# Edge-Aware Image Color Appearance and Difference Modeling

Abhinau K. Venkataramanan

###### Abstract

The perception of color is one of the most important aspects of human vision.
From an evolutionary perspective, the accurate perception of color is crucial
to distinguishing friend from foe, and food from fatal poison. As a result,
humans have developed a keen sense of color and are able to detect subtle
differences in appearance, while also robustly identifying colors across
illumination and viewing conditions. In this paper, we shall briefly review
methods for adapting traditional color appearance and difference models to
complex image stimuli, and propose mechanisms to improve their performance. In
particular, we find that applying contrast sensitivity functions and local
adaptation rules in an edge-aware manner improves image difference
predictions.

###### Index Terms:

Color Appearance, Contrast Sensitivity, Edge-Aware Filtering, Image Difference

##  I Introduction

A common notion of the defining property of “color” is generally the
wavelength of visible electromagnetic (EM) radiation. However, such a
description of color lacks a fundamental element - the (human) observer.
Indeed, color is a perceptual phenomenon that is related to, but not entirely
specified by, physical properties of surfaces such as reflectance spectra.
Rather, the properties of the human visual system (HVS) and its response to
the illumination of the environment, the brightness of the color, the presence
of other colors during viewing, etc. play a crucial role in how the color is
perceived. Under appropriately chosen conditions, the “same color”
(physically, in terms of reflectance/emission spectra) may be perceived as
being different, or vice versa.

A classic example of such a phenomenon is the color cube illusion [1], shown
in Figure 1, in which the “brown patch” at the center of the top face and the
“orange patch” at the center of the side face are identical, but appear to be
different. This may be attributed to our perception of the front face of the
cube being in a shadow. Furthermore, Lateral Inhibition (LI) leads to a
reduction in the sensitivity of the HVS to low-frequencies [2], which gives
rise to the characteristically band-pass contrast sensitivity function (CSF).

Therefore, when assessing differences in appearance between two images, it is
important to account for local phenomena, both regarding adaptation to
illumination and the sensitivity of the eye to spatial patterns of different
frequencies. This is precisely the approach taken by Image Color Appearance
[3] and image difference [4] models. In particular, these models apply local
adaptation rules and incorporate models of the CSF to adapt traditional color
appearance models, which have been developed for uniform color patches, to
predict the appearance of, and differences between, local regions in images.

In this paper, we posit that human perceptions of appearance are made somewhat
independently between (distinct regions) of objects. For example, when
applying the CSF to account for the visibility of various artifacts, we aim to
minimize the “leakage” of the influence of the CSF across object boundaries,
i.e., edges. A similar approach is adopted when estimating local adaptation
parameters, thereby reducing the influence of objects on each others’
appearance. Using this, we demonstrate that such “edge-aware” image difference
models better localize differences in the appearances of objects.

The rest of the paper is organized as follows. Section II first presents an
overview of the color difference modeling problem and Section III presents
Uniform Color Spaces (UCSs), which are simple models designed to predict color
differences. Section IV describes standard Color Appearance Models (CAMs),
which build on UCSs to account for how the HVS adapts to illumination
conditions, and Section V describes models of contrast sensitivity, which are
used to account for the change in visibility with spatial frequencies.

Section VI describes the iCAM02 [3] and iDiff [4] models, which are the image
color appearance and difference models that form the basis of this work.
Section VII describes the proposed edge-aware iCAM and iDiff frameworks, and
Section VIII describes the experiments used to validate our method. Finally,
we summarize the work presented in this paper and point out areas for future
work in Section IX.

Figure 1: The color cube illusion

##  II Color Difference Modeling

The simplest question that one can ask regarding the appearance of colors is
whether two colors appear identical. Such questions are answered using
psychophysical experiments, where subjects may be shown two colors and their
responses recorded. Psychometric functions are then used to build models of an
“average” observer, and these functions may depend on various factors such as
the color of the “reference” stimulus, the luminance to which the eye has
adapted, the background against which the stimulus was viewed, and in some
cases, the spatial frequency characteristics of the stimulus.

For instance, such experiments were carried out by [5, 6, 7] to develop models
of “chromatic contrast sensitivity”. In particular, [6] and [7] consider
several “chromatic-directions”, such as varying colors from red to green,
yellow-green to violet, etc. A detailed description of contrast sensitivity
functions is given in Section V

Given that we are able to predict whether two colors appear identical to an
average observer, under given viewing conditions, a more sophisticated task
would then be to quantify the difference between the colors. Such models are
termed as “color-difference” models. Finally, one may also desire “color
appearance models” (CAMs) that directly predict quantities that correlate with
human perception, called “appearance correlates”, such as brightness,
colorfulness, hue, etc. Note that while color-difference models typically use
color appearance models to predict appearance correlates and measure their
difference, this is not a necessary property of color-difference models. For
instance, the $\Delta E_{ab}$ metric computed using the CIELAB color space,
described in Section III, is computed from an opponent color space, and [7]
models a “probability of detection” function that quantifies the difference
between two colors directly from an LMS color space [8] that aims to model
cone responses.

