Acta Polytechnica 56(5):409–416, 2016 ©Czech Technical University in Prague, 2016
available online at http://ojs.cvut.cz/ojs/index.php/ap
BACK TO BASICS: TOWARDS NOVEL COMPUTATION
AND ARRANGEMENT OF SPATIAL SENSORY IN IMAGES
Wei Wen∗, Siamak Khatibi
Institute of Communication, Blekinge Institute of Technology, Karlskrona, Sweden
∗corresponding author: firstname.lastname@example.org
Abstract. The current camera has made a huge progress in the sensor resolution and the low-
luminance performance. However, we are still far from having an optimal camera as powerful as our
eye is. The study of the evolution process of our visual system indicates attention to two major issues:
the form and the density of the sensor. High contrast and optimal sampling properties of our visual
spatial arrangement are related directly to the densely hexagonal form. In this paper, we propose
a novel software-based method to create images on a compact dense hexagonal grid, derived from a
simulated square sensor array by a virtual increase of the ﬁll factor and a half a pixel shifting. After
that, the orbit functions are proposed for a hexagonal image processing. The results show it is possible
to achieve image processing operations in the orbit domain and the generated hexagonal images are
superior, in detection of curvature edges, to the square images. We believe that the orbit domain image
processing has a great potential to be the standard processing for hexagonal images.
Keywords: Hexagonal pixel, Square pixel, Hexagonal sensor array, Hexagonal processing, Fill factor,
Convolution, Orbit functions, Orbit transform.
Nowadays, the ubiquitous inﬂuence of cameras in our
life is undoubtable and this is thanks to the current
camera sensory technique, which has made a huge
progress on increasing the sensor resolution and the
low-luminance performance [
]. The progress achieve-
ments are due to the size reduction of the sensory
element (pixel), improvement in generation of the sig-
nal from collected light (quantum eﬃciency), and the
used hardware techniques of the sensor [
the image quality is not aﬀected only by the pixel size
or quantum eﬃciency of a sensor [
]. As the sensor
pixel size becomes smaller, a smaller die size detection,
gaining higher spatial resolution and obtaining lower
signal-to-noise ratio are required; all in cost of a lower
dynamic range and a fewer number of tonal levels.
The form, arrangement, and inter-element distance
of sensors play signiﬁcant roles in the image quality,
which is veriﬁed by a comparison between the current
sensor techniques and animal, especially human, vi-
sual systems. The eﬀect of the inter-element distance
on an image quality was studied in [
] which showed
that the inter-element distance could be decreased by
the means of a physical modeling and thus obtain-
ing higher quality images, higher dynamic range and
greater number of tonal levels. Anatomical and physi-
ological studies indicate that our visual quality related
issues, such as high sensitivity, high speed response,
high contrast sensitivity, high signal to noise ratio, and
optimal sampling are related directly to the form and
arrangement of the sensors in the visual system [
In the human eye, three types of color photoreceptors,
cones, are packed densely within a hexagonal pattern
form and are located mostly in fovea in the center
of retina [
]. Figure 1 shows a dramatic increase of
the cone cross sectional area and a decrease of cone
density within the eccentricity range represented in a
strip of the inner segments of photoreceptors from the
foveal center along the temporal horizontal meridian.
Rods, another type of photoreceptors with non-color
property, ﬁrst appear at about 100
m from the foveal
center and are smaller than the cones.
Hexagonal photoreceptors arrays are not found only
in human, but also often in compound eyes of insects
and other invertebrates. Actually, such hexagonal ar-
rays are more common for animals and plants than any
other geometric arrays, such as rectilinear orthogonal
arrays. This is due to the need of a motion detection,
which is obtained based on the local diﬀerence of light
intensity between adjacent neighboring photoreceptor
] using the lateral inhibition process [
Lateral inhibition is a contrast enhancement computa-
tion to exaggerate the light intensity diﬀerences of the
neighboring cells; it is useful in an edge detection. The
hexagonal array provides the best candidate for a con-
tiguous neighboring-cell computation of the local light
intensity diﬀerence between adjacent cells [
]. By us-
ing the contiguous neighboring cells at 60
ﬁrst-order computation of light intensity diﬀerences
is computed, and for the computation of ﬁner angu-
lar diﬀerences, such as 30
angle, the second-order
adjacent cells are used. Thus, this provides means
for a symmetric computation, using one set of com-
putational algorithms to compute the light intensity
diﬀerence of exactly 6 contiguous neighbors. Each suc-
cessive higher-order neighbor will compute the light
intensity diﬀerence with the angular direction rotated
degree successively. In the gradient based edge
Wei Wen, Siamak Khatibi Acta Polytechnica
Figure 1. The enhanced image of inner segments of a human fovea1 photoreceptor mosaic from the original image
printed in [
]. The mosaic strip is extending 575
m from the fovea1 center along the temporal horizontal meridian;
shown from upper left to lower right in the ﬁgure. Arrowheads indicate the edges of the sampling windows. Brackets,
shown as red, indicate a quadrant of the ﬁrst sampling window with the highest density of cones. The midpoint of
the boundary of this quadrant and a quadrant adjacent to it in the temporal direction (to the right) with similar
density and mean spacing was considered to be the point of 0.0 eccentricity. The strip contains proﬁles of only cones
up to the ﬁfth window, where the small proﬁles of rods begin to intrude. The large cells are cones and the small cells
are rods and the bar is 10 µm.
