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sensors

Article

A Common Assessment Space for Different

Sensor Structures

Wei Wen 1, * , Ondˇrej Kajínek 2, Siamak Khatibi 1, * and Goce Chadzitaskos 2

1Department of Technology and Aesthetics, Blekinge Institute of Technology, 37179 Karlskrona, Sweden

2Department of Physics, Czech Technical University, 11519 Prague 1, Czech Republic;

kajinond@fjﬁ.cvut.cz (O.K.); goce.chadzitaskos@fjﬁ.cvut.cz (G.C.)

*Correspondence: wei.wen@bth.se (W.W.); siamak.khatibi@bth.se (S.K.)

Received: 21 November 2018; Accepted: 22 January 2019; Published: 29 January 2019

Abstract:

The study of the evolution process of our visual system indicates the existence of variational

spatial arrangement; from densely hexagonal in the fovea to a sparse circular structure in the

peripheral retina. Today’s sensor spatial arrangement is inspired by our visual system. However,

we have not come further than rigid rectangular and, on a minor scale, hexagonal sensor arrangements.

Even in this situation, there is a need for directly assessing differences between the rectangular and

hexagonal sensor arrangements, i.e., without the conversion of one arrangement to another. In this

paper, we propose a method to create a common space for addressing any spatial arrangements and

assessing the differences among them, e.g., between the rectangular and hexagonal.

Such a space

is

created by implementing a continuous extension of discrete Weyl Group orbit function transform

which extends a discrete arrangement to a continuous one. The implementation of the space is

demonstrated by comparing two types of generated hexagonal images from each rectangular image

with two different methods of the half-pixel shifting method and virtual hexagonal method. In the

experiment, a group of ten texture images were generated with variational curviness content using

ten different Perlin noise patterns, adding to an initial 2D Gaussian distribution pattern image.

Then, the common space was obtained from each of the discrete images to assess the differences

between the original rectangular image and its corresponding hexagonal image. The results show

that the space facilitates a usage friendly tool to address an arrangement and assess the changes

between different spatial arrangements by which, in the experiment, the hexagonal images show

richer intensity variation, nonlinear behavior, and larger dynamic range in comparison to the

rectangular images.

Keywords:

software-based; common space; hexagonal image; pixel arrangement; pixel form;

continuous extension; resampling

PACS: J0101

1. Introduction

The visual sensory of some of biological species can easily outperform our conventional vision

technology. Inspired by such efﬁcient machines, we have built our electronic systems which aim

to capture a scenery with the same efﬁcient style of performance by emulating the structure and

function of biological counterparts. The sensor structure, sensor form, and surface shape of eye

show a wide range of adaptations to meet the requirements of the organisms which bear them. Eye

performance of different species vary in their visual acuity—the range of wavelengths they can detect,

their sensitivity in low light, their ability to detect motion or to resolve objects, and whether they can

discriminate colors [

1

]. The spatial sensor arrangement of the eyes plays a signiﬁcant role in such

Sensors 2019,19, 568; doi:10.3390/s19030568 www.mdpi.com/journal/sensors

Sensors 2019,19, 568 2 of 19

variational performances [

2

]. The study of the evolution process of our visual system indicates how

our spatial sensor arrangement is evolved and differentiated from other species and especially from

the closest ones, the primates, which has resulted in the existence of variational spatial arrangement;

from densely hexagonal in the fovea to a sparse circular structure in the peripheral retina. The high

contrast and optimal sampling properties of our visual system are directly related to the densely

hexagonal spatial arrangement.

Today’s sensor spatial arrangement is inspired by our visual system. However, we have not

come further than rigid rectangular and, on a minor scale, hexagonal sensor arrangements. Some of

the obstacles in developing new sensor arrangements are the difﬁculty in manufacturing, the cost,

and rigidity of hardware components. The virtual deformation of the sensor arrangement [

3

] provides

new possibilities for overcoming such obstacles. We need strong arguments to convince the involved

partners in sensor development to implement the virtual deformation ideas. It is not enough to only

show that the virtual deformation sensor arrangement is feasible, but also, that the addressing of new

arrangements can be achieved easily and smoothly, without need of deﬁning new grid structures

which generally results in heavy computation. Thus, we propose a new method in the paper which

eliminates the need for deﬁning new grid structures for addressing different sensor arrangements.

One direct application of the proposed method is its implementation as an assessment tool where

different sensor arrangements are compared with each other; i.e., without the need for conversion of

one arrangement to another one.

In this paper, we propose a method to create a common space which facilitates addressing and

assessing different spatial arrangements of sensors, e.g., between the rectangular and hexagonal

arrangements. Such a space is created by implementing a continuous extension of discrete

Weyl Group orbit function transform which extends a discrete arrangement to a continuous one.

The implementation of the space is demonstrated by comparing two types of generated hexagonal

images from each rectangular image with two different methods of the half-pixel shifting and virtual

hexagonal method. In the experiment, a group of ten texture images are generated with variational

curviness content using ten different Perlin noise patterns, adding to an initial 2D Gaussian distribution

pattern image. Then, the common space is obtained from each of the discrete images to address and

assess the differences between the original rectangular image and its corresponding hexagonal image.

This paper is organized as follows. In Section 2, the addressing of arrangement is explained.

Then the two types of image generation are explained in Section 3. Sections 4and 5present the

methodology of the common space and the experiment setup, respectively. Then the results are shown

and discussed in Section 6. Finally, we summarize our work in Section 7.

2. Arrangement Addressing

In relation to the assessment of two images having two different arrangements; e.g., one having

square and another hexagonal arrangement, the addressing of arrangement is the most important

issue by which it becomes possible to access each arrangement unit (the pixel). Such access property

for any arrangement should be easy and fast in implementation, in comparison to the popular square

arrangement. The problem of any arrangement, beside the square one, is manifested in ﬁnding new

deﬁnitions for grid structures. Here, we elaborate on the problem for the hexagonal arrangement,

which has been studied for more than four decades, and different addressing methods are suggested.

A hexagonal arrangement is addressed using two oblique axes [

4

], also referred to as skewed coordinate

system in [

5

], and h2 system in [

6

], where two basis vectors are not orthogonal. With such an oblique

coordinate system, each hexagonal pixel is addressed by an ordered pair of unit vectors. A symmetrical

hexagonal coordinate frame which uses three coordinates instead of two is used to represent each

pixel on a grid plane [

7

,

8

]. The major advantage of this coordinate system is that there is a one-to-one

mapping between hexagonal and square arrangements. Moreover, in [

9

], this symmetrical hexagonal

coordinate frame is used to derive various afﬁne transformations. The geometric transformations

on the hexagonal grid are conveniently simpliﬁed and the symmetry property of the hexagonal grid

Sensors 2019,19, 568 3 of 19

is successfully preserved. The three-axis coordinate system is also used in [

10

] for mathematically

handling the hexagonal arrangement. Spiral Architecture, inspired from anatomical consideration

of the primate’s vision system, is proposed by [

11

] which is a 1D addressing system. This address

grows from the center of image in powers of seven along a spiral-like curve. This addressing scheme

combined with two later proposed mathematic operations, spiral addition and spiral multiplication,

is the basic Spiral Architecture [

11

,

12

]. A similar single-index system for pixel addressing is proposed

by modifying the Generalized Balanced Ternary system [

13

,

14

]. A virtual hexagonal structure is

proposed by the authors of [

15

] where the hexagonal pixels do not physically exist but are recorded

during image processing in the memory space. The approach demands high computation for image

conversion (from one arrangement to another) for determining the locations (or the areas) of each pixel.

A reduced computational complexity method is derived from the virtual hexagonal structure proposal

by the authors of [16].

3. Image Generation

In this section, we explain generation of two types of images which have hexagonal arrangements.

The images are generated from an original image with square arrangement. An example of such

images is demonstrated in Figure 1.

Sensors 2019, 19, x FOR PEER REVIEW 3 of 20

from anatomical consideration of the primate's vision system, is proposed by [11] which is a 1D

addressing system. This address grows from the center of image in powers of seven along a spiral-

like curve. This addressing scheme combined with two later proposed mathematic operations, spiral

addition and spiral multiplication, is the basic Spiral Architecture [11,12]. A similar single-index

system for pixel addressing is proposed by modifying the Generalized Balanced Ternary system

[13,14]. A virtual hexagonal structure is proposed by the authors of [15] where the hexagonal pixels

do not physically exist but are recorded during image processing in the memory space. The approach

demands high computation for image conversion (from one arrangement to another) for determining

the locations (or the areas) of each pixel. A reduced computational complexity method is derived

from the virtual hexagonal structure proposal by the authors of [16].

3. Image Generation

In this section, we explain generation of two types of images which have hexagonal

arrangements. The images are generated from an original image with square arrangement. An

example of such images is demonstrated in Figure 1.

(a) (b) (c)

Figure 1. The images on three types of sensory arrangements. (a) The original square image (SQ); (b)

hexagonal image (Hex_E); (c) half-pixel shift image (HS_E).