##  III Uniform Color Spaces

While color representations such as $XYZ$ offer a device-independent
representation of color, and $LMS$ values characterize cone responses, neither
representation can be directly used to predict differences between colors. In
particular, (Euclidean) distances between two colors in the $XYZ$ and $LMS$
spaces are not “perceptually uniform”, i.e., equal distances between points do
not correspond to equal perceived differences in color.

This phenomenon is visualized in Figure 2, which shows the $xy$ chromaticity
plane with MacAdam’s ellipses [9] overlaid on top. Each ellipse represents the
set of colors that are “just-noticeably-different” from the color at the
center. Note that the ellipses have been scaled to 10x their size for
visibility. The green region at the top occupies a large region of slowly
varying colors, while colors change much quicker in the red and yellow regions
at the bottom-right.

Figure 2: MacAdam Ellipses on the $xy$ Chromaticity Plane

A Uniform Color Space (UCS) is one in which Euclidean distances between colors
predict the perceptual difference between them. In other words, two pairs of
colors that have the same Euclidean distance between them appear equally
different. The CIELAB model is arguably the most popular UCS in use today.
CIELAB transforms input tristimulus $XYZ$ values to an opponent-color space
[10] using information regarding the achromatic white reference illuminant,
also described using tristimulus values as $\textit{XYZ}_{n}$. The
transformation is carried out by applying a “compressive nonlinearity” $f$ to
the ratios $X/X_{n}$, $Y/Y_{n}$, and $Z/Z_{n}$, followed by transformation to
an opponent color space

$\displaystyle L^{*}$ $\displaystyle=116f\left(\frac{Y}{Y_{n}}\right)-16,$ (1)
$\displaystyle a^{*}$
$\displaystyle=500\left(f\left(\frac{X}{X_{n}}\right)-f\left(\frac{Y}{Y_{n}}\right)\right),$
(2) $\displaystyle b^{*}$
$\displaystyle=200\left(f\left(\frac{Y}{Y_{n}}\right)-f\left(\frac{Z}{Z_{n}}\right)\right),$
(3)

where

$f(x)=\begin{cases}\sqrt[3]{x}&\text{if }t>\delta^{3}\\\
\frac{x}{3\delta^{2}}+\frac{4}{29}&\text{else}\end{cases},$ (4)

and $\delta=6/29$.

CIELAB has proven to be a popular UCS, particularly due to the widespread
adoption of the $\Delta E_{00}$ color difference metric [11]. $\Delta E_{00}$
improves upon a previous version, called $\Delta E_{94}$, and the simple
Euclidean distance

$\Delta
E_{ab}=\sqrt{(L_{1}^{*}-L_{2}^{*})^{2}+(a_{1}^{*}-a_{2}^{*})^{2}+(b_{1}^{*}-b_{2}^{*})^{2}},$
(5)

which is the most straightforward application of the UCS construction.

A cylindrical transformation is typically applied to opponent color
representations to derive other attributes of color. A cylindrical
transformation applied to the CIELAB color space yields the CIELCh color
space, where

$C^{*}=\sqrt{a^{*2}+b^{*2}}$ (6)

denotes the “chroma” or “relative-saturation”, and

$h=\arctan\left(b^{*}/a^{*}\right)$ (7)

denotes the “hue angle”. When comparing two colors, in addition to the total
difference $\Delta E_{00}$, these quantities can be used to predict
differences in lightness, chroma, and hue [12], as

$\Delta L^{*}=L_{1}^{*}-L_{2}^{*}$ (8) $\Delta C^{*}=C_{1}^{*}-C_{2}^{*}$ (9)
$\Delta H=2\sqrt{C_{1}^{*}C_{2}^{*}}\sin\left(\frac{h_{1}-h_{2}}{2}\right)$
(10)

Modern UCS models modify the CIELAB framework by first transforming input
$XYZ$ tristimulus values to an $LMS$ space that predicts cone responses. Then,
a compressive non-linearity is applied to these LMS responses, and a
transformation is carried out to an opponent color space. IPT [13] and OKLab
[14] are two simple modern UCS models that follow this framework. IPT improves
on CIELAB in terms of its hue prediction, while OKLab improves over IPT in
terms of its lightness and chroma predictions.

These approaches have also been extended to high-dynamic range (HDR) colors
that cover a greater range of luminances and a wider gamut of chromaticities.
The typical approach remains the same, with modifications made to the
compressive non-linearities (HDR-CIELAB and HDR-IPT [15]), or a recalibration
of the entire model (ICtCp [16] and
$\text{J}_{\text{z}}\text{a}_{\text{z}}\text{b}_{\text{z}}$ [17]) while
following a similar sequence of operations.

##  IV Color Appearance Models

The CIELAB uniform color space [18], while not being a “true color appearance
model”, is worth discussing in this context since it is a precursor to current
widely-used CAMs. A key weakness of the CIELAB space is “color inconstancy”.
That is, CIELAB difference values are poor predictors of color constancy under
varying illuminations [19]. This behavior is attributed to the “improper”
method used by CIELAB to account for illumination conditions.