detection, a local process at pixel level, the edges are
discriminated by comparing the light intensity gradi-
ent of neighboring cells to ﬁnd sudden changes of light
intensity in adjacent pixels. It is shown in numeri-
cal studies that, when using the spatial computation,
the hexagonal pixel-arrays are more eﬃcient for edge
detection instead of the rectilinear arrays [
achieves a 40% computational eﬃciency by using a
hexagonal edge detection operator [
] and it can
eﬃciently detect motion in six diﬀerent directions at
60 degree [
]. Due to a biological inspiration and
aforementioned beneﬁtsl of hexagonal sensory arrays
implementation; generally, two main approaches are
followed by researchers to acquire hexagonal sampled
images. The ﬁrst approach is to manipulate the re-
sult of conventional acquisition sampling devices using
square sensor arrays, via software, to generate a hexag-
onally sampled image. The second approach is to use
dedicated hardware to acquire the image, such as the
super CCD from Fujiﬁlm whose sensor structure is
], or color ﬁlters in hexagonal shape for
the image sensors [
] to improve the quality of the
acquired color by the sensor.
In this paper, we propose a novel software-based
method to create images on a dense hexagonal sensor
grid by shifting the sensor array virtually. This is
derived from a simulated camera sensor array by a
virtual increase of ﬁll factor and transferring of the
shifted square pixel grid into a virtual hexagonal grid.
In our method, the images are ﬁrstly rearranged into
a new grid of virtual square grid composed by sub-
pixels with the estimated value of the ﬁll factor. A
statistical framework proposed in [
], consisting of a
local learning model and the Bayesian inference, is
used to estimate new subpixel intensities. Then the
subpixels are projected onto the square grid, shifted
and resampled in the hexagonal grid. Unlike the
previous hexagonal image processing methods, the
intensity of each hexagonal image pixel is estimated in
our method,which results to maintaining the density
of the arrangement. To the best of our knowledge,
our approach is the ﬁrst work that tries to create a
hexagonal image by a virtual increase of a ﬁll factor.
After the hexagonal image is generated, type A2 orbit
function from Weyl group is used to map the image
into the orbit domain. Then smoothing and gradient
ﬁltering, as typical image processing operations, are
done in the orbit domain on the hexagonal images.
We believe such hexagonal image processing has great
potential to be the standard processing for hexagonal
arrangement due to its practical usage. Such process-
ing is surprisingly more genuine and close to the way
how human brain works.
This paper is organized as follows. In Sections 2
and 3, the related researches of hexagonal resampling
and the methodology are discussed more in detail.
Section 4 presents the experiment setup. Then the
results are shown and analyzed in Section 5. Finally,
we summarized and discussed our work in this paper
in Section 6.
2. Related works to hexagonal
The software based approaches rearrange the conven-
tional square pixel grids to generate the hexagonal
images. There are three most common methods for
simulating hexagonal resampling :
The hexagonal grid is mimicked by a half-pixel
shift, which is derived from delaying the sampling
by a half a pixel on the horizontal direction as it
is shown in Figure 2 [
]. In Figure 2, the left
and right patterns are showing the conventional
square lattice and the new pseudo-hexagonal sam-
pling structure, whose pixel shape is still square;
see the six connected pixels by the dashed lines. In
such sampling structure, the distance between the
vol. 56 no. 5/2016 Back to Basics
Figure 2. The procedure from square pixels to hexag-
onal pixels by half-pixel shifting method.
two sampling points, pixels, are not the same; they
are one or √5/2.
The hexagonal pixel is simulated by a set of
square pixels, called hyperpel [
] which is widely
used for displaying a hexagonal image on normal
monitor. Figure 3 shows an example of the hyperpel,
composed of 20 ×20 square sub-pixels.
