponses of visual cells. For tone reproduction, the local adaptation effect is often simulated by computing the local adaptation luminance, and then using the infor- mation to extract the details and to reduce the dynamic range. Thus it is important to have a reliable way for pixel-wise estimating the local adaptation luminance

of an HDR image. And that in turn can be done by investigating the average log-luminance of a suitable neighborhood about each pixel. As we will see, a good choice of a suitable neighborhood to this purpose is the sparse decomposition derived in Section 2.2.

Let ˜V (x, y) be the local adaptation log-luminance at pixel (x, y). To compute ˜V , we consider a general- ized version of bilateral filtering (Durand and Dorsey, 2002; Tomasi and Manduchi,1998) by constructing a region-dependent scheme such that the computation of V (x˜ , y) takes account of bilateral effects from differ- ent regions. Specifically, for each pixel (x, y) in region Rk, we have

V (x, y) =˜ 1 Z˜x,y

 

(i, j)∈Rk

L(i, j) G˜ x,y(i, j) Kx,y(i, j)

+ 

(i, j)/∈Rk

L(i, j) G˜ x,y(i, j) Kx^,y(i, j)

 ,

(4)

where

Gx,y(i, j) = exp

− ((i − x)²+ ( j − y)²)/2σs²

, (5) Kx,y(i, j) = exp

− ( ˜L(i, j) − ˜L(x, y))²/2σr²

, (6) K_x^_,y(i, j) = exp

− ( ˜L(i, j) − ˜L(x, y))²/2σr^2

, (7)

and the normalization term is Z˜x,y = 

(i, j)∈Rk

Gx,y(i, j) Kx,y(i, j)

+ 

(i, j)/∈Rk

Gx,y(i, j) Kx^,y(i, j). (8)

Equation (4) comprises two aspects of bilateral fil- tering. The first part calculates the averaging in the same region, and the second evaluates the contributions from other regions. We have conveniently used the same spatial-domain filter Gx,yto regulate the effects to pixel (x, y) from all regions. Such an option makes it possible to apply the fast bilateral filtering described in Durand and Dorsey (2002), which downsamples the image and linearly interpolates the filter responses in range domain. Meanwhile, we haveσr ≥ σr^to ensure a flatter and more expanded range-domain filter Kx,y

for pixels in the same region of (x, y), and to lessen the influences from pixels of different regions with K^x,y. Ifσr = σr, our proposed scheme in (4) is reduced to the one used in Durand and Dorsey (2002). A suitable value forσscan be set to 4% of the image size, and we also haveσr = 2×σr^ = 0.4. Our discussion so far has been focused on the log-luminance domain. The local adaptation luminance, denoted as V, can now be recov- ered by V = exp( ˜V ), e.g., see the examples in Fig.3.

3. Piecewise Tone Mapping

Our approach to dynamic range reduction relies on region-wise constructing suitable tone-mapping func- tions that depend on the estimations of local adaptation luminance. It is also critical that the resulting piece- wise tone mappings could be smoothly pieced together to produce a good-quality displayable image without violating the overall impression of the original HDR radiance map. In this section, we will show that all these issues can be adequately addressed by consid- ering the region information encoded in a luminance- driven sparse decomposition.

3.1. Global and Local Mappings

A handy choice of simple functions for compressing the high luminance values into the displayable range [0, 1] is the nonlinear mapping used in Reinhard et al.

(2002)

ϕ(x) = x

1+ x. (9)

Ifϕ is considered for the whole luminance map, i.e., L^ = ϕ(L) = L/(1 + L), we will actually obtain a displayable but smoother image. This type of com- pression scheme is called global mapping or spatially uniform mapping. Another tone-mapping method is to extract from L the details H by H = L/V . Then, only the local adaptation luminance is compressed by V^= ϕ(V ) = V/(1 + V ). Recombining the details H with the compressed V^, we have

L^= H × V^=

L V



×

 V 1+ V



= L

1+ V, (10) which is a local mapping used by Reinhard et al. (2002) for preserving details. Equation (10) can also be inter- preted as adjusting the luminance through a position- dependent scale function. Let m(V )= 1/(1 + V ) be the position-dependent scale function. Then the local mapping (10) becomes L^ = m(V )L, conformed to that in Tumblin et al. (1999). Figure4illustrates the results of compressing garage by global and local mappings. The result of local mapping provides more details of the original HDR image.