Due to the normalization of $XYZ$ values against the reference white, the
CIELAB transform makes a simple attempt at accounting for the appearance of
the same color under various illumination conditions, hence its inclusion as a
precursor to modern CAMs. However, such a normalization is considered an
“incorrect” model of adaptation to reference whites, and has been shown to be
a significant source of the color constancy problem described above [19].

Rather, the “correct” method to account for the adaptation of the human eye to
varying illuminations is von-Kries adaptation [20]. von-Kries adaptation is
based on the von-Kries coefficient rule, which states that for any observer,
the cone responses to the same stimulus viewed under two different
illuminations are proportional to the cone responses for the reference white
under the two illuminations.

In other words, let $\textit{LMS}_{1}$, $\textit{LMS}_{2}$ denote the cone
responses for the same stimulus under two different illuminations.
Furthermore, let $\textit{LMS}_{w1}$ and $\textit{LMS}_{w2}$ denote the cone
responses for the reference white under the same two illuminations. Then,

$\begin{bmatrix}L_{2}\\\ M_{2}\\\ S_{2}\\\
\end{bmatrix}=\begin{bmatrix}L_{w2}/L_{w1}&0&0\\\ 0&M_{w2}/M_{w1}&0\\\
0&0&S_{w2}/S_{w1}\\\ \end{bmatrix}\begin{bmatrix}L_{1}\\\ M_{1}\\\ S_{1}\\\
\end{bmatrix}$ (11)

This equation, which predicts cone responses under varying illumination
conditions is called von-Kries adaptation. In general, similar “diagonal”
adaptation rules in the LMS color space are called generalized von-Kries
transforms [21], and are considered the “correct” method for performing
chromatic adaptation. In fact, a modification of the CIELAB UCS, called
Conelab2, that performs von-Kries adaptation in the LMS space was found to
exhibit better color constancy [22].

The success of CIELAB led to the first standardized color appearance model
published by CIE - CIECAM97s [23]. The general approach taken by CIECAM97s can
be described as follows, and this has been used as a template for modern CAMs.

  1. 1.

Convert the input $XYZ$ tristimulus values to an $LMS$ or “sharpened” $LMS$
color-space. “Sharpened” $LMS$ spaces are those that do not accurately reflect
cone responses but achieve better appearance prediction.

  2. 2.

Calculate, or set, a degree of adaptation $D$ based on the viewing conditions.

  3. 3.

Apply a von-Kries transform to adapt input colors to a reference illuminant,
typically the equal energy illuminant $XYZ=(100,100,100)$.

  4. 4.

Compress adapted $LMS$ values using a generalized Michaelis-Menten equation
[24].

  5. 5.

Transform compressed $LMS$ responses to an opponent color space.

Steps 1-3 together constitute the Chromatic Adaptation Transform (CAT) part of
the CAM, and CIECAM97s uses the Bradford CAM [25]. CIECAM97s was followed by a
simpler and improved model called CIECAM02 [26], which uses the CAT02 model of
chromatic adaptation. CAT02 is a modified version of the CMCCAT2000 [27]
model, and was proposed as a simpler linear alternative [28] to the Bradford
CAT used in CIECAM97s. As a result, CAT02 and CIECAM02 have emerged as, by
far, the most popular CAT and CAM models in use today. The CAT16 model
attempts to improve upon CAT02 by simplifying it further into a one-step
transform, and the corresponding CAM is called CAM16 [29].

##  V Models of Contrast Sensitivity

The color difference models described in the aforementioned settings have all
been developed based on psycho-visual experiments that recorded visible
differences between patches of colors. However, images are not “flat” or
“constant-value” patches. Rather, images are comprised of spatially varying
stimuli, which are most easily decomposed into spatial sine waves of varying
frequencies using the Fourier Transform.

Sine wave gratings have been considered a good method for analyzing the visual
systems, since they mirror the typical analysis of linear systems, and they
allow for the use of Fourier Analysis to understand the visibility of more
complex stimuli [30]. The simplest form of the experiment presents subjects
with achromatic sine wave gratings or Gabor patches having a given spatial
frequency, and subjects are allowed to change the contrast of each stimulus
until it is “just noticeable” [31, 30]. The minimum visible contrast is called
the detection threshold, and in its simplest form, is expressed as a function
of the spatial frequency $thresh(f)$. Then, the “contrast sensitivity
function” (CSF) is defined as

$CSF(f)=1/thresh(f)$ (12)

This analysis has been extended to opponent color spaces to also measure
chromatic CSFs in the red-green and blue-yellow channels [32], and along a
wider set of “chromatic directions” [33, 34]. The data obtained from such
experiments has been used to develop mathematical models of contrast
sensitivity [35, 36]. Another approach to measuring CSFs involves recording
the detectability of noise in stimuli, and this has been used to develop CSF
models of both sine wave gratings [37] and wavelet subbands [38]. Modern CSF
models also account for the background luminance, angular size of the object,
models of the optical system, etc [7, 39].