The hexagonal pixel is generated by mimicing
the Spiral Architecture method [
] by averaging
the pixel grey level values of four square pixels in
the structure. The method preserves the hexagonal
arrangement as each hexagonal pixel is surrounded
by six neighbour pixels. A reduction in the reso-
lution of the resulted hexagonal image is expected
due to averaging of four pixels for each hexagonal
generated pixel. Also, in this method the distance
between each of the six surrounding pixels and the
central pixel is not the same as it should be in a
The spiral architecture is a common method for the
hexagonal addressing and for the processing of the
hexagonal images [
]. In the spiral architecture, all
the hexagonal pixels are arranged on a spiral; it maps
the hexagonal image into a one-dimensional vector.
There are still many problems related to the hexagonal
arrangement and computation that cannot be solved
with all these approaches, e.g. the image resolution
and pixel intensity values are changed during the
resampling from the square grid to the hexagonal
grid. In the case of the hexagonal computation, the
hexagonal image still has to be mapped to a certain
architecture form ; as far as the Cartesian indexing
and coordinate is not yield in hexagonal computation.
For example, in the spiral architecture, the hexagonal
image is mapped into a one-dimensional vector to
achieve a faster addressing and processing. However,
such mapping results to a complicated computation
process for each image processing operation; e.g. due
to the reduction of the neighbour pixels from six to
two in the spiral architecture.
3.1. Hexagonal pixel
In this section, the method for generating a hexag-
onal image by resampling and half-pixel shifting a
square-pixel image is explained. According to the
resampling process in [
], the process is divided into
Figure 3. an example of hyperpel
Is <- Select Pixel Matrix
F <- Fill factor
S <- Active Area Size (F)
G <- Gaussian Distribution Estimation (I)
In <- Introduce Gaussian Noise (G)
P <- Active Area Size of Middle pixel
While P <= S
L <- Local Learning (In)
X <- Subpixel Intensity (L)
Ev <- Subpixel Gaussian Estimation (X)
H <- Hexagonal grid projection (Is)
E <- Histogram Distribution (Ev)
Es <- Subpixel Intensity Estimation (E)
I <- Pixel Intensity Estimation (Es)
Algorithm 1. Resampling process.
three steps: projecting the sampled signal to a new
grid of sub-pixels; estimating the values of subpixels
at the resampling positions; estimating the new pixel
intensity and arranging the data to the hexagonal
The three steps are elaborated in the following, as
well as in Algorithm 1.
A grid of virtual image sensor pixels is designed.
Each pixel is divided into 20
20 subpixels. Accord-
ing to the known ﬁll factor, the size of the active
area is S by S, where
. The intensity
value of every pixel in the image sensor array is
assigned to the virtual active area in the new grid.
The intensities of subpixels in the non-sensitive ar-
eas are assigned to be zero. An example of such
sensor rearrangement on sub-pixel level is presented
on the left in Figure 4, where there is a 3 by 3 pixels
grid, and the light and dark grey areas represent
the active and non-active areas in each pixel. The
active area is composed by 12 by 12 subpixels, and
thereby the ﬁll factor becomes 36 % according to
the above equation, and the intensities of active
areas are represented by diﬀerent greylevel values.
The second step is to estimate the values of sub-
pixels in the new grid of subpixels. Considering
the statistical ﬂuctuation of incoming photons and
their conversion to electrons on the sensor, there
is a need for a statistical model to estimate the
Wei Wen, Siamak Khatibi Acta Polytechnica
Figure 4. From left to right: the sensor rearrange-
ment onto the subpixel, the projection of the square
pixels onto the hexagonal grid by half pixel shifting
and the pixel intensities displayed by hyperpel.
original signal. The Bayesian inference is used for
the estimating every subpixel intensity, which is
considered to be in the new position of resampling.
Therefore, the more subpixels are used to present
one pixel, the more accurate the resampling will be.
By introducing the Gaussian noise into a matrix
of selected pixels and estimating the intensity val-
ues of the subpixels at the non-sensitive area with
diﬀerent sizes of active area (local modeling), a vec-
tor of intensity values for each subpixel is created.
Then the subpixel intensity will be estimated by
the maximum likelihood.
