Transformation of the Beta Distribution for Color Transfer

Hristina Hristova, Olivier Le Meur, R

emi Cozot and Kadi Bouatouch

University of Rennes 1, 263 Avenue G

eral Leclerc, 35000 Rennes, France

Keywords:

Beta Distribution, Bounded Distributions, Color Transfer, Transformation of Bounded Distributions.

Abstract:

In this paper, we propose a novel transformation between two Beta distributions. Our transformation progres-

sively and accurately reshapes an input Beta distribution into a target Beta distribution using four intermediate

statistical transformations. The key idea of this paper is to adopt the Beta distribution to model the discrete

distributions of color and light in images. We design a new Beta transformation which we apply in the context

of color transfer between images. Experiments have shown that our method obtains more natural and less

saturated results than results of recent state-of-the-art color transfer methods. Moreover, our results portray

better both the target color palette and the target contrast.

1 INTRODUCTION

The Gaussian distribution is a well-known and well-

studied continuous unbounded distribution with many

applications to image processing. The Gaussian

distribution is commonly adopted to ﬁt the distri-

butions of various image features, such as color

and light (Reinhard et al., 2001). The analyti-

cally tractable function and relative simplicity of

the Gaussian distribution reveal its signiﬁcance to

problems like transportation optimization (Olkin and

Pukelsheim, 1982), color correction for image mo-

saicking (Oliveira et al., 2011), example-based color

transfer (Faridul et al., 2014), etc.

Color transfer between images has raised a lot

of interest during the past decade. Color transfer

transforms the colors of an input image so that they

match the color palette of a target image. Color trans-

fer applications include image enhancement (Hristova

et al., 2015), time-lapse image hallucination (Shih

et al., 2013), example-based video editing (Bonneel

et al., 2013; Hwang et al., 2014), etc. Color trans-

fer is often approached as a problem of a transfer of

distributions, where the Gaussian distribution plays

a signiﬁcant role. Early research works on color

transfer assume that the color and light distributions

of images follow a Gaussian distribution. This as-

sumption has proved beneﬁcial for computing sev-

eral global Gaussian-based transformations (Reinhard

et al., 2001; Piti

e and Kokaram, 2006). However,

those global color transformations may produce im-

plausible results in cases when the Gaussian model

is not accurate enough. To tackle this limitation, im-

age clustering has been incorporated into the frame-

work of color transfer methods. A number of local

color transfer methods (Tai et al., 2005; Bonneel et al.,

2013; Hristova et al., 2015) adopt more precise mod-

els, such as Gaussian mixture models (GMMs), and

cluster the input and target images into Gaussian clus-

ters. This approach signiﬁcantly improves the results

of the color transfer.

So far, color transfer have been limited to

Gaussian-based transformations. Despite the fact that

color and light in images are bounded in a ﬁnite inter-

val, such as [0, 1], they are still modelled using the un-

bounded Gaussian distribution. Indeed, the Gaussian

distribution is commonly preferred over other types of

distributions thanks to its beneﬁcial analytical proper-

ties and its simplicity. Unfortunately, performing a

Gaussian-based transformation between bounded dis-

tributions may result in out-of-range values. Such val-

ues are simply cut off and eliminated, causing over-

/under-saturation, out-of-gamut values, etc., as shown

in results (a) and (b) in ﬁgure 1. Furthermore, as

a symmetrical distribution, the Gaussian distribution

cannot model asymmetric distributions. In practice,

the majority of the light and color distributions of im-

ages are left- or right-skewed, i.e. asymmetric. This

reveals an important limitation of the Gaussian model

when applied to image processing tasks and, in par-

ticular, to color transfer. To tackle these limitations of

the Gaussian-based transformations, in this paper we

adopt bounded distributions, and more speciﬁcally,

the Beta distribution. Figure 2 illustrates the beneﬁt

of using a bounded Beta distribution to model color

112

Hristova, H., Meur, O., Cozot, R. and Bouatouch, K.

Transformation of the Beta Distribution for Color Transfer.

DOI: 10.5220/0006610801120121

In Proceedings of the 13th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2018) - Volume 1: GRAPP, pages

112-121

ISBN: 978-989-758-287-5

Input image

Target image

(a) Reinhard et al. (b) Pitie et al.