##  VI Image Difference Modeling

While CSF models are more complex than simple CAMs due to the inclusion of
spatial frequency, sine-wave gratings and Gabor patches are still considered
simple stimuli compared to real-world images, since images exhibit a complex
interaction of spatially-varying stimuli. Furthermore, CSF models are not
directly equipped to predict either color appearance or differences.
Therefore, there is a need for combining visual adaptation and contrast
sensitivity effects with color appearance/difference models, particularly for
use in the context of image difference prediction.

The image Color Appearance Modeling (iCAM) framework was proposed by Fairchild
et al. in [3] that sought to adapt traditional CAM models to images and built
on previous approaches such as the Spatial CIELAB (S-CIELAB) [40] and Visible
Differences Predictor (VDP) [41] models. In short, the iCAM framework takes an
image as input and transforms it to the IPT UCS, after applying local
adaptation rules. The IPT values may be used to predict appearance correlates.
A detailed description of iCAM02 is provided in Algorithm 1. An updated
version of the model, named iCAM06 [42] has also been proposed. However, since
iCAM06 is focused on rendering HDR images, we do not include it in our
discussion.

Algorithm 1 The iCAM02 Image Color Appearance Model

Input:
$XYZ(i,j)$ \- Input image in CIE XYZ values.
Output:
$\textit{IPT}_{a}(i,j)$ \- IPT values that account for local adaptation.

  * •

Compute the local adaptation white point.

$XYZ_{white}\leftarrow GaussianFilt(XYZ)$

  * •

Use CAT02 to adapt $XYZ$ values to the D65 illuminant.

$XYZ_{a}\leftarrow CAT02(XYZ,XYZ_{white},D65)$

  * •

Convert adapted $XYZ$ values to an $LMS$ space.

$LMS_{a}\leftarrow M_{XYZ\rightarrow LMS}\times XYZ_{a}$

  * •

Compute the luminance-level adaptation factor, normalized to have value 1 at
1000 nits.

$k\leftarrow 1/\left(1+Y_{white}\right)$
$F_{L}\leftarrow\left((1/5)k^{4}Y_{white}+(1/10)\left(1-k^{4}\right)^{2}Y_{white}^{1/3}\right)/1.71$

  * •

Use $F_{L}$ to modulate the exponent of the IPT compressive nonlinearity.

$\alpha\leftarrow 0.43\max\left(F_{L},0.3\right)$

  * •

Apply the modulated compressive non-linearity.

$LMS_{a}^{{}^{\prime}}\leftarrow\text{sign}\left(LMS_{a}\right)\mid
LMS_{a}\mid^{\alpha}$

  * •

Convert to the opponent IPT UCS.

$IPT_{a}\leftarrow M_{LMS\rightarrow IPT}\times LMS_{a}^{{}^{\prime}}$

A simple extension of iCAM02 to predict image differences is to compute
$\Delta E$ metrics from the predicted IPT outputs when iCAM02 is applied to
the two images. We shall overload notation and refer to this metric also as
iCAM02, since we shall only discuss image difference models in this paper. A
more sophisticated approach to predicting image differences may be constructed
by combining the iCAM02 framework with the modular image difference (iDiff)
[4] framework. A detailed description of iDiff is provided in Algorithm 2.

Algorithm 2 The iDiff Modular Image Difference Algorithm

Input:
$\text{XYZ}_{1}(i,j),\text{XYZ}_{2}(i,j)$ \- Input images in CIE XYZ values.
Output:
$\Delta E(i,j)$ \- Total Error map.
$\Delta I(i,j)$ \- Brightness/Lightness difference map.
$\Delta C(i,j)$ \- Chroma difference map.
$\Delta H(i,j)$ \- Hue difference map.

  * •

Convert both inputs into an opponent color space [43]

$\textit{ACC}\leftarrow M_{XYZ\rightarrow ACC}\textit{XYZ}$

  * •

Apply the NSS-adapted CSF to all three channels of both inputs

$\widetilde{\textit{ACC}}\leftarrow\mathcal{F}^{-1}\left\\{\mathcal{F}\left\\{\textit{ACC}\right\\}\times
Flatten\left\\{f_{\theta}^{1/3}MovshonCSF(f_{\theta})\right\\}\right\\},$

where $f_{r}=\sqrt{f_{x}^{2}+f_{y}^{2}}$ is the radial frequency,
$\theta=\arctan\left(f_{y}/f_{x}\right)$ is the orientation, and
$f_{\theta}=f_{r}/\left(0.15\cos(4\theta)+0.85\right)$ accounts for the
“oblique effect” [41].

  * •

Preserve edges by performing “edge-enhancement” using the filter

$EdgeEnh(f_{\theta})\leftarrow 1+e^{-(f_{\theta}-30)^{2}/36}$

  * •

Apply local contrast enhancement using gamma curves.

$Mask\leftarrow GaussianFilter\left(\widetilde{A}\right)$
$\beta\leftarrow\text{clip}\left(2^{\frac{\text{median}\left(\widetilde{A}\right)-Mask}{\text{median}\left(\widetilde{A}\right)}},-10,10\right)$
$\textit{ACC}^{{}^{\prime}}\leftarrow\left(\max_{ij}\\{ACC\\}-\min_{ij}\\{ACC\\}\right)\left(\frac{ACC}{\max_{ij}\\{ACC\\}-\min_{ij}\\{ACC\\}}\right)^{\beta}$

  * •

Convert back to XYZ.