3.1. Generation of the Virtual Hexagonal Enriched Image (Hex_E)

The virtual hexagonal enriched image has a hexagonal pixel form on a hexagonal arrangement.

The generation process is similar to the resampling process in [17,18], which has three steps:

projecting the original image pixel intensities onto a grid of sub-pixels; estimating the values of

subpixels at the resampling positions; estimating each new hexagonal pixel intensity in a new

hexagonal arrangement where the subpixels are projected back to a hexagonal grid, which are shown

as red grids in Figure 2. In this arrangement the distance between each two hexagonal pixels is the

same and the resolution of the generated Hex_E image is the same as the original image.

3.2. Generation of the Virtual Half-Pixel Shift Enriched Image (HS_E)

The hexagonal grid in previous work [19,20] is mimicked by a half-pixel shift which is derived

from delaying sampling by a half pixel on the horizontal direction. The red grid, which is presented

in the middle of Figure 2, is the new pseudo hexagonal sampling structure whose pixel form is still

square. The new pseudo hexagonal grid is derived from a usual 2D grid by shifting each even row a

half pixel to the right and leaving odd rows unattached, or of course any similar translation. The

virtual Half-pixel Shift Enriched image (HS_E) is generated from the original enriched image [3]

which has a square arrangement.

Figure 1.

The images on three types of sensory arrangements. (

a

) The original square image (SQ);

(b) hexagonal image (Hex_E); (c) half-pixel shift image (HS_E).

3.1. Generation of the Virtual Hexagonal Enriched Image (Hex_E)

The virtual hexagonal enriched image has a hexagonal pixel form on a hexagonal arrangement.

The generation process is similar to the resampling process in [

17

,

18

], which has three steps: projecting

the original image pixel intensities onto a grid of sub-pixels; estimating the values of subpixels at the

resampling positions; estimating each new hexagonal pixel intensity in a new hexagonal arrangement

where the subpixels are projected back to a hexagonal grid, which are shown as red grids in Figure 2.

In this arrangement the distance between each two hexagonal pixels is the same and the resolution of

the generated Hex_E image is the same as the original image.

3.2. Generation of the Virtual Half-Pixel Shift Enriched Image (HS_E)

The hexagonal grid in previous work [

19

,

20

] is mimicked by a half-pixel shift which is derived

from delaying sampling by a half pixel on the horizontal direction. The red grid, which is presented in

the middle of Figure 2, is the new pseudo hexagonal sampling structure whose pixel form is still square.

The new pseudo hexagonal grid is derived from a usual 2D grid by shifting each even row a half

pixel to the right and leaving odd rows unattached, or of course any similar translation. The virtual

Half-pixel Shift Enriched image (HS_E) is generated from the original enriched image [

3

] which has a

square arrangement.

Sensors 2019,19, 568 4 of 19

Sensors 2019, 19, x FOR PEER REVIEW 4 of 20

(a) (b) (c)

Figure 2. Three types of sensory arrangements. (a) The sensor rearrangement onto the subpixel; (b)

the projection of the square pixels onto the hexagonal arrangement by half-pixel shifting method (i.e.,

HS_E image generation); (c) the projection of the square pixels onto the hexagonal grid in generation

of hexagonal image (Hex_E).

4. Common Space Based on Continuous Extension

To elaborate the common space, let us start with a simple 1D example. Assuming we have a

continuous 1D signal, it is not difficult to imagine that we can sample the signal with different time

intervals. However, the opposite way is not so easy; i.e., to obtain the continuous signal from different

time intervals. Further, this becomes even extremely difficult when we have sampled our data by a

certain time interval and try to use the data to resample according to another time interval. Here, for

the common space we have the last-mentioned condition where the sampled data is 2D and from the

image sensor. In this relation, the choice of spatial sensor arrangement affects the sampling results as

the choice of time interval in the 1D signal example. In the 2D sampling the data is sampled from a

continuous surface; i.e., each spatial sensor arrangement results in certain sampling data from certain

points on the continuous surface. By common space, we mean such continuous surface which is

created by continuous extension of spatial data; i.e., from sampling data from certain spatial sensor

arrangement a common space (a continuous surface) is generated. The common space is used to

estimate the sampling data according to another spatial sensor arrangement; i.e., a common space is

created by sampling data from hexagonal spatial arrangement and then the sampling data of a

rectangular spatial arrangement is estimated. In this way, on the common space, we have

correspondent points of each sampling point related to different spatial arrangements, which

facilitates the addressing and assessing of different spatial arrangements of sensors.

The common space is created by implementing a continuous extension of discrete Weyl Group

orbit function transform. Orbit functions on the Euclidean space are symmetrized exponential

functions. The symmetrization is fulfilled by a Weyl group corresponding to a Coxeter-Dynkin

diagram. The values of orbit functions are repeated on copies of a fundamental domain of the affine

Weyl group (determined by the initial Weyl group) in the entire Euclidean space. Recalling that the

exponential functions determine the Fourier transform on Euclidean space. Correspondingly, orbit

functions determine a symmetrized version of the Fourier transform which is also called an orbit

function transform. One of the key properties of orbit transform is that sequence of orbit transform,

and inverse orbit transform preserve the processed data. This property is preserved even when

discrete orbit function in the inverse orbit transform is replaced with a continuous orbit function of

the same family. In other words, for any symmetrical grid such as rectangular or hexagonal grid, in

frequency domain a continuous spectrum surface can be generated from the discrete information of

the grid. We call this continuous spectrum surface a common space. The creation and proof of such

common space is explicated in detail in Appendix A for interested readers.

The creation of continuous extension of the original data is independent of the data arrangement;

i.e., it is possible to create common space from any spatial arrangement, such as square or hexagonal

Figure 2.

Three types of sensory arrangements. (

a

) The sensor rearrangement onto the subpixel; (

b

) the

projection of the square pixels onto the hexagonal arrangement by half-pixel shifting method (i.e., HS_E

image generation); (

c

) the projection of the square pixels onto the hexagonal grid in generation of

hexagonal image (Hex_E).

4. Common Space Based on Continuous Extension

To elaborate the common space, let us start with a simple 1D example. Assuming we have

a continuous 1D signal, it is not difﬁcult to imagine that we can sample the signal with different

time intervals. However, the opposite way is not so easy; i.e., to obtain the continuous signal from

different time intervals. Further, this becomes even extremely difﬁcult when we have sampled our

data by a certain time interval and try to use the data to resample according to another time interval.

Here, for the common space we have the last-mentioned condition where the sampled data is 2D

and from the image sensor. In this relation, the choice of spatial sensor arrangement affects the

sampling results as the choice of time interval in the 1D signal example. In the 2D sampling the data is

sampled from a continuous surface; i.e., each spatial sensor arrangement results in certain sampling

data from certain points on the continuous surface. By common space, we mean such continuous

surface which is created by continuous extension of spatial data; i.e., from sampling data from certain

spatial sensor arrangement a common space (a continuous surface) is generated. The common space is

used to estimate the sampling data according to another spatial sensor arrangement; i.e., a common

space is created by sampling data from hexagonal spatial arrangement and then the sampling data

of a rectangular spatial arrangement is estimated. In this way, on the common space, we have

correspondent points of each sampling point related to different spatial arrangements, which facilitates

the addressing and assessing of different spatial arrangements of sensors.

The common space is created by implementing a continuous extension of discrete Weyl Group

orbit function transform. Orbit functions on the Euclidean space are symmetrized exponential

functions. The symmetrization is fulﬁlled by a Weyl group corresponding to a Coxeter-Dynkin

diagram. The values of orbit functions are repeated on copies of a fundamental domain of the afﬁne

Weyl group (determined by the initial Weyl group) in the entire Euclidean space. Recalling that

the exponential functions determine the Fourier transform on Euclidean space. Correspondingly,

orbit functions determine a symmetrized version of the Fourier transform which is also called an orbit

function transform. One of the key properties of orbit transform is that sequence of orbit transform,

and inverse orbit transform preserve the processed data. This property is preserved even when discrete

orbit function in the inverse orbit transform is replaced with a continuous orbit function of the same

family. In other words, for any symmetrical grid such as rectangular or hexagonal grid, in frequency

domain a continuous spectrum surface can be generated from the discrete information of the grid.

We call this continuous spectrum surface a common space. The creation and proof of such common

space is explicated in detail in Appendix Afor interested readers.

Sensors 2019,19, 568 5 of 19

The creation of continuous extension of the original data is independent of the data arrangement;

i.e., it is possible to create common space from any spatial arrangement, such as square or hexagonal

ones. We refer to these common spaces in relation to their original data arrangements, such as CSE_sq

or CSE_hex for the created common spaces from square and hexagonal arrangement, respectively.

On the common space, any grid structure is applied virtually; i.e., the corresponding addressing

of each pixel position from different arrangements are done on the common space. Thus, by knowing

the pixel form of each arrangement, the intensity value of each corresponding pixel is determined at

the pixel position on the common space.

