Flash and Storm: Fast and Highly Practical Tone Mapping based on

Naka-Rushton Equation

Nikola Bani

c and Sven Lon

cari

Image Processing Group, Department of Electronic Systems and Information Processing,

Faculty of Electrical Engineering and Computing, University of Zagreb, 10000 Zagreb, Croatia

Keywords:

HDR, Image Compression, Image Enhancement, LDR, Naka-Rushton Equation, Retinex, Tone Mapping

Operator.

Abstract:

Tone mapping operators (TMOs) are used to convert high dynamic range (HDR) images to their low dynamic

range (LDR) versions mostly to display them on standard display devices. The problem with many TMOs

that produce high-quality results is that they are too slow to be used in real-time applications. In this paper, a

new TMO is proposed whose steps are primarily designed to achieve high speed and to be practically imple-

mentable. Under this constraint the secondary goal is to produce low dynamic range images of high quality.

The proposed TMO is based on Naka-Rushton equation used in combination with additional improvements

and it has O(1) per-pixel complexity. The presented and discussed results show that, beside being faster and

more practical, the proposed TMO outperforms many state-of-the-art TMOs in terms of resulting LDR image

quality. To further demonstrate its practicality, the source code written in C++, Matlab, Python, Java, and

HTML+JavaScript is available at http://www.fer.unizg.hr/ipg/resources/color constancy/.

1 INTRODUCTION

The dynamic range of an image is the ratio between its

largest and smallest non-zero intensity. Even though

high dynamic range (HDR) images are used ever

more frequently (Reinhard et al., 2010), most display

devices are currently limited to show only low dyna-

mic range (LDR) images. To display an HDR image

on such devices, its dynamic range has to be com-

pressed in the process called tone mapping by means

of methods called tone mapping operators (TMOs).

Tone mapping usually processes the image luminance

channel, often calculated as the Y channel of the YUV

colorspace. For a pixel with a given RGB representa-

tion p = (R, G,B)

the value of Y is (Koschan and

Abidi, 2008):

Y = 0.299R + 0.587G +0.114B. (1)

Alternative luminance channels are found in HSV ,

HSL, and Lab colorspaces or custom deﬁniti-

ons (Bani

c and Lon

cari

c, 2014b; Nguyen and Brown,

2017). If the luminance value L of p is tone mapped

to L

, then p is changed to

p =





. (2)

If a TMO processes intensities based only on

their value and regardless of their location, then it

is a global TMO, otherwise it is a local one. Ex-

amples of global TMOs include application of Ste-

ven’s law (Tumblin and Rushmeier, 1993; Chiu et al.,

1993; Ward, 1994), imitation of human response to

light (Schlick, 1995; Pattanaik et al., 2000; Drago

et al., 2003; Reinhard and Devlin, 2005), histogram

adjustment (Larson et al., 1997), and sigmoidal con-

trast enhancement (Braun and Fairchild, 1999). For

local TMOs examples include application of aniso-

tropic diffusion (Tumblin and Turk, 1999), bilateral

ﬁltering of the image base layer (Durand and Dorsey,

2002), photographic practice (Reinhard et al., 2002),

luminance gradient ﬁeld manipulation (Fattal et al.,

2002; Mantiuk et al., 2006), Retinex theory (Mey-

lan and Susstrunk, 2006; Bani

c and Lon

cari

c, 2014a;

Bani

c and Lon

cari

c, 2016). Global TMOs are fas-

ter and usually simpler, but local TMOs tend to give

results of higher quality (Kuang et al., 2004; Kuang

et al., 2007; Urbano et al., 2010). The problem with

many TMOs that produce high-quality results is that

they are too slow to be used in real-time applications.

In this paper a new TMO is proposed whose steps

are primarily designed to achieve high speed and to

be practically implementable. Under this constraint

Bani

c, N. and Lon

cari

c, S.

Flash and Storm: Fast and Highly Practical Tone Mapping based on Naka-Rushton Equation.