Figure 1: Results (a) and (b) are obtained using two global color transfer methods (Reinhard et al., 2001; Piti

e and Kokaram,

2006). The input image consists of two lightness clusters (corresponding to the two main peaks in its lightness histogram),

whereas the target image is composed of three color clusters (as shown in its lightness-hue plot). Due to this fact, the global

color transformations fail to produce plausible results. Result (a) does not match the target colors, whereas result (b) is

signiﬁcantly over-saturated, which compromises its photo-realism. Result (c) is obtained using a local color transfer (Hristova

et al., 2015), which increases signiﬁcantly the quality of the color transfer and the result.

and light.

The Beta distribution is a bounded two-parameter-

dependent distribution, which can admit different

shapes and thus, ﬁt various data, bounded in a discrete

interval. Adopting the Beta distribution to model

color and light distributions of images is our key idea

and motivation.

In this paper, we propose a novel transformation

between two Beta distributions. Our transformation

consists of four intermediate statistical transforma-

tions which progressively and accurately reshape an

input Beta distribution into a target Beta distribution.

We apply our Beta transformation both globally and

locally in the context of a color transfer between im-

ages. The results, obtained using the proposed Beta

transformation, appear more natural and less saturated

than results of recent state-of-the-art methods. Ad-

ditionally, our results represent accurately the target

color palette and truthfully portray the target contrast.

The rest of the paper is organized as follows. Sec-

tion 2 presents state-of-the-art color transformations.

Section 3 introduces our Beta transformation. Results

from applying the proposed Beta transformation in

the context of color transfer are shown in section 4.

The ﬁnal section concludes the paper.

Figure 2: The right-hand plot illustrates the lightness his-

togram of the left-hand image as well as two approxima-

tions, namely Gaussian and Beta. The Gaussian distribution

provides a poor approximation of the asymmetric lightness

distribution. In contrast, the Beta distribution models the

image lightness more accurately by accounting for its right-

skewness.

2 RELATED WORK

This paper presents a novel Beta transformation ap-

plied to color transfer between images. Therefore,

hereafter we discuss existing transformations related

to color transfer. Color transfer methods are gener-

ally classiﬁed into statistical methods and geometric

methods (Faridul et al., 2014). In this paper, we focus

on statistical color transformations as they are more

general (i.e. can easily be applied to a variety of image

pairs), content-independent and do not require user

assistance.

Statistical color transformations modify speciﬁc

image features, such as color histograms and distribu-

tions. The very ﬁrst statistical color transfer method

is the non-parametric histogram matching. The lat-

ter transfers the target color palette to the input im-

age by matching the input and target cumulative den-

sity functions. The full histogram transfer often re-

sults in visual artifacts. To resolve this problem,

Pouli et al. (Pouli and Reinhard, 2010; Pouli and

Reinhard, 2011) propose two methods for partial his-

togram matching at different scales.

Furthermore, a number of parametric color trans-

formations have been proposed. They are computed

using the Gaussian distribution. In this sense, Rein-

hard et al. (Reinhard et al., 2001) are the ﬁrst to as-

sume that each color channel of the input and target

images can be well-described by a univariate Gaus-

sian distribution. They introduce a simple color map-

ping between two Gaussian distributions, which is ap-

plied between each pair of input/target color channels.

This method proves to be effective for natural scenes

but it may introduce false colors in the result.

Reinhard et al.’s transformation is a one-

dimensional transformation which does not take the

correlation between the channels of the color space

into account. To overcome this drawback, Piti

et al. (Piti

e and Kokaram, 2006) propose a three-

dimensional (3D) color transformation. Similarly to

Reinhard et al., Piti

e et al. describe the 3D color

Transformation of the Beta Distribution for Color Transfer

113

distributions of the input and target images using

the multivariate Gaussian distribution (MGD). This

helps for their color mapping to be computed as

a closed-form solution to the well-known Monge-

Kantorovich’s optimization problem (Evans, 1997).

Piti

e et al.’s color transformation is more robust than

Reinhard et al.’s, as the former minimizes the dis-

placement cost of the color transfer, preventing color

swaps and preserving the intended geometry of the

result.