5. Experimental Setup

Evaluating the proposed common space method in assessing different sensor structures is based

on using different generated images. In Section 3, the generated procedures of those types of image,

which are used in the evaluation, are all types of image that are originated from a rectangular

arrangement. Thus, generating images based on rectangular arrangement is essential for experimental

evaluation. On the other hand, to evaluate the addressing accuracy of the common space usage,

we need to generate such images which also have a content with random spatial variation in each

pixel. This is because by using the common space only one coordinate system is used to address each

pixel position and obtain its intensity value in two different arrangements; i.e., each pixel position and

intensity value of the originate arrangement to the common space is known, but the correspondent

position and intensity value on the other arrangement is estimated using the common space surface.

In relation to this, the evaluation of addressing accuracy can be achieved by measuring the estimations

error. The statistical validation of the estimations error requires the random spatial variation in each

pixel; i.e., as spatial variation in natural images. The estimations error can be measured for all pixels

of each two experimental images, using the common space addressing, or selected amount of their

correspondent pixels. In the experiments we used the latter option. To ensure that the selected pixels

represent different intensity levels it requires to generate the experimental images with a certain

intensity model; e.g., a Gaussian model.

An image dataset is created which consists of 10 high resolution (4096 by 2160) original images

(SQs) and their converted ones, of type of HS_E and Hex_E images with the same resolution;

i.e., the dataset has a total of 30 images, where the interval of subpixel is 30. The conversion process is

elaborated on in Section 3. Each of the ten original images is generated by adding a Gaussian image

(GI) to a random Perlin noise image (PI). The GI contributes to obtain all possible tonal levels in range

of 0–255 gray levels in each original image. Each GI is generated by

GI =255 ∗e−(x2

2σ2

1

+y2

2σ2

2

)(1)

where σ1and σ2are 1920 and 1280 respectively and the original images SQs is obtained by:

SQj=GI +P Ij(2)

where

j

is the image index number. The values of

σ1

and

σ2

are approximately half of the image

resolution in each direction. Based on the rule of thumb, GI represents fully a Gaussian intensity model

where the values of

σ1

and

σ2

are one third of image resolution in each direction. In this relation GI

is not fully a representative of a Gaussian intensity model. This is to prevent obtaining signiﬁcantly

lower level intensity values which can affect evaluation of addressing accuracy. By generating the

PI image, a pseudo-random spatial variation in each pixel is obtained which simulates variational

curviness content; i.e., we imitate the appearance of textures in natural images by a controlled random

process. In this way, using GI and PI, each original image of the dataset is generated to have natural

images properties and with wider range of variation than exists in a captured natural image. Each PI is

generated by implementing the Perlin noise algorithm [21,22] where each pixel of the image; PI(x, y),

Sensors 2019,19, 568 6 of 19

is computed by two major steps: (a) projection of pixel vector position on pseudorandom gradients of

→

g00 = [x00,y00]

,

→

g01 = [x01,y01]

,

→

g10 = [x10,y10]

, and

→

g11 = [x11,y11]

at integer points [0,0], [0,1], [1,0],

and [1,1], respectively, (b) interpolation and smoothing between points ‘value at the integer points by

a cubic spline function

S(x)=x2(3−2x)

and a linear interpolation function

L(ε,x,y)=x+ε(y−x)

as shown in Figure 3and explained by algorithm steps in Table 1. The PI contributes to obtain all

possible tonal levels in range of 0–255 gray levels. The range of

SQj

images; a combination of GI and

PI images where each has a range of 0–255 tonal levels, are normalized to obtain images with range of

0–255 tonal levels. The generation of SQ images is demonstrated in Figure 4.

Sensors 2019, 19, x FOR PEER REVIEW 6 of 20

steps in Table 1. The PI contributes to obtain all possible tonal levels in range of 0–255 gray levels.

The range of 𝑆𝑄 images; a combination of GI and PI images where each has a range of 0–255 tonal

levels, are normalized to obtain images with range of 0–255 tonal levels. The generation of SQ images

is demonstrated in Figure 4.

x

01

x

00

x

10

x

11

y

01

y

00

y

11

y

10

[0,0]

[1,1]

[0,1]

[1,0]

Figure 3. At integer grid points, 2D Perlin noise interpolates and smooths between pseudorandom

gradients.

Table 1. The algorithm of implemented 2D Perlin noise.

0. Input 𝑷

⃗

=[𝒙,𝒚]

1. 𝑺𝒙=𝑺(𝒙)

2. 𝑺𝒚=𝑺(𝒚)

3. 𝒖𝒂=𝑷

⃗

·𝒈

⃗

𝟎𝟎

4. 𝒗𝒂=𝑷

⃗

·𝒈

⃗

𝟏𝟎

5. 𝒂=𝑳(𝑺𝒙,𝒖𝒂,𝒗𝒂)

6. 𝒖𝒃=𝑷

⃗

·𝒈

⃗

𝟎𝟏

7. 𝒗𝒃=𝑷

⃗

·𝒈

⃗

𝟏𝟏

8. 𝒃=𝑳(𝑺𝒙,𝒖𝒃,𝒗𝒃)

9. Output 𝑳(𝑺𝒚,𝒂,𝒃)

(a) (b) (c)

Figure 4. Generation of an SQ image (a) is added to a Gaussian image: PI; (b): GI; (c): a random Perlin

noise image.

6. Results and Analysis

In this section, we show the addressing and assessment feasibility of three types of images of

SQ, HS_E, and Hex_E (i.e., having different pixel arrangements) using the common space. There are

ten of such triple types of images in the dataset and for each triple image type the results were

obtained in three stages of general preparation, case of CSE_sq and case of CSE_hex as it is shown in

the flowchart of Figure 5. The blue, green and red dash-line squares represent the image dataset

Figure 3.

At integer grid points, 2D Perlin noise interpolates and smooths between pseudorandom gradients.

Table 1. The algorithm of implemented 2D Perlin noise.

0. Input →

P=[x,y]

1. Sx=S(x)

2. Sy=S(y)

3. ua=→

P·→

g00

4. va=→

P·→

g10

5. a=L(Sx,ua,va)

6. ub=→

P·→

g01

7. vb=→

P·→

g11

8. b=L(Sx,ub,vb)

9. Output L(Sy,a,b)

Sensors 2019, 19, x FOR PEER REVIEW 6 of 20

steps in Table 1. The PI contributes to obtain all possible tonal levels in range of 0–255 gray levels.

The range of 𝑆𝑄 images; a combination of GI and PI images where each has a range of 0–255 tonal

levels, are normalized to obtain images with range of 0–255 tonal levels. The generation of SQ images

is demonstrated in Figure 4.

x

01

x

00

x

10

x

11

y

01

y

00

y

11

y

10

[0,0]

[1,1]

[0,1]

[1,0]

Figure 3. At integer grid points, 2D Perlin noise interpolates and smooths between pseudorandom

gradients.

Table 1. The algorithm of implemented 2D Perlin noise.

0. Input 𝑷

⃗

=[𝒙,𝒚]

1. 𝑺𝒙=𝑺(𝒙)

2. 𝑺𝒚=𝑺(𝒚)

3. 𝒖𝒂=𝑷

⃗

·𝒈

⃗

𝟎𝟎

4. 𝒗𝒂=𝑷

⃗

·𝒈

⃗

𝟏𝟎

5. 𝒂=𝑳(𝑺𝒙,𝒖𝒂,𝒗𝒂)

6. 𝒖𝒃=𝑷

⃗

·𝒈

⃗

𝟎𝟏

7. 𝒗𝒃=𝑷

⃗

·𝒈

⃗

𝟏𝟏

8. 𝒃=𝑳(𝑺𝒙,𝒖𝒃,𝒗𝒃)

9. Output 𝑳(𝑺𝒚,𝒂,𝒃)

(a) (b) (c)

Figure 4. Generation of an SQ image (a) is added to a Gaussian image: PI; (b): GI; (c): a random Perlin

noise image.

6. Results and Analysis

In this section, we show the addressing and assessment feasibility of three types of images of

SQ, HS_E, and Hex_E (i.e., having different pixel arrangements) using the common space. There are

ten of such triple types of images in the dataset and for each triple image type the results were

obtained in three stages of general preparation, case of CSE_sq and case of CSE_hex as it is shown in

the flowchart of Figure 5. The blue, green and red dash-line squares represent the image dataset

Figure 4.

Generation of an SQ image (

a

) is added to a Gaussian image: PI; (

b

): GI; (

c

): a random Perlin

noise image.

Sensors 2019,19, 568 7 of 19

6. Results and Analysis

In this section, we show the addressing and assessment feasibility of three types of images of SQ,

HS_E, and Hex_E (i.e., having different pixel arrangements) using the common space. There are ten of

such triple types of images in the dataset and for each triple image type the results were obtained in three

stages of general preparation, case of CSE_sq and case of CSE_hex as it is shown in the flowchart of Figure 5.

The blue, green and red dash-line squares represent the image dataset generation, case of CSE_sq and case

of CSE_hex respectively. The dot arrow shows the pixels are selected in the Hex_E, HS_E and SQ images.