DOI: 10.5220/0006621600470053

In Proceedings of the 13th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2018) - Volume 4: VISAPP, pages

47-53

ISBN: 978-989-758-290-5

the secondary goal is to produce low dynamic range

(LDR) images of high quality. The proposed TMO

is based on Naka-Rushton equation used in combi-

nation with additional improvements and it has O(1)

per-pixel complexity. The presented and discussed re-

sults show that, beside being faster and more practi-

cal, the proposed TMO outperforms many state-of-

the-art TMOs in terms of resulting LDR image qua-

lity. To further demonstrate its practicality and sim-

plify its usage by interested parties, the publicly avai-

lable source code of the proposed TMO is written in

several programming languages including JavaScript

with an HTML interface so that it can be used even

with only a browser.

The paper is structured as follows: in Section 2

the foundations for the new TMO are laid, in Secti-

ons 3 the TMO is extended to a local one, in Section 4

experimental results are presented and discussed, and

Section 5 concludes the paper.

2 FLASH - STARTING

GLOBALLY

2.1 Initial Tone Compression

Recently a high quality two-phase TMO called Puma

with O(1) per-pixel complexity has been propo-

sed (Bani

c and Lon

cari

c, 2016). During its ﬁrst phase

the actual tone compression is performed globally by

means of a method called Flash. It is fast and ef-

ﬁcient since it uses a simple to calculate curve, but

the resulting LDR image is crude and of low quality.

The second phase consists of enhancing this image by

applying Smart Light Random Memory Sprays Re-

tinex (SLRMSR) (Bani

c and Lon

cari

c, 2015; Bani

and Lon

cari

c, 2017), a brightness adjustment method,

whose parameters are set to speciﬁcally chosen values

for this purpose. SLRMSR produces the ﬁnal high

quality LDR image and it has O(1) per-pixel com-

plexity. Despite good overall theoretical complexity,

Puma is not especially fast due to a large constant

number of steps per pixel used in SLRMSR. Additio-

nally, SLRMSR has several parameters that need to be

tuned for various tone mapping effects. Hence, Puma

does not fully abide by the constraints laid out in the

introduction. Nevertheless, if SLRMSR is left out,

Flash can still be used as a good foundation for a de-

sired new TMO because of its effectiveness, while for

the enhancement of its results a more efﬁcient met-

hod has to be found. The core of Flash is the Naka-

Rushton equation (Shapley and Enroth-Cugell, 1984)

given in the form

+ a

L + aL

(3)

where L is the initial luminance, a is a scaling pa-

rameter, and L

is the image key, which is usually

approximated by calculating the geometric luminance

mean (Ward, 1994; Reinhard et al., 2002)

= exp

∑

ln(L(i) + ε)

(4)

where N is the number of pixels, L(i) is the i-th pixel

luminance, and ε is a small value to avoid logarithm of

zero. By being a special case of the perceptually ba-

sed Michaelis-Menten equation (Dowling, 1987), the

Naka-Rushton equation is to some degree inherently

also perceptually based, which makes Flash theoreti-

cally sound and even more attractive. To obtain high

quality results, Flash uses the V channel of the HSV

colorspace as the luminance channel where V is cal-

culated as V = max{R, G, B}. Flash

denotes the ap-

plication of Flash for a given value of a.

2.2 Leap - Simple Image Enhancement

The next step is to enhance the crude results of Flash

in a way that abides by the constraints laid out in

the introduction. A good starting point in designing

such an enhancement procedure partially motivated

by (Huang and Mumford, 1999) is to look for simple

properties common to many tone mapped LDR ima-

ges of high quality. Once some of these properties

are found, their incorporation into the results of Flash

should hopefully increase their quality.

The set of high quality tone mapped LDR ima-

ges needed to look for such properties was created

by turning the HDR images available at (NTUST,

2015) and (Fairchild, 2015) into LDR images by

using the procedure described in (Ma et al., 2015).