$\textit{XYZ}^{{}^{\prime}}\leftarrow M_{ACC\rightarrow
XYZ}\textit{ACC}^{{}^{\prime}}$

  * •

Convert to IPT and compute differences as in Equations 5 and 8-10.

The iDiff framework makes use of CSF models to enhance or suppress
frequencies, and therefore artifacts, based on their visibility. This is
achieved by filtering both input images with CSFs by multiplying in the
frequency domain. Curiously, the description of the CSF models used in iDiff
is inconsistent with the constraints that the authors wish to enforce on them,
and the plots showing the CSFs as a function of frequency. In particular, the
authors claim to normalize CSFs to have unit DC values. However, the
functional form of the “base CSF” prior to normalization is specified as

$MovshonCSF(f)=af^{c}e^{-bf}.$ (13)

Clearly, the response of this CSF at $f=0$ is 0, so it is unclear how
normalizing by any divisive factor can yield a non-zero DC response, such as
that in Figure 5 of [44]. A similar issue is faced by CSFs that have been
adapted to account for natural scene statistics, which involves multiplying
“base” CSFs by a factor of $f^{1/3}$.

To resolve this ambiguity, we propose using the following flattening rule
proposed by [45] to convert bandpass CSFs into lowpass CSFs, which is said to
account for supra-threshold effects.

$Flatten\left\\{CSF(f)\right\\}=\begin{cases}CSF(f_{max})&\text{if }f\leq
f_{max},\\\ CSF(f)&\text{else},\end{cases}$ (14)

where $f_{max}=\arg\max CSF(f)$. Therefore, all achromatic CSFs used in this
work have been modified using the flattening method, and chromatic CSFs have
been left unchanged since they are already low-pass. Furthermore, both classes
of CSFs have been normalized to a peak response of 1.

Finally, we also consider the image difference model obtained by integrating
iCAM02 and iDiff (say, iCAMDiff), which was proposed in [3]. This model
involves applying the local adaptation rules of iCAM02 first, followed by the
CSF-based difference model proposed by iDiff.

##  VII Edge-Aware Filtering

The primary level of complexity that an image contains over a uniform or sine-
wave patch is its spatial organization. Images contain structures at various
scales, such as textures and objects, that are not easily captured by
“stationary” models such as uniform and sine-wave patches. Indeed, even the
use of Fourier analysis and filtering directly does not adequately capture the
local spatial processing that occurs in the HVS. In particular, while the
contrast sensitivity to high-frequency sine-wave gratings is low, edges are
crucial visual cues that we use to “segment” images into objects and identify
textures.

The edge enhancement applied in iDiff aims to correct the low CSF weighting
applied to high frequencies by adding a Gaussian-shaped “bump” at high
frequencies. However, it is important to note that edges are not high-
frequencies, they merely contain them. The key property that defines edges is
their spatial localization and non-stationarity, which is not accounted for by
this model.

The key hypothesis of this paper is that the appearances of objects (and
distinct regions of objects) in images are generally assessed separately from
each other (though, not strictly independently). In other words, the “leakage”
of color appearance and difference mechanisms across edges is to be minimized.
To this end, we propose using edge-aware filtering, both to compute local
adaptation white points in iCAM and to apply CSF models in iDiff. Such
filtering may be interpreted as soft segmentation of regions that may
correspond to distinct (regions of) objects.

The most popular edge-aware filter is the bilateral filter [46], which
combines a Gaussian spatial weighting function $w_{s}(\cdot)$, which is the
“base filter”, with a Gaussian “influence function” that depends on pixel
intensities $w_{r}(\cdot)$ as

$\widetilde{I}(i,j)=\frac{\sum\limits_{(m,n)\in\Omega_{ij}}w_{s}\left(i-m,j-n\right)w_{r}\left(I\left(i,j\right)-I\left(m,n\right)\right)I\left(m,n\right)}{\sum\limits_{(m,n)\in\Omega_{ij}}w_{s}\left(i-m,j-n\right)w_{r}\left(I\left(i,j\right)-I\left(m,n\right)\right)}$
(15)

where $\Omega_{ij}$ denotes a spatial neighborhood centered on pixel location
$(i,j)$.

The influence function assigns lower weights to neighboring pixels that differ
significantly from the center pixel, thereby reducing the influence across
edges, which are characterized by abrupt changes in pixel values. Since local
adaptation white points are computed using a Gaussian filter in iCAM02, the
bilateral filter is a natural candidate as the edge-aware alternative.

To propose an edge-aware CSF filtering method, we generalize the spatial
weighting function in the bilateral filter to include any linear filter. That
is, we would like to replace the $w_{s}$ function by the CSF filter. However,
the most accurate method for applying the CSF is by multiplying in the
frequency domain, which is incompatible with the spatial domain definition in
Equation 15. Furthermore, no direct multiplication in the frequency domain
would be equivalent to the desired filter, since edge-aware filtering is non-
linear.