The thick and thin arrows represent the process of image generation and applying the selected pixels on the

images respectively. Table 2lists the symbols in Figure 5with their meanings. We explain the three stages

and then discuss and analyze the obtained results which indicate the feasibility and accuracy of addressing

and assessment of random pixels from one arrangement to another one.

Sensors 2019, 19, x FOR PEER REVIEW 7 of 20

generation, case of CSE_sq and case of CSE_hex respectively. The dot arrow shows the pixels are

selected in the Hex_E, HS_E and SQ images. The thick and thin arrows represent the process of image

generation and applying the selected pixels on the images respectively. Table 2 lists the symbols in

Figure 5 with their meanings. We explain the three stages and then discuss and analyze the obtained

results which indicate the feasibility and accuracy of addressing and assessment of random pixels

from one arrangement to another one.

Hex_E SQHS_E

Hex_E

p

24

×

200 random

positions

HS_E

p

SQ

CEhex

HS_E

CEhex

Hex_E

CEorg

SQ

CEorg

HS_E

CEsq

Hex_E

CEsq

Conversion

Conversion

CSE_hex CSE_sq

SQ

p

24 su b-

range

Image dataset

Case of

CSE_sq

Case of

CSE_hex

Figure 5. The flowchart to discuss and analyze the obtained results.

Table 2. Description of used symbols.

Symbol Full name and size Sensor

arrangement

Originated

from Method

SQ Square image

4096 × 2160 square - -

Hex_E

Hexagonal enriched

image

4096 × 2160

hexagonal SQ Conversion

HS_E Half-pixel shift image

4096 × 2160 square SQ Conversion

𝑺𝑸𝒑 Square matrix image

200 × 24 square SQ

Pixel selection

on SQ

𝑯𝒆𝒙_𝑬𝒑

Hexagonal enriched

matrix image

200 × 24

hexagonal Hex_E

Pixel selection

on Hex_E

𝑯𝑺_𝑬𝒑

Half-pixel shift matrix

image

200 × 24

square HS_E image

Pixel selection

on HS_E

CSE_sq Common Space surface continuous

extension SQ image New method,

see Section 4

Figure 5. The ﬂowchart to discuss and analyze the obtained results.

Table 2. Description of used symbols.

Symbol Full Name and Size Sensor

Arrangement Originated from Method

SQ Square image

4096 ×2160 square - -

Hex_E

Hexagonal enriched image

4096 ×2160 hexagonal SQ Conversion

HS_E Half-pixel shift image

4096 ×2160 square SQ Conversion

SQpSquare matrix image

200 ×24 square SQ Pixel selection

on SQ

Hex_Ep

Hexagonal enriched

matrix image

200 ×24

hexagonal Hex_E Pixel selection on

Hex_E

Sensors 2019,19, 568 8 of 19

Table 2. Cont.

Symbol Full Name and Size Sensor

Arrangement Originated from Method

HS_Ep

Half-pixel shift

matrix image

200 ×24

square HS_E image Pixel selection

on HS_E

CSE_sq Common Space surface continuous

extension SQ image New method,

see Section 4

CSE_hex Common Space surface continuous

extension Hex_E image New method,

see Section 4

SQCEorg

Estimated Square

matrix image

200 ×24

square CSE_sq Pixel selection on

the CSE_sq

Hex_ECEsq

Estimated Hexagonal

matrix image

200 ×24

hexagonal CSE_sq Pixel selection on

the CSE_sq

HS_ECEsq

Estimated Half-pixel shift

matrix image

200 ×24

square CSE_sq Pixel selection on

the CSE_sq

SQCEhex

Estimated Square

matrix image

200 ×24

square CSE_hex Pixel selection on

the CSE_hex

HexCEorg

Estimated Hexagonal

matrix image

200 ×24

hexagonal CSE_hex Pixel selection on

the CSE_hex

HSCEhex

Estimated Half-pixel shift

matrix image

200 ×24

square CSE_hex Pixel selection on

the CSE_hex

6.1. General Preparation

Each SQ image in the data set is an eight bits image; i.e., the range of intensity values is between

0 and 255. The pixels of each SQ image are partitioned by having 24 intensity sub-ranges (e.g., 10–19,

. . .

, 190–199, 240–250) to investigate in more detail the tonal variation. In each sub-range, 200-pixel

positions are selected randomly in each SQ image; i.e., 24 by 200 pixels are chosen randomly meanwhile

assuring to have different tonal levels and representative of the whole intensity range. The 24 intensity

sub-ranges are related to the statistical requirement of having a pixel population in which we can

select 200 pixels positions. According to our observation from the generated images, a binning

of 10 tonal levels could fulﬁll the requirement where each intensity sub-range has at least a pixel

population of 1%. Figure 6shows a typical pixel population for 25 intensity sub-ranges. The ﬁrst

intensity sub-range; with tonal levels between 0–9. And the last sub-range with tonal levels between

251–255 have less than the pixel population of 1% which accordingly will be discard in the pixel

selection process. The 200 random pixels in each intensity sub-range is because they contain sufﬁcient

spatial intensity variation information in a certain sub-range of tonal variations to underpin statistical

analysis. Using the pixel positions, the relative intensity values from SQ, HS_E and Hex_E images

are organized in new images of

SQp

,

HS_Ep

, and

Hex_Ep

respectively; each with size of 200 by 24.

The pixels of each column of such an image are ordered by sorting the linear indexing of the 200

random selected pixels in each intensity-subrange.

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Sensors 2019, 19, x FOR PEER REVIEW 9 of 20

Figure 6. A typical pixel population for 25 intensity sub-ranges.

6.2. In Case of CSE_sq

The common space of each SQ image, CSE_sq, is created according to Section 4. Using the

common space of CSE_sq and the pixel positions of a SQ image the corresponding pixel positions

and the related intensity values are estimated for SQ, HS_E, and Hex_E image types. Accordingly, in

correspondent to a 𝑆𝑄, three images of 𝑆𝑄,𝐻𝑆_𝐸

, 𝑎𝑛𝑑 𝐻𝑒𝑥_𝐸 are generated.

6.3. In Case of CSE_hex

The common space of each Hex_E image, CSE_hex, is created according to Section 4. As with

Case 6.1, by using the pixel positions of SQ image and the common space of CSE_hex,the

corresponding pixel positions and the related intensity values are estimated for SQ, HS_E, and Hex_E

image types. Accordingly, corresponding to a 𝐻𝑒𝑥_𝐸three images of 𝑆𝑄, 𝐻𝑆 and

𝐻𝑒𝑥 are generated.

6.4. Analysis of the Two Cases

In cases of CSE_sq or CE_hex, the images with a square or a hexagonal arrangement originate

the respective common spaces. Generally, in the process of obtaining the results by using a common

space and a pixel position in the originated image to the common space, the corresponding pixel

position and its intensity value are estimated for another type of image which has another

arrangement in comparison to the originated image. Here, we address the three questions of a) How

different are any two generated common spaces which are originated from two different

arrangements; e.g., the comparison of generated 𝑆𝑄 (representative of CSE_sq common space)

and 𝐻𝑒𝑥_𝐸 (representative of CSE_hex common space)? b) How similar are any generated

common space and its originated image; e.g., the comparison of 𝑆𝑄 to 𝑆𝑄 or 𝐻𝑒𝑥_𝐸 to

𝐻𝑒𝑥_𝐸? c) What is the accuracy of implementing any common space in addressing and

assessment between two types of arrangements; e.g., from SQ to Hex_E?

We generated ten CSE_sq and ten CSE_hex common spaces from the related images in the

dataset; i.e., each SQ image and its converted Hex_E image were used to create each related CSE_sq

and CSE_hex (a pair of common spaces). For each pair of the common spaces a pixel set of 200 chosen

pixels (see 6.1 and 6.2) of the originated images were chosen and organized as images. In this way,

ten 𝑆𝑄 and ten 𝐻𝑒𝑥_𝐸 images are obtained where each has size of 200 by 24 and represent

the relative common space. Question (a) is answered by comparison of the 𝑆𝑄 and 𝐻𝑒𝑥_𝐸

Figure 6. A typical pixel population for 25 intensity sub-ranges.

6.2. In Case of CSE_sq

The common space of each SQ image, CSE_sq, is created according to Section 4. Using the

common space of CSE_sq and the pixel positions of a SQ image the corresponding pixel positions

and the related intensity values are estimated for SQ, HS_E, and Hex_E image types. Accordingly,

in correspondent to a SQp, three images of SQCEorg ,HS_EC Esq ,and Hex_EC Esq are generated.

6.3. In Case of CSE_hex

The common space of each Hex_E image, CSE_hex, is created according to Section 4. As with

Case 6.1, by using the pixel positions of SQ image and the common space of CSE_hex, the corresponding

pixel positions and the related intensity values are estimated for SQ, HS_E, and Hex_E image types.

Accordingly, corresponding to a

Hex_Ep

three images of

SQCEhex

,

HSC Ehex

and

HexCEorg

are generated.