This procedure takes an initial LDR version of a gi-

ven HDR image and changes it iteratively in order to

increase the value of TMQI-II (Ma et al., 2015), an

adapted version of Tone Mapped image Quality In-

dex (TMQI) (Yeganeh and Zhou, 2013) that evaluates

structural ﬁdelity and statistical naturalness of a tone

mapped image by using the initial HDR image as a

reference. Although the procedure is far too slow to

be used in practical real-time systems, its ﬁnal results

are by deﬁnition of high quality. The procedure was

iteratively applied to HDR images until the TMQI-II

value of an image was very close to the maximum of

1 or the number of iterations reached 500.

The analysis of different properties across all of

the obtained high quality LDR images has shown that

the arithmetic mean of the grayscale image values is

VISAPP 2018 - International Conference on Computer Vision Theory and Applications

a stable property with a low standard deviation over

the images. For the ﬁrst dataset (NTUST, 2015) the

mean of the images’ grayscale means was 100.57 with

a standard deviation 3.85 and for the second data-

set (Fairchild, 2015) these values were 100.04 and

5.12, respectively. This suggests that an appropriate

mean grayscale value could also assure a higher qua-

lity. One of the simplest ways to achieve this is to

multiply a given image by a scalar in order to obtain

a desired mean grayscale value i.e. to jump from the

initial mean grayscale to a given one. In Section 4.2

it is shown that such a procedure may indeed signi-

ﬁcantly increase the image quality of a given LDR

image. For easier notation this procedure is named

Leap

with g being the target mean grayscale. For

some low key images their high intensity pixels may

appear ”burned” after increasing their brightness even

further by applying Leap, but as explained in (Rein-

hard et al., 2002), this can actually be desirable. As a

matter of fact, in (Reinhard et al., 2002) an extension

of Eq. (3) has been proposed for exactly that purpose.

The pseudocode for Leap is given in Algorithm 1. In

the Section 4.2 it is shown that Flash combined with

Leap is a fast and high quality global TMO.

After carrying out subjective assessment of a large

number of LDR images obtained by applying Flash

and Leap, it was concluded that round default values

for a and g that already produce visually appealing

results are 10 and 110, respectively.

Algorithm 1: Leap.

Input: image I, target mean grayscale g, upper

bound U

1: m = CalculateMeanGrayscale(I)

2: for all pixel i in I do

3: p = I(i)

4: p

= max





5: p

= max





6: p

= max





7: R(i) = p

8: end for

Output: image R

3 STORM - CONTINUING

LOCALLY

3.1 Extension to a Local TMO

Global TMOs are fast, but for results of the highest

quality local TMOs are used (Kuang et al., 2004; Ku-

ang et al., 2007). Hence a natural direction of further

Flash improvement is to extend it to a local version.

One of the common approaches for such an extension

is to perform tone mapping with pixel-based parame-

ters, which are determined by looking only at the local

area surrounding a given pixel instead of at the whole

image (Reinhard et al., 2010). There is evidence that

early stages of visual processing can be modelled by

ﬁltering of retinal image using ﬁlters of different sca-

les (Wilson, 1991) and similar approaches have been

used in various tone mapping and image enhancement

methods (Peli, 1990; Jobson et al., 1997; Pattanaik

et al., 1998). Thus the local extension of Flash propo-

sed here does the same thing. If R and C are the num-

ber of image rows and columns, respectively, then let

d be the smaller of these two i.e. d = min{R,C}. For

a pixel i Flash is applied to M squares of size s

d ×s

with pixel i in their center where s

is the scaling fac-

tor for the j-th square. If k

is the j-th square and

(

)

(i) the value of luminance of pixel i after ap-

plying Flash

to k

with i in its center, then the ﬁnal

luminance value for i is obtained as

S(i) =

∑

j=1

(

)

(i). (5)