To address this, we adopt an approach similar to the fast bilateral filter
proposed in [47], which we modify to improve numerical stability. Let us begin
with a simple assumption, that the set of luminance values is a discrete set
$\\{l_{1},\dots,l_{K}\\}$. Then, Equation 15 may be rewritten as

$\displaystyle\widetilde{I}(i,j)=\frac{\sum\limits_{l_{k}}\left(\sum\limits_{(m,n)}w_{s}\left(i-m,j-n\right)w_{r}\left(l_{k}-I\left(m,n\right)\right)I\left(m,n\right)\right)M_{k}\left(i,j\right)}{\sum\limits_{l_{k}}\left(\sum\limits_{(m,n)}w_{s}\left(i-m,j-n\right)w_{r}\left(l_{k}-I\left(m,n\right)\right)\right)M_{k}\left(i,j\right)}$
(16)

In short,

$\displaystyle\widetilde{I}=\frac{\sum\limits_{l_{k}}\left(w_{s}\otimes\left\\{W_{rk}\times
I\right\\}\right)M_{k}}{\sum\limits_{l_{k}}\left(w_{s}\otimes
W_{rk}\right)M_{k}}$ (17)

where $W_{rk}(i,j)=w_{r}\left(l_{k}-I(i,j)\right)I(i,j)$ is an “influence
map”, $M_{k}=\mathbbm{1}\left\\{I_{p}=l_{k}\right\\}$ is the binary mask
denoting whether the value of the center pixel is $l_{k}$, and $\otimes$
denotes convolution. In general, if the luminance can take continuous values
in the range $[0,L]$, we can divide the range into $K$ bins, each centered at
$l_{k}$, and replace the binary mask by an interpolation function.

In this manner, we have converted the computation of the non-linear bilateral
filter into a sequence of linear filters. We may now replace the convolution-
based filtering in Equation 17 with frequency domain filtering using the CSF,
to yield an edge-aware CSF filter. Note that the influence function is
computed using the intensity channel for all three channels, since edges are
best identified in the intensity/luminance domain.

##  VIII Experiments

In this section, we shall compare edge-aware iCAM, iDiff and iCAMDiff models
to their edge-unaware counterparts. We perform this analysis in two ways - by
visualizing error maps and by observing the behavior of the predicted
“aggregate error” for known distortions. In both cases, we shall rely on
visual inspection and qualitative evaluations, since quantitative subjective
data is not available. Moreover, as discussed in [4], using the mean value as
an aggregated difference metric is not preferred, since subjects are biased
towards regions of low quality. A detailed analysis of spatial aggregation
methods conducted in [48] arrived at a similar conclusion. Minkowski pooling
was identified in [48] as an improvement over mean pooling, since it assigns
higher weight to regions of higher distortion. Therefore, we compute the
aggregate difference value from error maps as

$\Delta E_{agg}=\left(\frac{1}{MN}\sum\limits_{i,j}\Delta
E(i,j)^{3}\right)^{1/3}.$ (18)

When using the flattening scheme proposed in Section VI, it was found that the
NSS adaptation using the $f^{1/3}$ factor did not change any of the results
significantly. Furthermore, we use the CAT16 chromatic adaptation transform
[29] and the OKLab UCS [14] in the place of CAT02 and IPT, since they have
been proposed as improvements over the two models respectively.

To evaluate the proposed models, we have used tone-mapped HDR images, since
tone-mapping is known to introduce both luminance and color artifacts. We use
the Durand TMO [47] and the Reinhard TMO [49] to generate two sets of videos,
with varying contrast and color desaturation parameters. Note that while the
two parameters are specific to achromatic and chromatic properties
respectively in theory, they tend to influence both aspects of image
appearance in practice.

The set of contrast-varied images were generated by setting the “base
contrast” parameter to $\\{10,100,1000,10000\\}$. To compute the differences,
images generated using a base contrast of 1000 were chosen as reference. The
set of saturation-varied images were generated by setting the desaturation
parameter to $\\{0.0,0.25,0.5,0.75\\}$, and the 0.0 desaturation images were
used as references for difference calculation. For both sets of images, the
aggregated differences predicted by iCAM02, iDiff, and iCAMDiff are shown in
Figure 3, and the predictions made by the proposed edge-aware counterparts are
shown in Figure 4.

(a) Contrast-Varied Images

(b) Desaturation-Varied Images

Figure 3: Image difference predictions made by the baseline iCAM02, iDiff and
ICAMDiff models

(a) Contrast-Varied Images

(b) Desaturation-Varied Images

Figure 4: Image difference predictions made by edge-aware iCAM02, iDiff and
ICAMDiff models

From these figures, we observe that the edge-aware iCAMDiff model improves
upon the baseline iCAMDiff model, which is the method recommended in [3], in
two ways. Firstly, the variation in quality over the contrast-varied images
was reduced significantly, signaling a greater confidence in the predictions.
Secondly, the baseline iCAMDiff model exhibits anomalous behavior at high
desaturation values, predicting a lower difference on average for a
desaturation parameter of 0.75 than for 0.5. Applying the model in an edge-
aware manner fixes the non-monotonicity issue, and reveals the asymptotic
behavior of perceived difference with increase in desaturation.