6.4. Analysis of the Two Cases

In cases of CSE_sq or CE_hex, the images with a square or a hexagonal arrangement originate the

respective common spaces. Generally, in the process of obtaining the results by using a common space

and a pixel position in the originated image to the common space, the corresponding pixel position

and its intensity value are estimated for another type of image which has another arrangement in

comparison to the originated image. Here, we address the three questions of (a) How different

are any two generated common spaces which are originated from two different arrangements;

e.g., the comparison of generated

SQCEorg

(representative of CSE_sq common space) and

Hex_ECEorg

(representative of CSE_hex common space)? (b) How similar are any generated common space and its

originated image; e.g., the comparison of

SQp

to

SQCEorg

or

Hex_Ep

to

Hex_ECEorg

? (c) What is the

accuracy of implementing any common space in addressing and assessment between two types of

arrangements; e.g., from SQ to Hex_E?

We generated ten CSE_sq and ten CSE_hex common spaces from the related images in the dataset;

i.e., each SQ image and its converted Hex_E image were used to create each related CSE_sq and

CSE_hex (a pair of common spaces). For each pair of the common spaces a pixel set of 200 chosen

pixels (see 6.1 and 6.2) of the originated images were chosen and organized as images. In this way,

ten

SQCEorg

and ten

Hex_ECEorg

images are obtained where each has size of 200 by 24 and represent

the relative common space. Question (a) is answered by comparison of the

SQCEorg

and

Hex_ECEorg

images. Figure 7shows the results of such comparisons where the absolute intensity value difference

of ten

SQCEorg

and

Hex_ECEorg

are measured. In the ﬁgure the colors from blue to yellow indicate that

Sensors 2019,19, 568 10 of 19

the difference value increases from 0 to 0.2. The total mean square error (MSE) between images shown

in Figure 7is 0.002 and multiple correlation among the images is 99.39%. The low MSE and high

correlation indicate that it is feasible to create almost the same common space for the two arrangements

of square and hexagonal. The created common spaces are close, but as expected, is not exactly the same;

e.g., a hexagonal arrangement has richer frequency spectrum than the square one which contributes to

obtain richer frequency spectrum on respective common space [23].

Sensors 2019, 19, x FOR PEER REVIEW 10 of 20

images. Figure 7 shows the results of such comparisons where the absolute intensity value difference

of ten 𝑆𝑄 and 𝐻𝑒𝑥_𝐸 are measured. In the figure the colors from blue to yellow indicate

that the difference value increases from 0 to 0.2. The total mean square error (MSE) between images

shown in Figure 7 is 0.002 and multiple correlation among the images is 99.39%. The low MSE and

high correlation indicate that it is feasible to create almost the same common space for the two

arrangements of square and hexagonal. The created common spaces are close, but as expected, is not

exactly the same; e.g., a hexagonal arrangement has richer frequency spectrum than the square one

which contributes to obtain richer frequency spectrum on respective common space [23].

Figure 7. Comparison of CSE_sq and CSE_Hex. The absolute intensity value difference of ten 𝑆𝑄

and 𝐻𝑒𝑥_𝐸 are shown.

Question (b) is answered by the comparison of 𝑆𝑄 to 𝑆𝑄 and 𝐻𝑒𝑥_𝐸 to 𝐻𝑒𝑥_𝐸

images. The results of such comparisons where the absolute intensity value difference of ten of 𝑆𝑄

to 𝑆𝑄 and 𝐻𝑒𝑥_𝐸 to 𝐻𝑒𝑥_𝐸 images are shown in Figures 8 and 9 respectively. The total

MSE between and multiple correlation among the images in Figure 8 is 0.0005 and 99.93%

respectively. In Figure 9, the total MSE between images is 0.00019 and multiple correlation among

them is 99.85%. The low MSE and high correlation in the results of the figures indicate that the

generated common spaces are very alike to their respective originated images but they are not strictly

the same.

Question (c) is answered by examining each case of CSE_sq and CSE_hex in addressing and

assessment between different types of arrangements. In case of CSE_sq ten of each 𝐻𝑒𝑥_𝐸,

𝐻𝑆_𝐸, and 𝑆𝑄 images are obtained, and they are compared to 𝐻𝑒𝑥_𝐸, 𝐻𝑆_𝐸, and 𝑆𝑄 (i.e.,

the representatives of the images of Hex_E, Hs_E, and SQ). In case of CSE_hex ten of each

𝐻𝑒𝑥_𝐸, 𝐻𝑆_𝐸, and 𝑆𝑄 images are obtained, and they are compared to 𝐻𝑒𝑥_𝐸, 𝐻𝑆_𝐸,

and 𝑆𝑄. Figures 10 and 11 show two examples of such comparison between 𝑆𝑄 to 𝑆𝑄 and

𝐻𝑒𝑥 to 𝐻𝑒𝑥_𝐸 respectively.

In Figure 10, the total MSE between the ten 𝑆𝑄 and 𝑆𝑄 is 0.0059 and multiple correlation

between them is 99.03%. In Figure 10 the total MSE between the ten of 𝐻𝑒𝑥 and 𝐻𝑒𝑥_𝐸is 0.0099

and correlation between the pixel sets is 98.26%. The results in the Figures 7–10 show that by

implementing the common space, it is feasible to address different arrangements where the intensity

difference between any random pixel which is addressed via common space or via conversion is very

small.

Figure 7.

Comparison of CSE_sq and CSE_Hex. The absolute intensity value difference of ten

SQCEorg

and Hex_ECEor g are shown.

Question (b) is answered by the comparison of

SQp

to

SQCEorg

and

Hex_Ep

to

Hex_ECEorg

images.

The results of such comparisons where the absolute intensity value difference of ten of

SQp

to

SQCEorg

and

Hex_Ep

to

Hex_ECEorg

images are shown in Figures 8and 9respectively. The total MSE between

and multiple correlation among the images in Figure 8is 0.0005 and 99.93% respectively. In Figure 9,

the total MSE between images is 0.00019 and multiple correlation among them is 99.85%. The low MSE

and high correlation in the results of the ﬁgures indicate that the generated common spaces are very

alike to their respective originated images but they are not strictly the same.

Question (c) is answered by examining each case of CSE_sq and CSE_hex in addressing and

assessment between different types of arrangements. In case of CSE_sq ten of each

Hex_ECEsq

,

HS_EC Esq

, and

SQCEorg

images are obtained, and they are compared to

Hex_Ep

,

HS_Ep

, and

SQp

(i.e., the representatives of the images of Hex_E, Hs_E, and SQ). In case of CSE_hex ten of each

Hex_ECEorg

,

HS_EC Ehex

, and

SQCEhex

images are obtained, and they are compared to

Hex_Ep

,

HS_Ep

,

and

SQp

. Figures 10 and 11 show two examples of such comparison between

SQCEhex

to

SQp

and

HexCEsq to Hex_Eprespectively.

In Figure 10, the total MSE between the ten

SQCEhex

and

SQp

is 0.0059 and multiple correlation

between them is 99.03%. In Figure 10 the total MSE between the ten of

HexCEsq

and

Hex_Ep

is

0.0099 and correlation between the pixel sets is 98.26%. The results in the Figures 7–10 show that by

implementing the common space, it is feasible to address different arrangements where the intensity

difference between any random pixel which is addressed via common space or via conversion is

very small.

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Figure 8. Comparison of the common space of CSE_sq and its originated image of SQ. The absolute

intensity value difference of ten 𝑆𝑄 and 𝑆𝑄 are shown.

Figure 9. Comparison of the common space of CSE_hex and its originated image of Hex_E. The

absolute intensity value difference of ten 𝐻𝑒𝑥_𝐸 and 𝐻𝑒𝑥_𝐸 are shown.

Figure 8.

Comparison of the common space of CSE_sq and its originated image of SQ. The absolute

intensity value difference of ten SQpand SQCEorg are shown.

Sensors 2019, 19, x FOR PEER REVIEW 11 of 20

Figure 8. Comparison of the common space of CSE_sq and its originated image of SQ. The absolute

intensity value difference of ten 𝑆𝑄 and 𝑆𝑄 are shown.

Figure 9. Comparison of the common space of CSE_hex and its originated image of Hex_E. The

absolute intensity value difference of ten 𝐻𝑒𝑥_𝐸 and 𝐻𝑒𝑥_𝐸 are shown.

Figure 9.

Comparison of the common space of CSE_hex and its originated image of Hex_E. The

absolute intensity value difference of ten Hex_Epand Hex_ECEorg are shown.

Sensors 2019, 19, x FOR PEER REVIEW 12 of 20

Figure 10. Comparison of ten 𝑆𝑄 and 𝑆𝑄 images.

Figure 11. Comparison of 𝐻𝑒𝑥 and 𝐻𝑒𝑥_𝐸.