The main computation cost of Eq. (5) is in calcula-

ting Eq. (4) for each i and k

, which is effectively

done by means of convolution over the luminance lo-

garithms. Such an approach was motivated by the

multiscale Retinex (MSR) algorithm (Jobson et al.,

1997), but there are two differences. First, while

MSR uses Gaussian kernels, having square kernels in

Eq. (5) allows faster ﬁltering by using a single inte-

gral image (Crow, 1984), which also brings the per-

pixel complexity of Eq. (5) to O(M). Second, unlike

in MSR, there are no weights in Eq. (5) for individual

kernels because using them has shown no signiﬁcant

beneﬁt in terms of quality. The contribution of vari-

ous kernels is shown in Fig 1. Like with Flash, higher

values of a for Storm give better contrast in the ﬁnal

image as it is shown in Fig. 2. Since the proposed

local TMO applies Flash multiple times and multiple

ﬂashes often occur during storms, it is named Storm

for easier notation, while its application with n speci-

ﬁed scaling factors and value of a for the underlying

Flash is denoted Storm

,...,s

)

. The pseudocode for

Storm is given in Algorithm 2.

3.2 Properties

Fig. 1 shows how more kernels make more details vi-

sible. With well chosen kernels Storm outperforms

Flash in terms of quality, but at the cost of the additi-

onal memory that is needed for the integral image.

In terms of complexity, Storm is also more com-

plex than Flash. Firstly, Storm calculates the integral

Flash and Storm: Fast and Highly Practical Tone Mapping based on Naka-Rushton Equation

(a) (b) (c) (d) (e)

Figure 1: Applications of Storm with different kernels followed by and Leap

110

and gamma correction: (a) Storm

(1)

(b) Storm

(

)

, (c) Storm

(

)

, (d) Storm

(

)

, and (e) Storm

(

256

)

(a) (b)

Figure 2: Results of application of (a) Storm

(

)

and (b)

Storm

(

)

to the same image.

Algorithm 2: Storm.

Input: image I, brightening parameter a, M kernel

sizes s

, upper bound U

1: for all pixel i in I do

2: p = I(i)

3: L = max{p

, p

}

4: L

= 0

5: for j = 1 to M do

6: L

= L

+ F

(

)

(i)

7: end for

8: R(i) =

9: end for

10: R = U · R/max(R(:))

Output: image R

image in O(1) per-pixel complexity. Next, to each

pixel Eq. (3) is applied M times and each time in O(1)

per-pixel complexity by using the integral image. To-

gether this gives O(M) per-pixel complexity, which

seemingly violates the constraint set up in the intro-

duction. However, M is supposed to be very small

because already for M ∈ {2,3,4} the results are of

high quality. Additionally, since M can be considered

a hyperparameter, once its value is chosen, it does not

change that often so it is effectively a constant. Thus,

taking all this into account, Storm can also be regar-

ded as having O(1) per-pixel complexity. Speed tests

in Section 4.2 also corroborate such reasoning.

After carrying out subjective assessment of a large

number of LDR images obtained by applying Storm

and Leap, it was concluded that the combination with

round default values that already produce visually

appealing results without showing unrealistically too

much details is Storm

(

)

+Leap

110

4 EXPERIMENTAL RESULTS

4.1 Image Quality Metrics

The best way to the assess the quality of tone map-

ped images would be to carry out subjective percep-

tual studies. However, such studies were omitted here

because they are time-consuming, they require spe-

cial environment and carefully calibrated equipment,

and they are not easy to reproduce. Instead, two

objective measures are used: Feature Similarity In-

dex For Tone-Mapped images (FSITM) (Ziaei Naf-

chi et al., 2015) and the already mentioned TMQI.

TMQI-II was not used here because while images

with a high TMQI-II are usually of high quality, a

lot of images with a low TMQI-II are actually of

high quality as well. This is because TMQI-II is

too susceptible to mean grayscale value as shown

in (Bani

c and Lon

cari

c, 2016). FSITM is based on

local phase information of images and it was shown

VISAPP 2018 - International Conference on Computer Vision Theory and Applications

(a) (b) (c)

Figure 3: Pairwise comparison between the results of Reinhard’s local TMO (Reinhard et al., 2002) on the left and

Storm+Leap on the right; for Reinhard’s TMO its Luminance HDR implementation with default parameter values was used.