Even though a global mapping like ϕ in (9) often has the drawback of losing the details in brighter ar- eas, its monotone property (dϕ/dx > 0) is desirable for preventing halos and other artifacts. It would be favorable if the monotonicity can be incorporated into a local tone-mapping method. Nonetheless, one still needs to figure out a reasonable way to monotonize a local mapping and determine an appropriate neighbor- hood for each such a monotonization.¹

Another issue is concerning the brightness of those large dark areas in Fig.4. In both results, the top-right and bottom-right parts look quite dark, and occupy a substantial proportion of each image. This impression is not very realistic because not enough details can be seen inside the large dark areas. (For comparison, refer to our result ofgarageshown in Fig.(1).) The alterna- tive illumination correction proposed by Kimmel et al.

(2003) can solve the brightness problem satisfactorily.

However, as to be described later, our method includes

Figure 7. Theρ values after kernel smoothing (e.g., Bishop,1995). Examples of correspondences to the log-luminance and to the region decomposition are illustrated. Note that largerρ values will amplify more the subtleties in luminance.

the brightness adjustment into a unified tone-mapping approach that a more general luminance-based bright- ness correction will be implicitly carried out subject to the overall impression constraint.

3.2. Modulating the Tone-Mapping Functions Perceptually, the proposed region decomposition cor- relates with an overall visual impression about the scene, including, for example, the contrasts and the order of brightness levels of different areas. And main- taining this impression after compressing the dynamic range should be a good criterion. We thus consider a piecewise tone mapping scheme that region-wise per- forms the monotonization, and globally retains the rela- tive brightness among different regions, i.e., thoseRks in Algorithm 1. The complete idea of deriving such tone mappings can be realized by the following steps.

3.2.1. Design a Local Mapping ψ. As remarked earlier, the local adaptation luminance V is the key

element in compressing the luminance L. We define a local mapping

ψ(L, V ; ρ, γ ) =

L V

_ρ

ϕ^γ(V )=

L V

_ρ V 1+ V

_γ , (11) where 0 < ρ < 2 and 0 < γ ≤ 1 are spatially- dependent parameters to adjust the image quality re- sulting from the HDR reduction. Specifically, when γ < 1, the darker areas in an HDR radiance map will be compressed into a larger dynamic range, com- pared with the caseγ = 1. On the other hand, since L/V contains the details, the ρ values would have di- rect impacts on preserving the image details after the dynamic-range reduction. Whenρ = γ , we have

ψ(L, V ; γ, γ ) =

L V

_γ V 1+ V

_γ

=

 L

1+ V

_γ . (12)

To illustrate the effect ofγ , we plot three global map- pings with γ = 0.3, 0.7 and 1.0 in Fig. 5(a), and show the results of directly applying (L/(1 + L))^0.3 and (L/(1 + V ))^0.3to thegarageimage in Figs.5(b) and5(c), respectively. While the visibility of dark ar- eas is boosted (cf. Figs.4(a) and4(b)), the overall out- comes are still unsatisfactory. (Notice that in this work we have used a spatially uniform value of γ = 0.3 for all the experiments.) We shall improve the results by investigating theρ values and other related factors through the succeeding steps.

3.2.2. Reshape the Global Mappingϕ^γ. Besides the local mappingψ in (11), we also need a global mapping likeϕ^γ to embed the overall impression into the result- ing region-wise tone mappings. It is therefore useful to make sure that for an arbitrary HDR image, the map- ping values of ϕ^γ take up the complete displayable range [0, 1]. We enforce this requirement through a reshapping step by

ϕ^γ → ˆϕ^γ and ϕˆ^γ(L)= αϕ^γ(L)+ β, (13)

where α and β are the scaling and the shifting vari- able, respectively, and their values can be analytically derived. More precisely, let Lmaxand Lminbe the max- imum and minimum luminance values of a given ra- diance map. We then solve the following simple lin- ear system to obtain the proper values of α and β in (13):