In each case of CSE_sq or CSE_hex, the intensity average and variance in the 24 tonal sub-ranges

of ten corresponding pixel sets of each 𝐻𝑒𝑥_𝐸, 𝐻𝑆_𝐸, 𝑆𝑄 or 𝐻𝑒𝑥_𝐸, 𝐻𝑆_𝐸,

𝑆𝑄 are shown in Figures 12 and 13 respectively. The figures show that it is feasible to assess

pixels on different arrangements due to the estimation of pixel position and the intensity value in

different arrangement by using common space and without the need for any conversion means (see

Section 4). The pixel sets from hexagonal arrangement show the highest average intensity value and

variance in each type of common space indicating richer intensity variation and larger dynamic range

compared to SQ the other pixel sets. Figure 14 shows the mean (a) and variance (b) of ratio values of

ten corresponding pixel sets between each SQ and 𝑆𝑄 to Hex_E image. The mean (a) shows the

nonlinear relation between SQ to Hex_E which was previously shown in [3,18]. The mean (a) also

shows that the relation between 𝑆𝑄 to Hex_E is similar to the relation between SQ to Hex_E and

behaves in a nonlinear manner. The variance (b) shows that the relation between SQ and 𝑆𝑄to

Hex_E are similar and nonlinear.

Figure 10. Comparison of ten SQCEhex and SQpimages.

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Figure 10. Comparison of ten 𝑆𝑄 and 𝑆𝑄 images.

Figure 11. Comparison of 𝐻𝑒𝑥 and 𝐻𝑒𝑥_𝐸.

In each case of CSE_sq or CSE_hex, the intensity average and variance in the 24 tonal sub-ranges

of ten corresponding pixel sets of each 𝐻𝑒𝑥_𝐸, 𝐻𝑆_𝐸, 𝑆𝑄 or 𝐻𝑒𝑥_𝐸, 𝐻𝑆_𝐸,

𝑆𝑄 are shown in Figures 12 and 13 respectively. The figures show that it is feasible to assess

pixels on different arrangements due to the estimation of pixel position and the intensity value in

different arrangement by using common space and without the need for any conversion means (see

Section 4). The pixel sets from hexagonal arrangement show the highest average intensity value and

variance in each type of common space indicating richer intensity variation and larger dynamic range

compared to SQ the other pixel sets. Figure 14 shows the mean (a) and variance (b) of ratio values of

ten corresponding pixel sets between each SQ and 𝑆𝑄 to Hex_E image. The mean (a) shows the

nonlinear relation between SQ to Hex_E which was previously shown in [3,18]. The mean (a) also

shows that the relation between 𝑆𝑄 to Hex_E is similar to the relation between SQ to Hex_E and

behaves in a nonlinear manner. The variance (b) shows that the relation between SQ and 𝑆𝑄to

Hex_E are similar and nonlinear.

Figure 11. Comparison of HexCEsq and Hex_Ep.

In each case of CSE_sq or CSE_hex, the intensity average and variance in the 24 tonal sub-ranges

of ten corresponding pixel sets of each

Hex_ECEsq

,

HS_EC Esq

,

SQCEorg

or

Hex_ECEorg

,

HS_EC Ehex

,

SQCEhex

are shown in Figures 12 and 13 respectively. The ﬁgures show that it is feasible to assess pixels

on different arrangements due to the estimation of pixel position and the intensity value in different

arrangement by using common space and without the need for any conversion means (see Section 4).

The pixel sets from hexagonal arrangement show the highest average intensity value and variance in

each type of common space indicating richer intensity variation and larger dynamic range compared

to SQ the other pixel sets. Figure 14 shows the mean (a) and variance (b) of ratio values of ten

corresponding pixel sets between each SQ and

SQCEhex

to Hex_E image. The mean (a) shows the

nonlinear relation between SQ to Hex_E which was previously shown in [

3

,

18

]. The mean (a) also

shows that the relation between

SQCEhex

to Hex_E is similar to the relation between SQ to Hex_E and

behaves in a nonlinear manner. The variance (b) shows that the relation between SQ and

SQCEhex

to

Hex_E are similar and nonlinear.

Sensors 2019, 19, x FOR PEER REVIEW 13 of 20

Figure 12. Intensity average of ten corresponding pixel sets of each 𝐻𝑒𝑥_𝐸, 𝐻𝑆_𝐸, 𝑆𝑄 or

𝐻𝑒𝑥_𝐸, 𝐻𝑆_𝐸, 𝑆𝑄

Figure 13. Variance of ten corresponding pixel sets of each 𝐻𝑒𝑥_𝐸, 𝐻𝑆_𝐸, 𝑆𝑄 or

𝐻𝑒𝑥_𝐸, 𝐻𝑆_𝐸, 𝑆𝑄.

Figure 12.

Intensity average of ten corresponding pixel sets of each

Hex_ECEsq

,

HS_ECEsq

,

SQCEorg

or

Hex_ECEor g,HS_ECEhex,SQCEhex .

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Figure 12. Intensity average of ten corresponding pixel sets of each 𝐻𝑒𝑥_𝐸, 𝐻𝑆_𝐸, 𝑆𝑄 or

𝐻𝑒𝑥_𝐸, 𝐻𝑆_𝐸, 𝑆𝑄

Figure 13. Variance of ten corresponding pixel sets of each 𝐻𝑒𝑥_𝐸, 𝐻𝑆_𝐸, 𝑆𝑄 or

𝐻𝑒𝑥_𝐸, 𝐻𝑆_𝐸, 𝑆𝑄.

Figure 13.

Variance of ten corresponding pixel sets of each

Hex_ECEsq

,

HS_ECEsq

,

SQCEorg

or

Hex_ECEor g,HS_ECEhex,SQCEhex .

Sensors 2019, 19, x FOR PEER REVIEW 14 of 20

(a) (b)

Figure 14. The mean (a) and variance (b) of ratio values of ten corresponding pixel sets between each

SQ and 𝑆𝑄 to Hex_E.

The pixel sets on corresponding arrangements via two types of common spaces are compared

and shown in Table 3. The comparison shows the correlation and MSE relation between each pair of

pixel sets. The results in the table indicate the feasibility of addressing each type of common space to

the same type of arrangement due to small MSE and high correlation values. The similar results of

correlation and MSE in Table 4 shows the assessment feasibility of different arrangements by

comparison of the pixel sets on different arrangement and via two types of common spaces.

Table 3. Comparison of pixel sets on corresponding arrangements via two types of common spaces.

Image Index

Pair of

pixel

sets

No.1 No.2 No.3 No.4 No.5 No.6 No.7 No.8 No.9 No.10

Corre-

lation

𝑺𝑸𝑪𝑬𝒐𝒓𝒈

𝑺𝑸𝑪𝑬𝒉𝒆𝒙 99.63% 99.63% 99.62% 99.61% 99.66% 99.64% 99.63% 99.63% 99.61% 99.63%

𝑯𝑺_𝑬𝑪𝑬𝒔

𝒒

𝑯𝑺_𝑬𝑪𝑬

𝒉

98.25% 98.45% 98.28% 98.39% 98.79% 98.32% 98.41% 97.76% 98.49% 99.79%

𝑯𝒆𝒙_𝑬𝑪

𝑬

𝑯𝒆𝒙_𝑬𝑪

𝑬

98.23% 98.44% 98.26% 98.39% 98.78% 98.32% 98.41% 97.77% 98.49% 99.78%

MSE

𝑺𝑸𝑪𝑬𝒐𝒓𝒈

𝑺𝑸𝑪𝑬𝒉𝒆𝒙 0.0024 0.0021 0.0038 0.0046 0.0040 0.0038 0.0038 0.0044 0.0039 0.0005

𝑯𝑺_𝑬𝑪𝑬𝒔

𝒒

𝑯𝑺_𝑬𝑪𝑬

𝒉

0.0070 0.0054 0.0092 0.0085 0.0080 0.0091 0.0055 0.0065 0.0060 0.0080

𝑯𝒆𝒙_𝑬𝑪

𝑬

𝑯𝒆𝒙_𝑬𝑪

𝑬

0.0104 0.0080 0.0136 0.0125 0.0119 0.0135 0.0081 0.0095 0.0088 0.01193

Table 4. Assessment by comparison of the pixel sets on different arrangements via two types of

common spaces.

Image Index

Pair of

pixel

sets

No.1 No.2 No.3 No.4 No.5 No.6 No.7 No.8 No.9 No.10

Corre-

lation

𝑺𝑸𝑪𝑬𝒐𝒓𝒈

𝑯𝑺_𝑬𝑪𝑬

𝒉

98.19% 98.17% 98.00% 98.00% 97.96% 97.89% 98.02% 97.96% 97.99% 98.07%

𝑺𝑸𝑪𝑬𝒐𝒓𝒈

𝑯𝒆𝒙_𝑬𝑪

𝑬

98.23% 98.16% 98.01% 97.94% 97.95% 97.88% 98.04% 97.98% 97.97% 98.07%

𝑯𝑺_𝑬𝑪𝑬𝒔

𝒒

𝑺𝑸𝑪𝑬𝒉𝒆𝒙 95.99% 96.19% 96.12% 96.20% 96.67% 96.16% 96.03% 95.49% 96.08% 97.76%

Figure 14. The mean (a) and variance (b) of ratio values of ten corresponding pixel sets between each

SQ and SQC Ehex to Hex_E.