Despite the efﬁciency of Piti

e et al.’s method, the

global Gaussian assumption may become too strong

to ensure a good color transfer, as shown in ﬁg-

ure 1. To improve the quality of the color transfer, lo-

cal color transfer methods have been introduced (Tai

et al., 2005; Bonneel et al., 2013; Hristova et al.,

2015). Such local methods ﬁrst partition the input and

target images into several clusters which can be ﬁtted

by an MGD. The clustering is commonly performed

using GMMs (Tai et al., 2005; Hristova et al., 2015).

After the image clustering, either Reinhard et al.’s or

Piti

e et al.’s color transformations are carried out be-

tween the corresponding input/target clusters. The re-

sults of the local color transfer methods strongly de-

pend on how the target clusters are mapped to the in-

put ones. For instance, Bonneel et al. (Bonneel et al.,

2013) propose a mapping function based only on the

lightness of the input and target images. Hristova et

al. (Hristova et al., 2015) go further and introduce four

new mapping policies, which are functions of both the

lightness and the chroma of the input and target im-

ages.

Local color transfer methods adopt more pre-

cise distribution models, i.e. GMMs, to ﬁt the input

and target color distributions, which improves signif-

icantly the results of the color transfer, as shown in

ﬁgure 1. Evidently, this indicates that the more ac-

curate the adopted distribution model, the better the

results of the color transfer. We believe that we can

further improve the precision of the model by using

the Beta distribution. The Beta distribution can ac-

count for the skewness of the color and light distri-

butions. Moreover, the Beta distribution, which is

bounded, is a more appropriate model for the discrete

color and light distributions of images than the con-

tinuous Gaussian distribution. The following section

presents our novel transformation between two Beta

distributions, which is later applied to color transfer

between images.

3 BETA TRANSFORMATION

The present section consists of two parts: a ﬁrst

part presenting both the Beta distribution and well-

known statistical transformations of the Beta distribu-

tion, and a second part introducing our Beta transfor-

mation.

3.1 Beta Distribution

First, we present the density function of the Beta

distribution as well as several well-known statistical

transformations which play an important role in the

derivation of our Beta transformation.

3.1.1 Density Function

The density function f (·) of a Beta distributed ran-

dom variable x with shape parameters α, β > 0 (de-

noted x ∼ Beta(α,β)) is given as follows:

f (x) =

B(α,β)

α−1

(1 −x)

β−1

, (1)

where x ∈ [0,1], and B(α,β) denotes the Beta func-

tion. The Beta distribution is a univariate distribution.

The variable x, distributed according to the Beta law,

can be directly transformed into a Fisher variable us-

ing a simple statistical transformation, as presented

hereafter.

3.1.2 Beta-Fisher Relationship

Let x ∼ Beta(α,β). Then, a variable y, obtained as

y = f

(x,α,β), where function f

(·) is deﬁned as

follows:

(x,α,β) =

βx

α(1 −x)

, (2)

is a Fisher variable with shape parameters 2α and

2β (denoted y ∼ F (2α, 2β)). Equation (2) maps the

bounded interval [0, 1] into the semi-bounded interval

[0, ∞) with lim

x→1

y = ∞.

Reversely, a variable z = f

(y,α,β), where

(·) is obtained by the following formula:

(y,α,β) =

αy

β + αy

, (3)

and y = f

(x,α,β), is a Beta variable, e.g. z ∼

Beta(α,β). Equation (3) maps the semi-founded in-

terval [0, ∞) into the bounded interval [0, 1] with

lim

y→∞

z = 1.

GRAPP 2018 - International Conference on Computer Graphics Theory and Applications

114

u~Beta(α

, β

)

Beta-to-Fisher:

:= T

(u)

~F(2α

,2β

)

Fisher-to-Beta:

g := T

)

g~Beta(α

,β

)

v~Beta(α

, β

)

Fisher-to-Standard-Normal:

:= T

)

~N(0,1)

Standard-Normal-to-Fisher:

:= T

(s)

~F(2α

,2β

)

Input image u

Target image v

Output image g

Figure 3: Our Beta transformation consists of four intermediate transformations (steps). The ﬁrst two steps progressively

transform the input distribution into a standard normal distribution. Then, using the target distribution, the last two trans-

formations reshape the standard normal distribution into a Beta distribution with shape parameters close to the target shape

parameters. For each step of our method, the ﬂowchart shows the transformed distribution and its corresponding approxima-

tion. For illustration’s sake, the input and target distributions are extracted from the lightness channels of two images. The

same sequence of steps can be applied independently to each of the two chroma channels of the input and target images. The

ﬂowchart illustrates one iteration of our method.