 Lmax

1+ Lmax

_γ 1

 L_min 1+ Lmin

_γ 1







α β



=

1

0



. (14)

3.2.3. Estimateρ by Kernel Smoothing. Appropriate values ofρ have to be assigned for the local mapping ψ(L, V ; ρ, γ ) in (11). We plan to give larger ρ values to most pixels for revealing details, as long as no halos will occur. For each region Rk, we first construct a gridDkwithinRk. TheDkis assumed to be the largest grid of resolution ×  with all its grid points at least ε-pixel away from the region boundary ∂Rk. (The idea can be better understood by referencing the illustration in Fig.6.) We then assign some preliminaryρn value

to each pixel n∈ Dkby

ρn =









γ, if log(Ln/Vn)≤ −1, 0.5 × (γ + ρmax), if log(Ln/Vn)≥ 1,

ρmax, otherwise.

(15)

The first two rules of (15) simply reflect that if Ln

is already very different from its local adaptation lu- minance Vn, using a larger ρn could only cause an inconsistent overemphasis by further amplifying the log-ratio. In addition, for all pixels on∂Rk, theirρn

values are also set to γ , and those of the pixels be- tweenDkand∂Rkcan be computed by interpolations.

It will later become clear that putting such constraints near∂Rk is useful in constructing an overall smooth ρ surface across regions. With all these ρnvalues pre- defined, we now apply kernel smoothing to adjust and to derive theρ values for all pixels in Rk. See Fig.7for an example ofρ surface of thegarageimage. Note that we haveρmax= 1.8 and  = 4 for all the experiments.

3.2.4. Monotonize Local Tone Mappings. For each regionRk, a monotonization based on the global map- ping ˆϕ^γ in (13) is to estimate the variablesαkandβk

so that the local mappingψ in (11) can be elevated to ψ(L, V ; ρ, γ ) = αˆ kψ(L, V ; ρ, γ ) + βk, (16)

and to account for both local and global factors in region-wise compressing the luminance. We begin by sampling N pixels from∂Rk according to the sorted

| log(Ln/Vn)| values in ascending order. In our exper- iments, using the first 5% of boundary pixels will be sufficient to give good results. For each sample pixel n, we have itsψnfrom (11) and ˆϕ^nγfrom (13). The values ofαk andβk can be obtained by calculating the least squares solution of

ψ1 ψ2 · · · ψN

1 1 · · · 1

T αk

βk



=

ϕˆ₁^γ ϕˆ^γ₂ · · · ˆϕ^γ_NT

. (17) With (16) and (17), ˆψ becomes a proper tone map- ping for reducing the dynamic range inRk, and com- putes the displayable luminance value of each pixel (x, y) ∈ Rkby

L^(x, y) = ˆψ(L(x, y), V (x, y); ρ(x, y), γ )

= αk

L(x, y) V (x, y)

_ρ(x,y)

V (x, y) 1+ V (x, y)

_γ

+ βk. (18) Once the foregoing procedures have been performed for eachRk, the resulting LDR image is ready for dis- playing. This is accomplished by multiplying the com- pression ratio L^/L to each of the HDR RGB channels with suitable gamma correction for a specific CRT or LCD.

We now give some reasoning for the proposed piece- wise tone mapping, especially for the justifications of monotonizations. Notice that only pixels on the re- gion boundaries are considered for the monotoniza- tion in (17), and theirρ values after kernel smoothing are still close to γ . It implies that ψ(L, V ; ρ, γ ) ≈ (L/(1 + V ))^γ at and near each region boundary. These properties justify our compression scheme in two ways.

First, the least squares fitting (17) to reshapeψ into ˆψ can be reasonably solved owing to that the underly- ing process is indeed to fit a general function of the form (L/(1 + V ))^γ, valid to all the sampled points, to the global mapping ˆϕ^γ(L) = α(L/(1 + L))^γ + β.