Table 1: Mean TMQI and FSITM

TMQI obtained on images from (Ward, 2015) with cumulative execution time.

TMO TMQI FSITM

TMQI t(s)

Ashikhmin (Ashikhmin, 2002) 0.6620 0.7338 225.23

Drago (Drago et al., 2003) 0.7719 0.8158 30.69

Durand (Durand and Dorsey, 2002) 0.8354 0.8405 225.14

Fattal (Fattal et al., 2002) 0.7198 0.7810 64.78

Mantiuk (Mantiuk et al., 2006) 0.8225 0.8266 88.03

Mantiuk (Mantiuk et al., 2008) 0.8443 0.8494 36.20

Pattanaik (Pattanaik et al., 2000) 0.6813 0.7635 46.91

Reinhard (Reinhard et al., 2002) 0.8695 0.8581 33.41

Reinhard (Reinhard and Devlin, 2005) 0.6968 0.7679 30.01

Flash

0.8072 0.8315 21.19

Flash

+Leap

110

0.8755 0.8625 21.26

Storm

(

)

0.7675 0.8004 24.35

Storm

(

)

+Leap

110

0.8782 0.8551 24.59

to give better results than TMQI. If combined with

TMQI, it performs a better assessment than both

TMQI and TMQI-II and this combination is deno-

ted as FSITM

TMQI where C is a color channel. In

this paper the green (G) color channel is used since it

was shown to give good results (Ziaei Nafchi et al.,

2015). All these measures are in range [0,1] with a

higher number meaning higher quality. These metrics

are well established, easily reproducible, they have a

sound theoretical background, their values can be cal-

culated fast and automatically, and, all of them were

shown to be well correlated with subjective measures.

4.2 Tone Mapping Quality and Speed

The quality of the results of the proposed TMO was

evaluated by applying them to images in the HDR da-

taset given at (Ward, 2015). For TMOs with ﬁxed

parameters this dataset is challenging since it con-

tains HDR images from different sources including

artiﬁcially generated ones. Table 1 shows the obtai-

ned values of objective quality measures for the pro-

posed and existing TMOs with default parameter va-

lues. For results of other TMOs the open source Lu-

minance HDR software was used like in (Ma et al.,

2015). This was also an opportunity to compare the

proposed TMOs to easily available implementations

of other well-known TMOs. The results for Flash dif-

fer from the ones given in (Bani

c and Lon

cari

c, 2016)

because here the gamma correction was carried out

after tone mapping as it is supposed to be done. The

values of objective metrics are higher for Flash and

Storm than for all other tested TMOs and although

the differences are small, they still demonstrate the

quality of the proposed TMOs. Since Reinhard’s lo-

cal TMO (Reinhard et al., 2002) is considered to be

among the best, in Fig. 3 its results are compared to

the results of Storm.

Table 1 also shows that Flash and Storm are faster

than other TMOs, which makes them good candidates

for real-time applications.

5 CONCLUSIONS

A new TMO has been proposed in a global and local

variant under the constraints of O(1) per-pixel com-

plexity and a practical design. Both variants were

Flash and Storm: Fast and Highly Practical Tone Mapping based on Naka-Rushton Equation

shown to outperform state-of-the-art TMOs in terms

of resulting LDR image quality even though obtaining

high quality was only their secondary goal. Another

important fact is that the proposed TMO was shown

to be signiﬁcantly faster than other TMOs. Its success

demonstrates how having implementation constraints

during the design phase of a TMO can nevertheless

lead to fast and practical TMOs that also produce re-

sults of the highest quality. Additionally, the steps

of the proposed TMO have their roots in the results

of perceptual experiments. Future work will consider

solutions that are less dependant on structures such as

integral images in order to decrease TMO’s memory

consumption and to make it more hardware-friendly.

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