3.2 Transformation of the Beta

Distribution

Hereafter, we introduce our one-dimensional (1D)

Beta transformation (it is a transformation between

two univariate Beta distributions). Let u and v be 1D

input and target random variables, following a Beta

distribution, i.e. u ∼ Beta(α

, β

) and v ∼ Beta(α

). We aim to transform variable u so that its dis-

tribution becomes similar to the target distribution

(i.e. the distribution of v).

Transforming one Beta distribution into another

one in a single pass could be challenging. To the

best of our knowledge, no such transformation has yet

been proposed. In this paper, we progressively trans-

form the input Beta distribution by beneﬁting from

the potential of well-known statistical transformations

and approximations. We propose a transformation of

the Beta distribution, consisting of four intermediate

transformations, as follows:

1. Transformation of the input variable u into a

Fisher variable f

∼ F (2α

,2β

);

2. Transformation of the Fisher variable f

into a

standard normal variable s

;

3. Transformation of the standard normal variable s

into a Fisher variable f

∼ F (2α

,2β

);

4. Transformation of the Fisher variable f

into a

Beta variable g ∼ Beta(α

,β

The key idea of our transformation consists in

transforming the input Beta distribution into a stan-

dard normal distribution. The standard normal dis-

tribution can then be easily reshaped using the shape

parameters of the target Beta distribution. However,

the Beta distribution cannot be directly transformed

into a standard normal distribution. To this end, we

ﬁrst perform an intermediate transformation to com-

pute a Fisher distribution from the input Beta distri-

bution. Then, we use the computed Fisher distribu-

tion to approximate the standard normal distribution.

The output of the proposed transformation is the Beta

variable g which has a distribution similar to the tar-

get Beta distribution. The steps of our transformation

are illustrated in ﬁgure 3 and described hereafter.

Transformation of the Beta Distribution for Color Transfer

115

3.2.1 Beta-to-Fisher

During the ﬁrst step of our transformation, we trans-

form the input Beta variable u into a Fisher variable

= T

(u) with shape parameters 2α

and 2β

follows:

: u → f

(u,α

,β

), (4)

where function f

(·) is deﬁned in (2). Once we have

obtained the Fisher variable f

, we transform it into a

standard normal variable s

in the second step of our

transformation.

3.2.2 Fisher-to-Standard-Normal

The transformation of the Fisher variable f

, obtained

during the ﬁrst step of our method, is based on Paul-

son’s equation (Paulson, 1942; Ashby, 1968).

In general, Paulson’s equation transforms a given

Fisher variable f ∼ F (α, β) into a standard normal

variable s ∼ N(0,1) as follows (see appendix A for

a detailed derivation of Paulson’s equation):

s = f

(f,µ

,µ

,σ

, p) =

−µ

+ σ

, (5)

with σ

= f

(α, p), σ

= f

(β, p), µ

= f

(σ

) and

= f

(σ

), where p ∈N, and the functions f

(·) and

(·) are deﬁned as follows:

(γ, p) =

γp

and f

(σ) = 1 −σ

. (6)

We adopt Paulson’s equation (5) to transform the

Fisher variable f

∼ F (2α

,2β

), computed with

transformation (4), into a standard normal variable

= T

). The transformation T

is deﬁned as fol-

lows:

: f

→ f

,µ

,σ

, p), (7)

where p ∈ N and (σ

)

= f

(2α

, p), (σ

)

(2β

, p), µ

= f

(σ

) and µ

= f

(σ

). Function

is deﬁned in (5).

3.2.3 Standard-Normal-to-Fisher

Once we have computed the standard normal variable

, we inverse Paulson’s equation (5) to transform s

into a Fisher variable f

. We carry out the inversed

Paulson’s equation using the target shape parameters

and β

instead of the input shape parameters α

and β

Let α and β be any two shape parameters. We

ﬁrst present the inversed Paulson’s equation for trans-

forming any standard normal variable s into a Fisher

variable f ∼ F (2α,2β):

−µ

+ 2µ

+ s

−µ

= 0, (8)

where σ

= f

(2α, p), σ

= f

(2β, p), µ

= f

(σ

)

and µ

= f

(σ

). The solutions of the inversed Paul-

son’s equation are derived in appendix B.