Had we arbitrarily sampled pixels within a region, thoseρ values of the selected points could vary signifi- cantly (ρ ≈ γ ). The least squares fitting would become both difficult to interpret its meaning and sensitive to the sampling perturbations. (In this case, the mono- tonization is no longer implicitly concerning a func- tion of a general form but of the comprehensive form (L/V )^ρ(V/(1 + V ))^γ.) Second, since ˆϕ^γ is monotone, the reshapping of a local mapping ψ into ˆψ should approximately retain the global brightness consistency among the regions of a given sparse decomposition.

Of course, the validity of our claim establishes upon the assumption that the perimeter of an image region would be brighter than that of a less bright region.

This property is generally true because the region de- composition for an HDR image is obtained through a grouping process based on the brightest-block-first rule. Therefore, it is fair to conclude that compress- ing the dynamic-range with (18) has the advantage of emphasizing the local details of each region without breaking the global visual consistency.

4. Experiments and Discussions

Having detailed each step of our approach, depicted in Fig.8, we are now in a position to go through some of our experimental results and comparisons with other related works. In all our experiments, the HDR im-

Figure 8. The steps of our tone reproduction method.

ages are downloaded from the Web, and stored in the radiance map format. To illustrate, a typical radiance map with multiple exposure values is shown in Fig.9, where, in this example, the dynamic range ofStanford memorialis about 250,000 : 1.

4.1. Related Issues and Experimental Results The efficiency bottlenecks of our method lie in the steps of EMD perceptual grouping and region-wise bilateral filtering. For thememorialimage of size 512 by 768 in Fig.9, the elapsed time of grouping and of filtering on a 3.4 GHz PC are 2.70s and 4.17s, respectively. Clearly, the time complexity of region-wise bilateral filtering should depend on the number of regions. We handle this issue by generalizing the fast bilateral filtering in Durand and Dorsey (2002) to incorporate region

Table 1. Timing results for grouping and filtering.

Size No. of Grouping Filtering HDR Image (pixels) regions (seconds) (seconds)

garage 352× 224 8 0.58 0.69

memorial 512× 768 10 2.70 4.17

door 224× 336 3 0.23 0.83

park 744× 405 6 0.95 2.25

chairs 344× 224 14 0.29 1.22

groveC 712× 480 17 1.89 5.27

groveD 712× 480 18 4.63 4.94

office 1000× 656 68 2.71 22.36

Tahoe 1024× 768 5 2.74 3.24

clock 768× 1024 34 5.70 15.38

designCenter 656× 1000 13 2.06 7.76

Tintern 768× 1024 14 5.08 4.97

Belgium 1024× 768 32 3.20 15.58

synagogue 1024× 768 9 3.05 3.48

The corresponding results are shown in Figs.11,13,14,15and17.

support. As a result, a region-wise bilateral filter can be implemented in a way that the slowdown is not proportional to the number of regions. For instance, ten regions are constructed by perceptual grouping for Stanford memorial, but region-wise bilateral filtering simply doubles the time needed for a typical bilateral filtering (∼1.86s). More timing results are listed in Table1.

We like to emphasize most of the parameters for producing the results in this paper are fixed. In partic- ular, for region-wise bilateral filtering, we have set the spatial scaleσsto 4% of the image size, and the range- domain scalesσr = 2 × σr^ = 0.4; for tone-mapping,

we haveγ = 0.3 and ρmax= 1.8. Therefore, in effect, only two parameters need to be tuned: the EMD group- ing thresholdsθ and δ. Moreover, to select the values ofθ and δ is not complicated at all. In our experiments, typical values ofθ are chosen between [1.5, 2.0], and those ofδ are between [0.5, 1.0]. Within these ranges the choice of values forθ and δ is not crucial to achiev- ing satisfactory results. Figure10shows the outcomes of applying different values ofθ and δ.

It can happen that, with bilateral filtering, pixels around large gradients may still undergo small halos due to that too few neighboring pixels are included by the scale of range-domain filter in evaluating proper range-domain responses. This problem is remarked by Durand and Dorsey (2002) as uncertainty, a conse- quence of computing a statistical estimator on insuf- ficient data, and thus causing a fairly high variance value. We have experimented with a similar scheme as in Durand and Dorsey (2002) to resolve the uncertainty by considering a low-pass version ˜Vl of ˜V . More pre- cisely, in our experiments a 3×3 small-scale Gaussian kernel withσ < 1 is used to slightly blur ˜V into ˜Vl

as the improved local adaptation log-luminance. This tactic not only fixes the uncertainty problem but also gives a bit more details without generating perceptible halos.