In the third step, the subpixels are projected back
to the original grid and then transformed onto a
hexagonal grid shown as red grids on the left and
in the middle of Figure 4 respectively. The red
hexagonal grid, which is presented in the middle
of Figure 4, is used for estimation of the pixel in-
tensities of the hexagonal image. As the middle
row of pixels is shifted to the right by a half a
pixel, on subpixel level the sampling position is also
shifted by a half a pixel as the method in [
yellow grid represents the pixel connection in the
new hexagonal sampling grid. In comparison to
the method in [
], in our method the subpixels
in each square area are estimated with respect to
the virtual increase of the ﬁll factor. The inten-
sity value of a pixel in the hexagonal grid is the
intensity value which has the strongest contribu-
tion in the histogram of belonging subpixels. The
corresponding intensity is divided by the ﬁll factor
for removing the FF eﬀect to obtain the hexagonal
pixel intensity; as illustrated on the right in Figure 4
by means of hyperpels.
3.2. Hexagonal computation
The discrete transform of Lie group provides a possibil-
ity for frequency analysis of discrete functions deﬁned
on a triangle or hexagonal grids [
]. Figure 5 shows
an example of fundamental domain
, the fundamen-
tal weights (
)of the orbit function in the form
of a hexagon. The similarity of the grid in the funda-
mental domain of a certain orbit function from Weyl
group to the grid of hexagonal images makes the orbit
functions interesting for the hexagonal computation in
which it becomes possible to process hexagonal images
without any further transformation. In [
], the dis-
crete orbit functions convolution was deﬁned and was
Figure 5. The fundamental domain
weights (ω1, ω2), and simple roots (α1, α2).
Figure 6. The ﬂow chart of hexagonal computation
of image processing operations.
used for image processing in the spatial domain. Here
we propose hexagonal computational for the image
processing operation using directly the orbit domain.
The ﬂow chart of such a hexagonal computation for
typical operations is shown in Figure 6. A hexagonal
image, generated by the proposed method in § 3.1, is
transformed to an orbit domain using the hexagonal
grid related to the image. Then two ﬁlter kernels, con-
structed in the spatial domain, are transformed to the
orbit domain using the same hexagonal grid related to
the image. The process of the convolution in the orbit
domain is a multiplication operation, shown as X in
the middle dash box in Figure 6. According to the
convolution theory in the orbit domain, the multiple
convolutions can be combined to one by multiplication
operations, which implies that all the image process-
ing operations remain in the orbit domain and use
the same hexagonal grid. The hexagonal image is
obtained by the inverse transformation of results in
the orbit domain in which the scaling factor plays a
key role and it is deﬁned by
, where the
constant scaling factors for each operation and the n
is the number of operations.
vol. 56 no. 5/2016 Back to Basics
4. Experimental setup
A group of optical images with assumption of known
ﬁll factor were simulated using our own codes and
Image Systems Evaluation Toolbox (ISET) [
MATLAB. The ISET is designed to evaluate how im-
age capturing components and algorithms inﬂuence
image quality, and has been proved to be an eﬀective
tool for simulating the sensor and image capturing [
The ﬁll factor value of 36% was chosen for the simu-
lated image sensor, having the resolution of 128
and 8-bits quantization levels. The sensor had a pixel
area of 8
m, with well capacity of 10000 e
The read noise and the dark current noise were set
to 1 mV and 1 mV/pixel/sec respectively. The im-
age sensor was rearranged to a new grid of virtual
square sensor pixels, each of which was composed of
20 subpixels. All the image simulation setup is
the same as in [
]. The optical images are from COIL-
20 (Columbia University Image Library) [
is a database of grayscale images of 20 objects. For
generation of sensor images, the luminance of optical
images was set to 200 cd/m
and the diﬀraction of the
optic system was considered limited to ensure that the
brightness of the output is as close as possible to the
brightness of the scene image. The exposure time was
also set to what is used for the sensor with 100 % ﬁll
factor correspondingly for the simulated sensor. For
capturing images, the sensor exposure time is set to
1 ms. All the processing is programmed and done by
Matlab2015 on a HP laptop with an Intel i7-5600U
CPU and a 16GB RAM memory to keep the process
stable and fast.
5. Results and discussion
Three of the hexagonal images generated according
to the proposed method mentioned in § 3.1 are shown
next to the bottom row of Figure 7, where the cor-
responding optical images, simulated sensor images,
recovered square-pixel images are shown from top to
bottom rows respectively. The bottom row in Fig-
ure 7 shows the images that are zoomed in the red
rectangle area. For visualization purposes each pixel
in the hexagonal images is mapped according to the
hyperpel method in [
] and is composed of 20
subpixels. The corresponding logarithm of histograms
of the hexagonal and square-pixel images in Figure 7
is shown in Figure 8, which indicates that in both
recovered square-pixel images and generated hexag-
onal images the tonal levels are extended, and also
the tonal ranges are wider in comparison to the simu-
lated camera sensor image. The generated hexagonal
images are quite similar to the square-pixel images in
intensity values according to their histograms shown
in Figure 8. This is reasonable as far as the hexago-
nal images are obtained by half-pixel shifting of the
square-pixel image recovered with the enhanced ﬁll
In our test, the type A2 orbit function from Weyl
group is used for the transformation of the hexago-
Figure 7. From top to bottom, the optical images,
simulated sensor images, recovered square-pixel im-
ages, hexagonal images and the zoomed regions, shown
with red rectangles, in hexagonal images.