Now, we solve the inversed Paulson’s equation (8)

using the target shape parameters α

and β

. In (8),

we replace s by s

(computed with transformation (7))

and the functions σ

, σ

, µ

and µ

by the following

functions respectively: (σ

)

= f

(2α

, p), (σ

)

(2β

, p), µ

= f

(σ

) and µ

= f

(σ

After solving equation (8) using the aforemen-

tioned parameters, we obtain a Fisher variable f

∼

F (2α

,2β

). Each sample f

of f

is computed using

a transformation T

, i.e. f

= T

) ∀i ∈{1,...,n}:







→



,µ

,σ

,1)



,if s

< 0,

→



,µ

,σ

,−1)



,if s

≥ 0,

(9)

where p ∈N and function f

(·) is deﬁned in equation

(20) (appendix B).

3.2.4 Fisher-to-Beta

In the ﬁnal step of our transformation, we transform

the Fisher variable f

into a Beta variable g using the

following transformation T

: f

→ f

,α

,β

), (10)

where function f

(·) is deﬁned in (3). The variable

g = T

) is approximately distributed according to

a Beta distribution with shape parameters α

and β

i.e. its distribution is similar to the target distribution.

3.2.5 Choice of p

The inverse Paulson’s equation (8) has two solutions

(as shown in appendix B), which can be both negative

and positive, depending on the parameters µ

, µ

, σ

and σ

. In contrast, the values of the variables, dis-

tributed according to the Fisher law, are non-negative.

To ensure that each component f

of the Fisher vari-

able f

is non-negative (see (9)), we make the param-

eter p equal to 4, following Hawkins et al.’s proposi-

tion (Hawkins and Wixley, 1986). By choosing p = 4,

we also make sure that Paulson’s equation (5) holds

for small values of the shape parameters α

and β

as discussed in (Hawkins and Wixley, 1986).

3.2.6 Iterations

For the sake of robustness, our Beta transformation

can be performed iteratively using a dynamic input

variable. In this case, at the beginning of each itera-

tion (after the ﬁrst one) the input variable is updated

GRAPP 2018 - International Conference on Computer Graphics Theory and Applications

116

using the resulting variable from the previous itera-

tion, i.e.:

(t)

(

Beta

(u,v),if t = 1;

Beta

(t−1)

,v),if t ∈ [2,N],

(11)

where g

(t)

denotes the result after iteration t, T

Beta

de-

notes our Beta transformation (which is a combina-

tion of the four steps, presented earlier in this section)

and N denotes the number of iterations. The beneﬁt of

iterating the proposed transformation and the choice

of an appropriate value for N are discussed in the fol-

lowing section.

3.3 Evaluation of Beta Transformation

To evaluate the performance of our method, we carry

out our Beta transformation on 111 different pairs of

input and target data samples, which we model using

the Beta distribution. The data samples are extracted

from a collection of images containing both indoor

and outdoor images.

We compute the percentage error between the

shape parameters of each resulting (from the trans-

formation) distribution and the target shape parame-

ters. The Box-and-Whisker plots in ﬁgure 4 show the

distributions of the percentage errors (for 1 and 5 it-

erations of our transformation). After 1 iteration, the

mean percentage error is less than 0.05 and it con-

verges towards 0 after 5 iterations.

The plots in ﬁgure 4 illustrate the beneﬁt of per-

forming our Beta transformation iteratively. After a

single iteration, the mean percentage error is already

signiﬁcantly small. The more we iterate, the smaller

the mean percentage error. For the application, shown

in this paper (i.e. color transfer), our experiments have

indicated that the change in the distribution of the re-

sult g

(t)

becomes negligible after the ﬁfth iteration,

i.e. for t > 5, as illustrated in ﬁgure 5. Therefore, the

results, shown in this paper, are obtained using ﬁve

iterations of our Beta transformation, i.e. N = 5.

1 iteration

0.00 0.05 0.10 0.15

P ercentage error

P ercentage erro r

0e+00 4e−06 8e−06

5 iterations

Figure 4: Box-and-Whisker plots of the percentage errors

for the two Beta shape parameters α and β (computed after

1 and 5 iterations of our transformation).