The pixel sets on corresponding arrangements via two types of common spaces are compared

and shown in Table 3. The comparison shows the correlation and MSE relation between each pair

of pixel sets. The results in the table indicate the feasibility of addressing each type of common

space to the same type of arrangement due to small MSE and high correlation values. The similar

results of correlation and MSE in Table 4shows the assessment feasibility of different arrangements by

comparison of the pixel sets on different arrangement and via two types of common spaces.

Table 3. Comparison of pixel sets on corresponding arrangements via two types of common spaces.

Image Index

Pair of

Pixel Sets No.1 No.2 No.3 No.4 No.5 No.6 No.7 No.8 No.9 No.10

Corre-lation

SQCEorg

SQCEhex

99.63% 99.63% 99.62% 99.61% 99.66% 99.64% 99.63% 99.63% 99.61%

99.63%

HS_ECEsq

HS_ECEhex

98.25% 98.45% 98.28% 98.39% 98.79% 98.32% 98.41% 97.76% 98.49%

99.79%

Hex_ECEsq

Hex_ECEorg

98.23% 98.44% 98.26% 98.39% 98.78% 98.32% 98.41% 97.77% 98.49%

99.78%

Sensors 2019,19, 568 14 of 19

Table 3. Cont.

Image Index

Pair of

Pixel Sets No.1 No.2 No.3 No.4 No.5 No.6 No.7 No.8 No.9 No.10

MSE

SQCEorg

SQCEhex

0.0024 0.0021 0.0038 0.0046 0.0040 0.0038 0.0038 0.0044 0.0039 0.0005

HS_ECEsq

HS_ECEhex 0.0070 0.0054 0.0092 0.0085 0.0080 0.0091 0.0055 0.0065 0.0060 0.0080

Hex_ECEsq

Hex_ECEorg 0.0104 0.0080 0.0136 0.0125 0.0119 0.0135 0.0081 0.0095 0.0088 0.01193

Table 4.

Assessment by comparison of the pixel sets on different arrangements via two types of

common spaces.

Image Index

Pair of

Pixel Sets No.1 No.2 No.3 No.4 No.5 No.6 No.7 No.8 No.9 No.10

Corre-lation

SQCEorg

HS_ECEhex

98.19% 98.17% 98.00% 98.00% 97.96% 97.89% 98.02% 97.96% 97.99%

98.07%

SQCEorg

Hex_ECEhex

98.23% 98.16% 98.01% 97.94% 97.95% 97.88% 98.04% 97.98% 97.97%

98.07%

HS_ECEsq

SQCEhex

95.99% 96.19% 96.12% 96.20% 96.67% 96.16% 96.03% 95.49% 96.08%

97.76%

HS_ECEsq

Hex_ECEorg

98.22% 98.42% 98.22% 98.37% 98.77% 98.28% 98.39% 97.72% 98.47%

99.75%

Hex_ECEsq

SQCEhex

95.98% 96.19% 96.13% 96.20% 96.66% 96.18% 96.02% 95.52% 96.07%

97.76%

Hex_ECEsq

HS_ECEhex

98.24% 98.46% 98.30% 98.40% 98.79% 98.35% 98.41% 97.79% 98.49%

99.79%

SQCEorg

HS_ECEsq

96.43% 96.78% 96.55% 96.51% 96.93% 96.59% 96.55% 96.20% 96.51%

98.05%

SQCEorg

Hex_ECEsq

96.42% 96.78% 96.56% 96.51% 96.91% 96.59% 96.54% 96.22% 96.50%

98.04%

SQCEhex

HS_ECEhex

97.93% 97.84% 97.88% 97.86% 97.87% 97.77% 97.68% 97.70% 97.70%

97.99%

SQCEhex

Hex_ECEorg

97.98% 97.84% 97.90% 97.80% 97.86% 97.76% 97.72% 97.72% 97.68%

97.99%

HS_ECEsq

Hex_ECEorg

99.98% 99.98% 99.98% 99.98% 99.98% 99.98% 99.98% 99.98% 99.98%

99.98%

HS_ECEhex

Hex_ECEorg

99.90% 99.90% 99.89% 99.90% 99.90% 99.89% 99.90% 99.90% 99.89%

99.91%

MSE

SQCEorg

HS_ECEhex 0.0044 0.0037 0.0072 0.0071 0.0075 0.0068 0.0039 0.0047 0.0043 0.0069

SQCEorg

Hex_ECEorg 0.9823 0.9816 0.9801 0.9794 0.9795 0.9788 0.9804 0.9798 0.9797 0.9807

HS_ECEsq

SQCEhex

0.0089 0.0080 0.0096 0.0100 0.0086 0.0100 0.0100 0.0107 0.0099 0.0032

HS_ECEsq

Hex_ECEorg 0.0031 0.0027 0.0032 0.0029 0.0022 0.0031 0.0034 0.0036 0.0034 0.0012

Hex_ECEsq

SQCEhex

0.0255 0.0236 0.0274 0.0283 0.0259 0.0283 0.0278 0.0288 0.0274 0.0120

Hex_ECEsq

HS_ECEhex 0.0238 0.0204 0.0288 0.0274 0.0270 0.0287 0.0201 0.0220 0.0212 0.0279

SQCEorg

HS_ECEsq 0.0051 0.0048 0.0049 0.0050 0.0043 0.0049 0.0048 0.0052 0.0050 0.0030

SQCEorg

Hex_ECEsq 0.0148 0.0148 0.0136 0.0131 0.0123 0.0141 0.0138 0.0137 0.0135 0.01257

SQCEhex

HS_ECEhex 0.0018 0.0020 0.0021 0.0019 0.0022 0.0021 0.0023 0.0023 0.0021 0.0078

SQCEhex

Hex_ECEhex 0.0053 0.0064 0.0042 0.0051 0.0044 0.0046 0.0089 0.0083 0.0079 0.0029

HS_ECEsq

Hex_ECEsq 0.0061 0.0061 0.0061 0.0061 0.0061 0.0061 0.0061 0.0061 0.0061 0.0061

HS_ECEhex

Hex_ECEorg 0.0039 0.0040 0.0036 0.0036 0.0036 0.0036 0.0041 0.0040 0.0041 0.0035

Sensors 2019,19, 568 15 of 19

7. Conclusions

In the paper we proposed a method to create a common space, which eliminates the need for

deﬁning new grid structures for addressing different sensor arrangements. We showed the feasibility of

addressing and assessing different spatial arrangements of sensors, speciﬁcally between the rectangular

and hexagonal arrangements. We explained how the common space is created by implementing

a continuous extension of discrete Weyl Group orbit function transform, which extends a discrete

arrangement to a continuous one. The results indicate that the common space facilitates an easy tool

for addressing any pixel position on any arrangement and speciﬁcally we showed such facilitation

on square and hexagonal arrangements. It was also shown that the tool has signiﬁcant property to

assess the changes between different spatial arrangements by which, in the experiment, the pixel sets

on hexagonal images show richer intensity variation, nonlinear behavior, and larger dynamic range in

comparison to the pixel sets on rectangular images.

Author Contributions:

Data curation, W.W. and S.K.; Formal analysis, W.W. and S.K.; Methodology, O.K.;

Software, O.K.; Supervision, G.C.; Writing—original draft, W.W. and S.K.; Writing—review and editing, W.W.

and S.K.

Funding: This research received no external funding.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

Appendix A

A.1. Root System

A root system is a conﬁguration of vectors in a Euclidean space satisfying certain geometrical

properties. Let us deﬁne a root system as a ﬁnite set of non-trivial vectors

∆={αi∈Rn}

that fulﬁl

three conditions:

•Roots αi∈∆spanRn

•

If

αi∈∆

, then

λα ∈∆⇔λ∈{−1, 1}

: every root system contains only two scalar

multiples of each root: the root itself and its reﬂection,

•α,β∈∆⇒γαβ∈∆

: root system is closed under reﬂection with respect to

hyperplanes orthogonal to roots.

γαβ

denotes reﬂection of root

β

with respect

to hyperplane orthogonal to root α.

So-called crystallographic root systems also fulﬁl the fourth condition: ∀α,β∈∆:2(α,β)

(α,α)∈Z

We can unambiguously choose a set of simple roots

Σ⊂∆

. Simple roots fulﬁl two extra conditions:

•all simple roots are linearly independent,

•

every root

αi∈∆

, can be expressed as a linear combination of simple roots, such

that all coefﬁcients of this linear combinations are either all non-negative (such root is

called positive root), or are all non-positive (negative root).