nal image to the orbit domain. By using the orbit
functions, there is no need to map the hexagonal im-
age to certain coordinates system or an addressing
system. We believe such hexagonal image process-
ing has great potential to be the standard processing
for hexagonal arrangement due to its practical us-
age. Such processing is surprisingly more genuine
and close to the way how human brain works [
considering certain pathways of mapping of the in-
formation from the hexagonal grid in the visual sys-
In our experiment, three basic ﬁlter kernels in the
spatial domain are used to operate on the generated
hexagonal images. These ﬁlters are a mean ﬁlter, a
gradient or an edge detector ﬁlter, and the combina-
tion of the two mentioned ﬁlters. According to the
orbit convolution deﬁnition and a hexagonal grid, the
mean and edge detector ﬁlter kernels are as follows:
fmean = 1/3
Wei Wen, Siamak Khatibi Acta Polytechnica
Figure 8. Log of histograms of the three object images which are shown in Figure 7: Bird (left), Cylinder (right),
where the two ﬁlter kernels in the spatial domain are
fmean = 1/9
0 0 −1
−1 3 0
0 0 −1
The ﬁltering results according to the hexagonal com-
putation, see § 3.2, are shown in Figure 9. From the
left to right, the ﬁltered images by the mean, the edge
detector, and the combination ﬁlters are shown, where
the edges are shown as a binary image. The results of
the edge detector ﬁlter are improved by applying the
smoothing ﬁlter duo to the used edge detector ﬁlter
which is a ﬁrst order gradient computation.
Each pixel in the hexagonal image has six contigu-
ous neighbors which results to more eﬀective and
Figure 9. From the left to right, the ﬁltered images
by the mean ﬁlter, the edge detector ﬁlter, and the
vol. 56 no. 5/2016 Back to Basics
Images Object 1
Optical square 4858 3528 1998
Recovered square 3169 2357 4858
Hexagonal 5135 3295 3436
Table 1. The result of ratio images for diﬀerent type
eﬃcient responsivity to gradient-based edge detection.
To show this responsivity, two diﬀerent mean ﬁlters,
whose sizes are 3
3 and 5
5, are used for smoothing
of the optical square-pixel images, recovered square-
pixel images and hexagonal images, then the edge
detector is used on the smoothed images to detect the
edges in the images. The two resulted edge-detected
images, using the two smoothing ﬁlters on each of the
optical square-pixel images or the recovered square-
pixel images or the hexagonal images, are gray level
images. The pixel-based intensity ratio of each pair of
the edge detected images is computed and ﬁnally the
sum of such ratio images is obtained. Apparently, the
blurring eﬀect of more smoothed images reduces the
detected edges. In Table 1, the result of the ratio im-
ages for diﬀerent type of images is shown. According
to the table, although it is impossible to detect the
vertical edges due to the hexagonal form of our hyper-
pel, the hexagonal images still can preserve the edges
superior in comparison to the square-pixel images.
In this paper, a novel approach is proposed for gener-
ating the hexagonal images from a simulated sensor
based on a known ﬁll factor of the sensor. The results
show that the generated hexagonal images have the
same tonal range and tonal levels as the square-pixel
image. Also, their histogram distribution is very sim-
ilar, indicating that our approach keeps the source
information as much as possible during the genera-
tion. The orbit functions from Weyl group are used
for hexagonal computation. Using orbit functions, it
is possible to process hexagonal images with multiple
operators, and at the same time, in the orbit domain
without further need of mapping to a diﬀerent coor-
dinate system or addressing system. Although the
processing speed of the orbit function convolution is
still very slow, it provides a special domain for the
hexagonal computation and for the image processing
operations. The results also show that the edge de-
tection on the hexagonal images preserves the edges
superiorly, in comparison to the square-pixel images;
with a signiﬁcant curvature detection ability. In the
future, our work would be to improve the processing
speed of the orbit functions and to test our hexago-
nal images generation method with real cameras, to
develop a way to create a hexagonal image from a
conventional rectangle image sensor.
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