4 RESULTS

We apply our Beta transformation in the context of

color transfer between input and target images. We

carry out our 1D Beta transformation independently

on each pair of input/target color channels. To this

end, we use CIE Lab, as the color space provides ef-

ﬁcient channel decorrelation. Each of our results in

this paper is obtained using 5 iterations of our trans-

formation. We perform the Beta transformation glob-

ally (i.e. on the distributions of entire images) as well

as locally (i.e. on the distributions of image clusters).

4.1 Global Color Transfer

To perform a global color transfer, we ﬁrst model the

distribution of each input/target channel by a Beta dis-

tribution and then, we carry out our Beta transforma-

tion between the channel distributions. In ﬁgure 6,

our global Beta-based color transfer is compared to

the Gaussian-based 3D linear color transformation by

Piti

e et al. (Piti

e and Kokaram, 2006) (also carried

out in CIE Lab). Target image (a) is a highly bright

image, characterized by a low contrast (i.e. there is an

absence of strongly deﬁned highlights). Similarly, our

ﬁrst result in ﬁgure 6 does not contain highly contrast-

ing regions, i.e. the illumination of our result appears

uniform (much like the illumination of target image

(a)). Moreover, our ﬁrst result has a lower contrast

than the input image but it looks sharper than its cor-

responding result by Piti

e et al. We observe that our

result (a) preserves more details from the input im-

age (note the sharpness of the girl’s face and hair in

our ﬁrst result). In contrast, Piti

e et al.’s ﬁrst result

contains regions of over-saturated pixels, appearing

on the girl’s face, shoulder and ﬂower (snippets (1)

and (2)), causing a loss of details. Such highlights are

uncharacteristic for target image (a) and their pres-

ence increases the contrast of Piti

e et al.’s result.

Furthermore, target image (b) in ﬁgure 6 is char-

acterized by a presence of strong highlights and deep

shadows. Likewise, highly contrasting regions appear

in our result (b), whereas the absence of strong high-

lights in Piti

e et al.’s result (b) inﬂuences the contrast

decrease and makes the image appear ﬂat.

4.2 Local Color Transfer

Instead of modeling the entire distributions of color

and light by Beta distributions, we can build an

even more accurate model using Beta mixture models

(BMMs) (Ma and Leijon, 2009). We use BMMs to

cluster the input and target images according to one

of two components, i.e. lightness or hue. We adopt

Transformation of the Beta Distribution for Color Transfer

117

1 iteration

2 iterations

5 iterations

10 iterations

Figure 5: Results of our Beta transformation (applied iteratively in the context of color transfer between images) for different

number of iterations. Bottom-right corner: images of the observed difference between each result and the result, obtained after

the ﬁfth iteration. As observed, the more we iterate, the smaller the difference. The difference between the result, obtained

after the ﬁfth iteration, and the result, obtained after the tenth iteration, is negligible and therefore, we can assume that our

method converges after the ﬁfth iteration (this conclusion has been drawn using various results of color transfer). The input

and target images for the color transfer are presented in ﬁgure 3.

Input imageTarget image

[Pitie et al., 2007] [Pitie et al., 2007]

Our result Our result

(a)

(b)

(a) (a)

(b) (b)

(1) (2) (1) (2)

(2) (2)

(1) (1)

(2)(2)

Figure 6: Global color transfer. Our results portray the contrast of the target image better than the Piti

e et al.’s results and they

appear sharper than Piti

e et al.’s results. Snippets (1) and (2) illustrate differences between our results and Piti

e et al.’s results

(best viewed on screen).

the classiﬁcation method, proposed in (Hristova et al.,

2015), to determine the best clustering component for

each image. Additionally, we let clusters overlap and

we use the BMMs soft segmentation to compute the

overlapping pixels. Then, we apply the mapping poli-

cies, proposed in (Hristova et al., 2015), to map the

target to the input clusters and we carry out our Beta

transformation between each pair of corresponding

clusters (see (Hristova et al., 2015)). The overlapping

pixels are inﬂuenced by more than one Beta transfor-

mation. To determine the ﬁnal value of an overlap-

ping pixel, we compute the average of all transforma-

tions, containing the pixel, using exponential decay

weights like in (Hristova et al., 2015).