When each root is expressed as a linear combination of simple roots, we can introduce ordering of

roots. So-called highest of roots is denoted

ξ

and is expressed as

ξ=m1α1+m2α2

, where

α1

,

α2

are

simple roots. Coefﬁcients

m1

,

m2

are called marks. There are several signiﬁcant sets of vectors that

are related to each root system: set of co-roots

(a∨

i)

, weights

(ωi)

and co-weights

(ω∨

j)

. Co-roots and

co-weights are normalized variants of roots and weights, respectively:

a∨

i=2αi

hαi,αii(A1)

ω∨

j=2ωi

hωi,ωii(A2)

Sensors 2019,19, 568 16 of 19

Roots and weights are dual to each other, in the following sense:

hαj,ω∨

ji=ha∨

i,ωii=δij (A3)

These four sets of vectors are used to form four lattices (root lattice Q, co-root lattice

Q∨

weight

lattice P and co-weight lattice

P∨

) which will be used in the deﬁnition of discrete orbit function. All four

lattices are deﬁned in the following manner:

Q=Zα1+Zα2Q∨=Zα∨

1+Zα∨

2

P=Zω1+Zω2P∨=Zω∨

1+Zω∨

2

Each of these lattices can have its non-negative part (denoted with superscript +) and positive

part (denoted with superscript ++).

A.2. Weyl Groups

When having root system

∆

composed of roots

αi

, we deﬁne

ri

, as a reﬂection with respect to

root

αi

. Set of reﬂections

ri

will generate so-called Weyl group W. Afﬁne Weyl group is an extension

of Weyl group, it is generated by reﬂections

ri

plus reﬂection

r0

, which is a reﬂection with respect to

highest root

ξ

Weyl group orbit of point x is a ﬁnite set of points generated by all actions of Weyl group

W. Similarly, the afﬁne orbit of point x is generated by all actions of

Waff

on point x, however, afﬁne

orbit is an inﬁnite set, due to the reﬂection

r0

. Fundamental region of

Waff

is a closed subset of

Rn

such

that it contains exactly one point of each afﬁne Weyl group orbit. The fundamental region for afﬁne

Weyl groups in R2space can be chosen a convex hull of points n0, ω∨

1

m1,ω∨

2

m2o.

The dual root system

∆∨

is obtained as system of co-roots. Reﬂections related to dual root system

∆∨

generate dual Weyl group

ˆ

W

. Dual Weyl group

ˆ

W

has its fundamental region

F∨

and can be

extended to afﬁne dual Weyl group

ˆ

Waff

. Since roots and co-roots differ only with their lengths, both

W and

ˆ

W

generated by the same sets of reﬂections. However, highest co-root

η=m∨

1α∨

1+m∨

2α∨

2

differs from highest root

ξ

in both length and direction, and thus the dual afﬁne Weyl group

ˆ

Waff

is not

the same as

Waff

. As a consequence,

F6=F∨

. Root systems are not the only way how to generate Weyl

groups. Roots of simple Lie algebras coincide with simple roots-designation of Lie algebras are often

used to designate Weyl groups generated by reﬂections with respect to roots of given Lie algebra.

A.3. Orbit Functions and Orbit Transforms

Weyl group orbit functions were deﬁned for all simple Lie algebras

(An, Bn, Cn,

Dn, G2, F4, E6, E7, and E8)

and they can be used for generalized Fourier analysis of data on the

fundamental region F of the corresponding Weyl groups. This theory allows for similar discretization

as in the case of common Fourier discrete analysis studies, and can be used for the analysis of digitized

data on the fundamental region.

Sine, cosine functions, plus

eix

are generalized to systems with nonorthonormal basis through

orbit functions. Moreover, certain Weyl groups provide more types of functions, e.g.,

C2and G2

Weyl

groups allow us to deﬁne

Cs

,

Cl

,

Sland Ss

functions, as described in [

23

]. A_2 Weyl group provide

only straightforward generalization of cosine, sine and complex exponential functions. These orbit

functions are generally, i.e., regardless of underlying Weyl group, deﬁned as:

Φλ(x)=∑

ω∈W

ei2πhωλ,xi(A4)

ϕλ(x)=∑

ω∈W

det(ω)ei2πhωλ,xi(A5)

Ξλ(x)=∑

ω∈W

ei2πhωλ,xi(A6)

Sensors 2019,19, 568 17 of 19

where parameter

x∈Rn

and label

λ∈Q

Since the orbit functions are invariant to operations of

W or We

, respectively, we can restrict the parameter x to the fundamental region F or even fundamental

region

Fe

, respectively. Since S-orbit function

ϕ

is anti-symmetric, it vanishes for x on boundary of F

and for

λ

on reﬂection hyperplane. Through these two facts, the restriction of x and

λ

looks as follows:

Φλ(x): x ∈F, λ∈P+,

ϕλ(x): x ∈e

F, λ∈P++,

Ξλ(x): x ∈Fe,λ∈P+∪r1P++,

the e

F denotes the interior of fundamental region F.

For the discretization of orbit functions, we choose arbitrary ﬁxed positive integer M that deﬁnes

the density of the lattice. The discrete fundamental region

FM

is constructed as an intersection of

fundamental region F and stretched subset of lattice P∨:

FM=1

MP∨/Q∨∩F=(s1

Mω∨

1+· · · +sn

Mω∨

ns0+

n

∑

i=1

s1mi=M, s0, s1, . . . , sn∈Z≥0)(A7)

For discrete orbit functions we use set

ΛM

, which is a set of discrete labels

λ

. The parameter M

has the same meaning as for discrete fundamental region. The set ΛMis expressed as follows:

FM=P/MQ ∩MF∨=(s1ω1+· · · +snωns0+

n

∑

i=1

s1m∨

i=M, s0, s1, . . . , sn∈Z≥0)(A8)

Due to the invariance of functions to the actions of Weyl group W, and the (anti-)symmetry of

functions, discrete orbit functions can be restricted in the following way:

Φλ(x): x ∈FM,λ∈ΛM,

ϕλ(x): x ∈f

FM,λ∈g

ΛM,

Ξλ(x): x ∈Fe

M,λ∈Λe

M,

For further relations, scalar product over discrete fundamental region is crucial. Having two

discrete functions

f(x)

and

g(x)

, deﬁned over discrete fundamental region with density M, we deﬁne

their scalar product as

hf, giFM=∑

x∈FM

ε(x)f(x)g(x)(A9)

Note that the region of

FM

, may change depending on the used orbit function. E.g.,

when computing

hf

,

ϕλiFM

, we can omit boundary of

FM

since

ϕλ=

0 on the boundary of

FM

,

thus hf, ϕλiFM=hf, ϕλif

FM.

For λ,λ0∈ΛM, the orbit functions hold the orthogonality relation:

hΦλ,Φλ0iFM=cM2|W||stabW(λ)|δλλ0

hϕλ,ϕλ0if

FM=cM2|W||stabW(λ)|δλλ0

hΞλ,Ξλ0iFe

M0=cM2|We||stabWe(λ)|δλλ0

(A10)

M is the density of the discrete fundamental region, c denotes the determinant of Cartan matrix

for the underlying group W, Cartan matrix

C=cijn

i,j=1

,

cij =hαi

,

αji

.

|W|

is the order of group W,

stabW(λ)

is the stabilizer of the point

λ

under W. Generally speaking, the

stabG(x)

is a maximum

subgroup of G, such that it holds g(zx)=x∀g∈stabG(x).

Sensors 2019,19, 568 18 of 19

Since orbit functions are pairwise orthogonal over the ﬁnite region, we can expand the discrete

functions f(x), g(x) and h(x) into a ﬁnite series of orbit functions:

f(x)=∑

λ∈ΛM

F(Φ)

λΦλ(x)x∈FM,Φ−orbit transform

g(x)=∑

λ∈f

FM

G(ϕ)

λϕλ(x)x∈f

FM,ϕ−orbit transform

h(x)=∑

λ∈Fe

M

H(Ξ)

λΞλ(x)x∈Fe

M,Ξ−orbit transform

(A11)

Function f(x) needs to be deﬁned on

FM

, g(x) must be deﬁned on

f

FM

and h(x) is deﬁned on

Fe

M

.

The F denotes the spectrum of discrete function f. The superscripts

Φ

,

ϕ

and

Ξ

are used for distinction

between different kinds of spectra and are not commonly used.

The spectra points are given by

F(Φ)

λ=hf,ΦλiFM

hΦλ,ΦλiFM

,λ∈ΛM

G(ϕ)

λ=hg,ϕλig

FM

hϕλ,ϕλig

FM

,λ∈g

ΛM

H(Ξ)

λ=hh,ΞλiFe

M

hΞλ,ΞλiFe

M

,λ∈Λe

M

(A12)

A.4. Continuous Extension

One of the key properties of orbit transform is that sequence of orbit transform and inverse orbit

transform preserve the processed data, e.g.,

f=∑λ∈ΛM

hf,ΦλiFM

hΦλ,ΦiFM

Φλ

. This property is preserved even

when discrete orbit function in the inverse orbit transform Equation (A11) is replaced with continuous

orbit function of the same family:

f(x)=∑

λ∈ΛM

F(Φ)

λΦλ(z)x∈Rn

g(x)=∑

λ∈g

ΛM

G(ϕ)

λϕλ(z)x∈Rn

h(x)=∑

λ∈Λe

M

H(Ξ)

λΞλ(z)x∈Rn

(A13)

In this case, we obtain a continuous extension of the original data. As proven, see [

2

], certain

families of orbit functions can provide high-quality approximation with quick convergence to the

original continuous data.

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