In Fig.11, we demonstrate several LDR images de- rived by our tone-mapping method. (The respective sparse decompositions are shown in Fig.12.) The com- pression results are characterized by two types of visu- ally pleasing effects: maintaining overall impressions of luminance and preserving details and local contrasts.

To examine the first effect we can compare the relative brightness levels of different parts in each pair of an HDR and its compressed LDR images. The quality of

Figure 9. By setting a fixed range and cutting off out-of-range luminance values, theStanford memorialradiance map can only be displayed like these images.

Figure 10. The only two parameters we need to select are the EMD grouping thresholdsθ and δ. Typical values of θ are from 1.5 to 2.0, and those ofδ are from 0.5 to 1.0. Left column, from top to bottom: the results of choosing {θ = 2.0, δ = 1.0}, {θ = 1.55, δ = 0.85}, and {θ = 1.5, δ = 0.5}. Right column: the corresponding sparse decompositions of the left column. The respective number of regions is 4, 8, and 12 from top to bottom.

the second effect can be inferred, say, from those im- ages in Fig.11—local contrasts are not degraded and fine details in dark areas are revealed.

4.2. More on the Sparse Decomposition

It should be clear now the sparse decomposition plays an important role in our formulation for solving tone

reproduction problems. By encoding the overall im- pression this way, we gain the advantages of focusing on performing local tone mappings over (a few) re- gions, and meanwhile keeping the global visual con- sistency through the proposed monotonizations. This effect of retaining relative brightness among different regions is depicted in Figs.13and14.

Suppose we disregard the region information, and use only the local adaptation luminance and the local

Figure 11. HDR tone-reproduction results. From left to right and top to bottom:chairs,groveC,groveD,office,Tahoe,clock,designCenter, Tintern,Belgium, andsynagogue.

Figure 12. Sparse decompositions. From left to right and top to bottom (with the number of derived regions):chairs(14),groveC(17), groveD(18),office(68),Tahoe(5),clock(34),designCenter(13),Tintern(14),Belgium(32), andsynagogue(9).

Figure 13. The effect of sparse decomposition and piecewise tone mapping. (a) The original HDR image is displayed with different exposures.

(b) An eight-region sparse decomposition is obtained after the perceptual grouping stage. (c) Without piecewise tone mapping, the brightness around some areas is somewhat abnormal, e.g. the wall on the left and the inside of the garage. (Here we apply the local mapping (_V^L)^ρmax(_1+V^V )^γ to the whole image withρmax= 1.8 and γ = 0.3.) (d) A close-up of (c). (e) Based on the region information of (b), the relative brightness is monotonized after piecewise tone mapping. Nonetheless, the details of the dark areas are still noticeable. (f) A close-up of (e).

mapping defined in (10), the tone reproduction results are like the example in Fig. 13(c). That is, we apply the local mapping (_V^L)^ρmax(_1+V^V )^γ to the whole image rather than individual regions. (Recall thatρmax= 1.8 andγ = 0.3.) As can be seen in Figs.13(c) and (d), the

yellow exterior on the left is darkened, and the inside area is too bright and lack of contrast. On the contrary, the results in Figs.13(e) and(f)yielded by applying our piecewise tone mapping to the sparsely decomposed image reveal a good compromise between local and

Figure 14. (a) The original HDR image is displayed with an acceptable exposure setting. (b) A three-region sparse decomposition. (c) Without piecewise tone mapping, the brightness around some areas is somewhat abnormal, e.g. the cloud and the indoor scene. (Directly mapped by (_V^L)^ρmax(_1+V^V )^γwithρmax= 1.8 and γ = 0.3.) (d) After piecewise tone mapping.