Figure 7 illustrates the beneﬁt of applying a local

color transfer between images. In ﬁgure 7, our global

Beta transformation fails to transfer properly the tar-

get color palette to the input image. This is due to the

fact that the color distributions of the input and target

images in ﬁgure 7 cannot be modelled well enough

by a single Beta distribution. To improve the result of

the global transfer, we apply our Beta transformation

locally, using 2 clusters. That way, BMMs, as more

precise distribution models for the input/target distri-

butions, help to transfer the target colors more accu-

rately. We compare our local method with Bonneel et

al.’s (Bonneel et al., 2013) and Hristova et al.’s (Hris-

tova et al., 2015) local methods, both of which per-

form clustering and carry out Gaussian-based trans-

formations. As shown in ﬁgure 7, Bonneel et al.’s

method fails to match the target ﬂoral color (making

the ﬂower in the result appear less white and slightly

saturated), whereas Hristova et al.’s method causes

overexposure of the foreground pixels in the result.

In contrast, our local result portrays the natural cream

white color of the target rose without overexposing it.

Furthermore, our local Beta transformation can

also inﬂuence the contrast of the result. For instance,

ﬁgure 8 shows that when we apply our Beta trans-

formation globally, we accurately transfer the target

color palette. However, that way we also decrease

the contrast of the result, making it appear unnaturally

under-saturated. Due to the speciﬁc contents of the in-

put/target images in ﬁgure 8, users may expect a more

saturated result with a higher contrast. To preserve the

input contrast, we perform our Beta transformation

locally using two lightness clusters, i.e. highlights and

shadows. That way, we can transfer the target colors

without compromising the input contrast.

As the input and target images in ﬁgure 8 consist

of a single dominant color, the average hue provides

a decent statistic for measuring the similarity of each

GRAPP 2018 - International Conference on Computer Graphics Theory and Applications

118

Our result, 2 clusters

[Bonneel et al., 2013]

[Hristova et al., 2015]

Input image

Target image

Our result, 1 cluster

Figure 7: Local color transfer. The local Beta transformation is more efﬁcient than the global Beta transformation in cases

when the input/target color and light distributions cannot be well-modelled by a single Beta distribution. We use 2 clusters to

obtain a naturally-looking result, matching accurately the target color palette and contrast.

Input image

Target image

[Bonneel et al., 2013]

[Hristova et al., 2015]

Our result, 2 clusters

Our result, 1 cluster

Figure 8: Our local color transfer may signiﬁcantly inﬂu-

ence the contrast of the resulting image. Our global method

accurately transfers the target color palette, but it also de-

creases the contrast, resulting in an unnaturally ﬂat image.

On the other hand, our local Beta transformation preserves

the input contrast.

result to the target color palette. Figure 9 presents a

plot of the lightness-hue distributions of the input and

target images appearing in ﬁgure 8. In the plot of ﬁg-

ure 9, the mean lightness and the mean hue of each

result in ﬁgure 8 are displayed in circles. We observe

that our local result, obtained using two clusters, has

the same mean hue as the target image. In contrast,

Hristova et al.’s result has an out-of-gamut mean hue.

Indeed, a closer visual comparison between our local

result and Hristova et al.’s result (ﬁgure 8) indicates

that our local method transfers the target color palette

more accurately (note the slightly different (from the

target image) shade of blue in Hristova et al.’s result).

The hue statistic supports this observation: our local

result matches the target colors better than Hristova

et al.’s result. Furthermore, the difference between

our local result and the target image in terms of mean

lightness is expected and can be explained by the in-

tentional preservation of the input contrast. Finally,

Our result, 1 cluster

Our result, 2 clusters

[Hristova et al., 2015]

[Bonneel et al., 2013]

Target

Input

Figure 9: Lightness-hue distribution plot of the input and

target images from ﬁgure 8. The left cluster corresponds to

the color distribution of the input image, whereas, the right

cluster corresponds to the color distribution of the target im-

age. The red circle illustrates the mean target lightness and

hue, whereas the dashed red line visualizes the mean target

hue.

more results, obtained with our local Beta transfor-

mation, are presented in ﬁgure 10.

5 CONCLUSION AND FUTURE

WORK

In this paper, we have presented a transformation be-

tween two Beta distributions. Our main idea involved

modelling color and light image distributions using

the Beta distribution (or BMMs) as an alternative to

the Gaussian distribution (or GMMs). We have ap-

plied our Beta transformation in the context of color

transfer between images, though the transformation

can be applied to any bounded data, following the

Beta distribution law. Our results have shown the ro-

bustness of our method for transferring given target

color palette and contrast.