The key advantage of the edge-aware model is expected to be in providing
higher quality error maps that localize differences better. To demonstrate
that we do indeed achieve this behavior, we present sample distortion maps
from both distortion classes for the baseline and edge-aware models in Figures
5 and 6.

In Figure 5, we observe that the proposed edge-aware model greatly reduces the
spurious CSF filtering-related artifacts present in the baseline model’s
output, such as the oblique patches emerging out of the buildings at the top
of columns 750 and 1600 (approximately). Secondly, the baseline model predicts
a significant difference around row 400 and column 1200, which corresponds to
the lights on the side of the building. However, the actual difference is not
so pronounced, and the edge-aware model rightly predicts that the main
difference in the region is the halo of the light.

In Figure 6, we again observe two main improvements that the edge-aware model
makes over the baseline. Firstly, the baseline model overestimates the
difference above the lights (row 400, column 1200) and above the dome (row
200, column 500). We attribute this to the CSF “leakage” from the lights
nearby. The edge-aware model corrects this overestimation. Moreover, the edge-
aware model provides a more uniform estimation of the error on the building
around column 1500, and provides cleared outlines around errors corresponding
to cars on the road.

In summary, our experiments demonstrate that the edge-aware iCAMDiff model
provides more robust predictions of both local and aggregate image differences
across two distortion types.

(a) Reference Image

(b) Test Image

(c) Total Error (Baseline)

(d) Total Error (Edge-Aware)

(e) Intensity Error (Baseline)

(f) Intensity Error (Edge-Aware)

(g) Chroma Error (Baseline)

(h) Chroma Error (Edge-Aware)

(i) Hue Error (Baseline)

(j) Hue Error (Edge-Aware)

Figure 5: Image difference maps for a pair of contrast-distorted images

(a) Reference Image

(b) Test Image

(c) Total Error (Baseline)

(d) Total Error (Edge-Aware)

(e) Intensity Error (Baseline)

(f) Intensity Error (Edge-Aware)

(g) Chroma Error (Baseline)

(h) Chroma Error (Edge-Aware)

(i) Hue Error (Baseline)

(j) Hue Error (Edge-Aware)

Figure 6: Image difference maps for a pair of desaturation-distorted images

##  IX Conclusion

In this paper, we have reviewed color appearance, color difference, and
contrast sensitivity models. We have motivated the need for image color
appearance and color difference models that combine aspects of all the
aforementioned models, and described two state-of-the-art models that attempt
to tackle the problem. We then motivated and derived edge-aware image color
appearance and difference models, and evaluated their performance for two
distortion classes, demonstrating their utility.

While this work offers a promising case for edge-aware image difference
models, there are still gaps that must be addressed. Firstly, a more robust
evaluation must be conducted using a larger dataset, after obtaining
subjective data specific to such distortions. In addition, a key drawback of
our model is that the iterative edge-aware filtering method used is slow. This
may be accelerated using better fast approximations to bilateral filters [50],
or simpler isotropic filtering techniques [51]. This work may also be extended
to HDR images using the HDR UCSs and CSFs mentioned above.

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# The Nudging Effect on Tracking Activity

Ruochun Wang<EMAIL_ADDRESS>0000-0002-1097-1906 The University of
SydneySydneyNSWAustralia2000 ,  Amani Abusafia<EMAIL_ADDRESS>0000-0001-9159-6214 The University of SydneySydneyNSWAustralia2000 ,  Abdallah
Lakhdari<EMAIL_ADDRESS>0000-0001-8005-1534 The University
of SydneySydneyNSWAustralia2000 and  Athman Bouguettaya
<EMAIL_ADDRESS>0000-0003-1254-8092 The University of
SydneySydneyNSWAustralia2000

(2022)

###### Abstract.

Wearables activity trackers are becoming widely adopted to understand
individual behavior. Understanding behavior may help in self-regulation such
as self-monitoring, goal-setting, self-corrective, etc.; Nevertheless,
challenges exist in attaining consistent use and adoption of wearables, which
hinders behavior understanding. Research has suggested that nudging strategies
may change and sustain human engagement. However, it is still unknown how
nudging may affect human wearing behavior on an individual level. We conducted
a six-month study in which we tested several nudging techniques on the same
participants. The preliminary results of our research show that participants
perform better when a nudging strategy is applied. In addition, participants
responded differently to different nudging techniques. Future research can
focus on developing an individual-based nudging mechanism to encourage users
to wear their devices consistently.