Figure 15. Stanford memorial.From left to right, top to bottom: the respective compression results derived by bilateral filtering (Durand and Dorsey,2002), photographic tone reproduction (Reinhard et al.,2002), gradient domain (Fattal et al.,2002), and our method. Our method can reveal the fine details of the scene and maintain a global impression similar to that of the photographic (Reinhard et al.,2002).

global contrasts. Taking the door image as another example, a similar effect produced by piecewise tone mapping can again be observed in the indoor area, cf.

Figs.14(c) and(d).

4.3. Comparisons

Some recent methods (Durand and Dorsey, 2002;

Fattal et al.2002; Reinhard et al.,2002) on displaying

Figure 16. (a) The sun is too dark in the result of Reinhard et al. (2002). (b) Corrected by piecewise tone mapping. (c) Bilateral filtering (Durand and Dorsey,2002) produces some halos near the skyline and the shadows. (d) Our method not only preserves more details but prevents most halos.

HDR images also can eliminate halos and preserve de- tails. Despite different quality of visual impressions, their results generally look good. It is indeed hard and biased to rank the performances of different methods.

So far, the fairest but impractical way to evaluate a tone- reproduction method may only depend on the feeling of someone who has been present when the photos were taken. Thus, in what follows, we shall focus on the typical differences and characteristics between the outcomes of our method and others, illustrated with the examples shown in Figs.15–17.

The gradient-domain compression by Fattal et al.

(2002) performs well in preserving local contrasts and details. However, their method does not explicitly deal

with the balance between local contrasts and global consistency. Thereby a noticeable difference between their compression outcomes and ours is that in their results global contrasts are more often distorted. For instance, brighter areas may not be bright enough as they should be. A good example is the circular window of Stanford memorial in Fig. 15. Note that, in the original radiance map, the area of the circular window is at least 200 times brighter than the top-right area of the image.

While our tone-mapping approach relates to pho- tographic tone reproduction (Reinhard et al., 2002) and fast bilateral filtering (Durand and Dorsey,2002) in computing the local adaptation luminance, the

Figure 17. (a) The result of Ward et al. (1997). (b) The details of the medallion and the carvings are preserved by our method. (c) The result of our method (cf. Fig. 3 of Yee and Pattanaik,2003).

proposed method usually yields better contrasts within the displayable range and preserves more image de- tails. The examples in Fig.15should well demonstrate these common phenomena. In addition to the foregoing differences in quality, we further discuss another two noticeable distinctions between our results and those produced by Durand and Dorsey (2002) and Reinhard et al. (2002), using the examples in Fig.16.

The first distinction is that although the photographic method of Reinhard et al. (2002) basically maintains overall impressions, sometimes the approximate lo- cal adaptation luminance is too local—the scales of Gaussian kernels are not large enough. Consequently, only those pixels with similar luminance values are in- cluded for computing the local adaptation luminance, and thus the contrast of a nearby area cannot be fully expressed. In particular, this effect often occurs in very bright areas, and causes the brightness of these areas to decrease. For instance, the sun in Fig.16(a) appears to be too gray. Alternatively, owing to piecewise tone mapping, our results do not have this anomaly, e.g., Fig.16(b). The second distinction is that the bilateral filtering method of Durand and Dorsey (2002) still gen- erates some halos near the boundaries between bright

and dark regions. The differences can be observed near the skyline and near the shadows in Figs.16(c) and(d).

Via a smooth ρ-surface adjustment, our method pro- duces a bit lower contrasts along boundaries, and can avoid the problem of halos.

Our approach differs from global operators, e.g., Ward et al. (1997), by the effectiveness on preserving details. Ward et al. (1997) adopt histogram adjustment to design the tone-mapping function. They use a linear ceiling that constrains visible contrast distortions and thus prevents halos. The resulting function is global and monotonically increasing. Although an operator of this kind is fast and halo-free, it tends to blur details. The proposed piecewise tone mapping instead maintains the global contrasts via regulating local mapping func- tions. Local contrasts are preserved by (18), which in- cludes the local adaptation luminance V (x, y) and the ρ surface. These differences between global mappings and our method are illustrated in Figs.17(a) and(b).