The proposed 1D Beta transformation has been

Transformation of the Beta Distribution for Color Transfer

119

Target image (a) Target image (b)

Result (a) Result (b)

Input image

Figure 10: More results, obtained with our Beta transformation. We apply our Beta transformation locally using two clusters

(either lightness or hue clusters). The mapping between the input/target clusters is carried out using the mapping policies,

proposed in (Hristova et al., 2015).

applied on each input channel separately without ac-

counting for the channels inter-dependency. The lat-

ter highlights the main limitation of our method. To

improve the results of our method, we can extend the

dimensionality of our Beta transformation by consid-

ering the Dirichlet distribution. The Dirichlet distri-

bution (Kotz et al., 2004) is the multivariate general-

ization (for more than two shape parameters) of the

Beta distribution. The Dirichlet distribution with four

shape parameters could be used to model any three-

dimensional data, such as the joint 3D distribution of

color and light in images. Given the promising color

transfer results, obtained using our Beta transforma-

tion, we believe that the Dirichlet distribution would

also prove beneﬁcial to image editing and that is why,

we consider it as a future avenue for improvement.

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APPENDICES

A: Paulson’s Equation

Hereafter, we derive Paulson’s equation (5). First,

we present a well-known transformation of the Fisher

distribution, which plays a key role for the derivation

of Paulson’s equation.

Fisher-Chi-square Relationship. A Fisher variable

can be expressed as a ratio of two Chi-square vari-

ables. Let x and y be two Chi-square random vari-

ables with α and β degrees of freedom respectively

(x ∼ χ

, y ∼ χ

). Then:

x/α

y/β

∼ F (α,β). (12)

Normal-Chi-square Relationship. Let z ∼ N(µ,σ

)

be a normal variable and w ∼χ

be a Chi-square vari-

able with γ degrees of freedom. Then, z can be ex-

pressed as follows (Hawkins and Wixley, 1986):

z =





, (13)

where p ∈ N and µ and σ are functions of γ and p:

= f

(γ, p) and µ = f

(σ). (14)

Functions f

(·) and f

(·) are deﬁned in (6).

Derivation of Paulson’s Equation. We derive Paul-

son’s equation using Fieller’s approximation (Fieller,

1932). Fieller’s approximation transforms the ratio

of two normally distributed variables into a standard

normal variable. Let x ∼N(µ

,σ

) and y ∼ N(µ

,σ

)

be two normally distributed random variables. Then,

Fieller approximation (Fieller, 1932) admits the fol-

lowing form:

s = f

(x,y,µ

,µ

,σ

) =

−µ





+ σ

(15)

Variable s, where s = f

(x,y,µ

,µ

,σ

), is a

standard normal variable.

Paulson’s equation can be derived from (15) by

computing the normal variables x and y using the Chi-

square distribution. In this sense, we express the nor-

mal variables x and y in (15) using equation (13) as

follows:

x = (w

/α)

and y = (w

/β)

, (16)

where w

∼ χ

and w

∼ χ

, and p ∈ N. Then, we

compute a new variable f as follows:

f =

/α

/β

. (17)

From (12), it becomes clear that f is a Fisher variable

with shape parameters α and β, i.e. f ∼F (α,β). Once

we have replaced the ratio x/y in Fieller’s approxima-

tion (15) by its equivalence from equation (17), we

obtain Paulson’s equation (5). Paulson’s equation (5)

transforms a Fisher variable into a standard normal

variable.

B: Inversed Paulson’s Equation

For each pair of shape parameters (α, β) and each

standard normal variable s, the inversed Paulson’s

equation (8) can be solved as a quadratic equation for

t = f

. Let t = (t

,...,t

) and s = (s

,...,s

), where

and s

are the samples of t and s respectively. Then,

the two solutions t

and t

of (8) are computed ∀i ∈

{1,...,n} using a function f

,µ

,σ

,C)

(Ashby (Ashby, 1968) presents the solutions for p =

3):

= t

= f

,µ

,σ

,1), (18)

= t

= f

,µ

,σ

,−1), (19)

where the function f

(·) is deﬁned as follows:

,µ

,σ

,C) =

−µ

√

−µ

(20)

for determinant D: D = s

(σ

+σ

(σ

+σ

−

−4)), and coefﬁcient C = ±1.

Transformation of the Beta Distribution for Color Transfer

121