Nudging; motivation; incentives

††copyright: none††journalyear: 2022††doi: …††price: 15.00††isbn:
978-1-4503-XXXX-X/18/06††ccs: Human-centered computing Empirical studies in
HCI

##  1\. Introduction

Wearable activity trackers (i.e., Fitbit) have been used to report health-
related activities (Gulotta et al., 2016; Shih et al., 2015; Nuss and Li,
2021). The wearables’ regular use will lead to consistent reporting, which may
help understand the users’ behavior. Understanding the behavior may help in
several aspects, including activity recommendations, self-regulation, and
health intervention (Epstein et al., 2016). However, attaining consistent use
and adoption of wearables is a challenge (Heatherton et al., 1994). Indeed,
people often tend to forget or stop wearing their devices over time (Epstein
et al., 2016). The inconsistency in using the wearables is a common challenge
that hinders the full understanding of users’ behavior (Shih et al., 2015). A
crucial question is how to change behaviors to increase and sustain regular
activity tracking?

Nudging is a key mechanism to increase and motivate regular human engagement
(Caraban et al., 2019). In this paper, we aim to study the impact of nudging
strategies on individuals’ adherence to wearing activity trackers. Even though
several studies have explored the use of nudging theory to change human
behavior (Caraban et al., 2019). Most of these studies focused on changing
instant behavior in the short term. In addition, the activity of wearing a
tracker requires daily motivation and remembrance. Hence, it may require
different nudging strategies. Additionally, most of these studies would try a
different nudging strategy for each group. This suggests that people will
respond to nudging strategies similarly (Caraban et al., 2019; Tseng et al.,
2018). However, in reality, individuals react differently based on their
individual characteristics and preferences. Therefore, this paper aims to
study the effect of several nudging strategies on each individual over time.
For instance, all participants were studied using the same nudging strategies.
Furthermore, our study used different email reminders as a nudging tool
reminding participants to wear and use their gadgets once every month. Then we
assess the change in their device-wearing behavior in terms of the number of
days they wore their devices.

##  2\. Study Method

Overall, we have recruited 28 participants to wear the Fitbit smartwatch and
report health data for around 6 months (gender: female = 13, male = 15; age: M
= 21.71, SD =3; BMI: M = 23.46, SD = 3.95; education: High diploma = 10,
Bachelors = 11, Masters = 3, PhD = 1). Our data was collected through the
Fitbit smartwatch remote back-end database. Fitbit smartwatch tracks user
health data such as average breathing rate, steps count, and consumption of
calories to the remote server in real-time. At the beginning of this study, we
provided all participants with a Fitbit smartwatch and requested them to wear
it daily. We computed their active rate as the percentage of the days they
wore the devices over a month. A device is considered worn if the number of
steps is more than one. Based on their first-month reporting active rates
(baseline period), 28 participants were evenly divided into the experimental
and control groups. Starting from the second month, we sent short emails
containing a nudging message to the experiment group at the end of each month.
In total, there are five different nudging strategies that we have used in our
emails: 1) Neutral - Which contains the basic monthly activity data without
emotional bias; 2) Positive - Motivational message, which explains the
positive impact of their consistency in wearing the devices on their health
and our research ; 3) Negative - A message that informs the participants that
third of them have not worn their devices the past 30 days; 4) Rank - A
message which mentions each participant their rank among all participants in
the previous month and how consistent where the top three participants; 5)
Monetary - A message which reminds participants that a monetary award would be
obtained at the end of the month for their participation.

##  3\. Results

Figure 1. Active rates of both groups each month

Figure 2. Average active rates difference between experiment and control
groups each month

Figure 3. Active rates distribution in each month for (A) Experiment group (B)
Control group

The first experiment studies the impact of each nudging strategy by comparing
the average active rates between the experiment and the control groups in each
month (See Fig.2). Note that the experiment group’s participants who did not
open their email in a month were dropped from that month. The preliminary
results show that the experimental group had a higher active rate than the
control group, except when the message was negative. According to the results,
all nudging methods increased the active rates of participants by 5-20% except
for the negative message, which slightly reduced the participation by 3% (See
Fig.2). Additionally, the results show that the positive message has the
highest impact. Another observation is that, generally, the active rates of
the participants of both groups have declined over time. This is due to the
natural habit of losing motivation in humans as time passes (Epstein et al.,
2016).

We analyze the participants’ active rates distribution to ascertain the
previous experiment. The previous experiment only considers the average active
rate in each month. We visualize the distribution of both groups as shown in
Fig.3. In general, both figures show a wide distribution of active rates
across the months except for the first baseline month. This distribution
indicates that (1) Nudging techniques impact people differently, e.g., some
will be more motivated by the positive messages while others will be motivated
by ranking. This observation gives us an insight that individuals’ behavior
varies with nudging strategies. A customized nudging mechanism is needed to
identify the right nudging for each individual based on their previous
behavior. (2) people’s behavior in consistently wearing their Fitbit is
different; this is reflected in the active rates distribution of the control
group (Fig.3 (B)). Lastly, the distribution of the active rates in the first
month is condensed. This may be justified by the excitement of participants at
the start of the project (Epstein et al., 2016).

##  4\. Conclusion

We have shown that different nudging strategies impact people’s consistency in
wearing their activity trackers. Our future work will be to build a nudging
mechanism that considers the individual behavior of participants. We will also
recruit more participants to have an extensive study.

## Acknowledgment

This research was partly made possible by LE220100078 and LE180100158 grants
from the Australian Research Council. The statements made herein are solely
the responsibility of the authors.

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