Yee and Pattanaik (2003) develop an algorithm con- sisting of segmentation, grouping, and assimilation for estimating the local adaptation luminance: Pixels are categorized by their log-luminance values based on a given bin-width. The grouping process is akin to

computing connected components for pixels that be- long to the same category. Small singleton groups are then merged into their larger neighbors, and the result- ing groups form a layer. These processes are repeated several times with different bin-widths to yield mul- tiple layers of an HDR image. Hence every pixel in each layer belongs to some group, and has its aver- aged group luminance value. The local adaptation lu- minance at each pixel location is obtained by averaging its corresponding group luminance values from all lay- ers. Like bilateral filtering, the formulation in Yee and Pattanaik (2003) is aimed to compute the local adap- tation luminance for preserving details and preventing halos. Indeed Yee and Pattanaik do not propose a new tone-mapping operator, but employ existing ones (e.g.

Tumblin and Rushmeier,1993), to display HDR im- ages with the local adaptation luminance derived by their scheme. However, we use the grouping process to obtain region information (in order of luminance) for adjusting piecewise tone-mapping functions, not just to estimate the local adaptation luminance. One of our experimental results for comparing the two methods is given in Figs.17(c) (cf. Fig. 3 of Yee and Pattanaik (2003).

5. Conclusions

We have thoroughly investigated the tone-reproduction problem in two aspects: (1) deriving local adaptation luminance for preserving details, and (2) region-wise reducing the high dynamic range into a displayable range without breaking overall impressions. For a vi- sual model, the correctness of local adaptation mainly depends on the definition of locality. Only by an overall consideration of both luminance and perception factors can one reproduce local high contrasts in the form of visually pleasing details rather than halos.

Our proposed sparse decomposition for the lumi- nance has been playing a pivotal role in our formula- tion. Through the image decomposition, we are able to analyze both the local adaptation luminance and the tone-mapping functions. Specifically, we formulate a region-wise bilateral weighting scheme to enhance the local adaptation effect inside a region. Pertaining to the tone-mapping functions, the decomposition implic- itly encodes the globally relative contrasts into a small number of perceptually significant regions. Since the grouping process is driven by a brightest-block-first rule, the derived regions are arranged roughly in or- der of the luminance value. These properties are useful

in modulating an arbitrary local tone mapping of a re- gion into the one strengthened with the desirable global property by our so-called monotonization step.

The piecewise tone-mapping scheme indeed cul- tivates various tone-reproduction qualities. Besides maintaining global contrasts among different regions by fitting to the global mapping ˆϕ^γ(L), it includes the two parameters, γ and ρ, in the mapping functions (18) to regulate brightness and local contrasts of tone- reproduction results. We have well illustrated this point by using an appropriateγ value to effectively elevate the brightness levels in dark areas. And theρ surface tempers high contrasts in too dark and too bright areas, as well as alongside region boundaries (which usually imply strong edges). The near-boundaryρ values are uniformly close toγ , so that we can smoothly stitch regions and prevent possible halos.

Finally, as expected and verified by our other experiments, directly applying the proposed tone- reproduction algorithm to LDR images would also considerably enhance image qualities. Therefore, the piecewise tone mapping is effective in not just com- pressing high dynamic range but also expanding low dynamic range. However, in both cases the dynamic range is considered to be continuous. It is possible that a radiance map has a bipolar distribution of extremely large and small luminance values, i.e., there is a large gap in the dynamic range. For future work, it will be interesting to devise specific algorithms for this kind of situation. Another extension we are interested in as future work is to apply graphical models to the tone reproduction problem. Graphical models are shown to be useful tools for low-level vision modeling. Their application to high dynamic range reduction appears to be a new and promising direction.

Acknowledgments

We thank Paul Debevec, Fredo Durand, Dani Lischin- ski, Shree K. Nayar, Sumanta N. Pattanaik, Erik Reinhard, and Greg Ward for making their HDR radiance maps and output images available. This work was supported in part by grants 93-2213-E-001-010 and 94-EC-17-A-02-S1-032.

Note

1. By a monotonization within an appropriate neighborhood, we mean to monotonize a local mapping so that it almost has the monotone property ofϕ in that region.