Subpixel Edge Reconstruction using Aliased Pixel Brightness

Abstract

One of the most common image features used in machine vision are edges, and there is a substantial body of research oil various techniques for performing edge detection. Edges are useful in many applications as image comparing, recognition and other. Here is presented edge detection method with subpixel accuracy. Method is based on decision that different intensity and size areas influence pixel brightness with some relation function. Hear presented functions to calculate one dot of edge going through the pixel. Test results show that with 0.01 standard deviation is estimated 47% of dots, with 0.05 standard deviation is estimated 88% of dots and 94% with 0.06 standard deviation. Also it is defined, that linearity decrease is more than 5% when edge Cut triangle which area is less then 10% of pixel area. III. 7, bibl. 6 (in English; summaries in English, Russian and Lithuanian).

Full text

43
ELECTRONICS AND ELECTRICAL ENGINEERING
ISSN 1392 – 1215 2008. No. 8(88)
ELEKTRONIKA IR ELEKTROTECHNIKA
ELECTRONICS
T170 ELEKTRONIKA
Subpixel Edge Reconstruction using Aliased Pixel Brightness
V. Vyšniauskas
Šiauliai University,
Vilniaus str. 141, LT – 76353 Šiauliai, Lithuania
Introduction
One of the most common image features used in
machine vision are edges, and there is a substantial body of
research on various techniques for performing edge
detection. Edge is an imaginable line that separates two
regions with different luminosity. When luminosity hops
sharply, edge is well visible or imaginary, but when change
of luminosity is slight – edge is very light or nearly
invisible.
There are many methods for edge detection, but most
of them can be grouped into two categories, zero-crossing
and search-based. The search-based edge detects methods
looking for maximum and minimum in the first derivative
of the image. The zero-crossing based methods search for
zero crossings in the second derivative of the image [1].
A drawback with using edges is that not only do edge
detectors but also extract meaningful and useful edges. In
addition, many other spurious ones that arise from noise
and small changes in intensity values. If all such edges are
kept then the resulting image is hard for subsequent
processing. Large number of edge points can seriously
increase computational amount and decrease result quality
with remaining fraction of noise. The alternative is to
select a subset of edges for further analysis and ignore the
remaining fractions. Generally, a threshold on the gradient
magnitude of pixels solves this problem. Unfortunately, in
practice edge thresholding often done intuitively and
frequently requiring user tuning of parameters [2]. Higher
threshold level usage results lost of some necessary edges
contrariwise lower threshold level leaves more
unnecessary fractions. Optimal threshold level defined for
each image individually sometimes separately for different
parts of the same image [2, 3].
Image projection and digitizing
Image cameras have a lens to gather the incoming
light and focus all or part of the image on the image sensor
surface. Image sensor is flat panel, with sensitive to light
elements. Image sensors grouped in two categories,
analogue and digital. Analogue sensors output analogue
signal, which is used in analogue television or digitalized
to obtain digital image. Digital image sensors are made
from millions square shape light sensors that are capturing
light and converting it into electrical signals. Sensors are
organized in rows and columns as rectangular matrix. One
light sensor represents a single dot in the image with some
luminosity and are named pixel. Pixel of gray scale image
represents with one brightness digit, each pixel of colour
image represented as three colors – red, green and blue
brightness digits. Most popular quantization is 8 bits or 1
byte but also are used 10 or more bits quantization. Image
quality depends on quantization directly.
Each pixel of such image is neither a dot nor a square
but an abstract sample. Pixels could be reproduced at any
size and shape as a slight visible dot or a square.
Actually image sensor consists of square shape pixels,
which have some micrometers in size. Each pixel of image
has a fixed position and variable brightness which
represents average luminosity on pixel area (Fig. 1). Black
diagonal line shows edge between light and dark areas.
Therefore, a projected image light and dark part
covers some pixel surface partially. These pixels obtain
average brightness from both light and dark parts. This
phenomenon is named pixel aliasing [4] and presents in
grayscale and color images. Aliased pixel brightness
depends on light and dark areas ratio and luminosity of
these areas.
It is evident that this pixel brightness is directly
proportional to dark and light area ratio and difference of
these areas luminosity is a rate factor.
Fig. 1. Pixel aliasing phenomenon

44
Edge reconstruction from aliased pixel brightness
Decision about aliased pixel brightness is described as
formula (1).
 
D
P
L
DLP B
S
S
BBB 







 , (1)
where P
B– average (visible) brightness of pixel, L
B–
brightness of light area, D
B– brightness of dark area, P
S–
whole pixel area, L
S– pixel light area.
Brightness of the light L
Band dark D
Bareas is
obtainable from nearest neighbour pixels in opposite sides
of aliased pixel.
Light area L
Sis an integral of commonly unknown
function of an edge )(xfy

.
dx
xfSL

1
0
)( . (2)
Fig. 2. Trapezium area
To simplify task, let’s replace function segment with
line (Fig. 2). It is possible because pixel is a smallest peace
of the image represented as a dot or a square with
monotonous brightness. Yet another assumption that edge
line cross two opposing sides of pixel. Assumed the
decisions light area L
Sis calculated as trapezium.


2bahSL (3)
or
m
hSL , (4)
where 2bam  is average of vertical length of the
trapezium. Pixel side length is marked h. Pixel light area
and a whole pixel area ratio is the same as mand hratio.
h
m
S
S
P
L. (5)
Value mcan be formulated from formulas (1) and (5)











DL
DP
BB
BB
hm . (6)
To reconstruct edge position with subpixel accuracy
four steps must be done: 1) to find aliased pixel; 2) to
select two nearest opposite pixels with highest and lowest
brightness; 3) to calculate mline value by formula (6), that
is a distance from brightest pixel border to estimated dot
on edge; 4) to draw a line through these dots that estimates
real edge with subpixel accuracy (Fig. 3).






75
.040200401601
1m;






50.040200401201
2m;






25
.04020040801
3m.
Calculation is illustrated in (Fig. 3). Here 9 pixels are
shown and edge (dash and line) goes through a middle
pixel row. Light pixel brightness is 200, dark – 40 and
aliased with brightness 160, 120, 80. From pixel brightness
calculated distances mand these dots used to draw an
estimated edge line. The distance mare calculated from the
pixel brightness and these dots are used to draw estimated
edge.
Different situation is when an edge crosses adjacent
sides of pixel and intercepts triangle area. This situation is
more complicated because triangle area (Fig. 4) is not
linear function when edge dot travels via diagonal from
one corner to diagonally opposite corner. This area also
depends on angle between edge and pixel horizontal or
vertical side. That additionally complicates problem
decision.
To simplify the task take assumption that edge is
parallel to pixel diagonal, then


yx mm .
Fig. 3. Subpixel edge dots calculation
Fig. 4. Edge intercepts triangle area

45
Two different formulas (7) are used to calculate
estimated edge dot coordinates x
m,y
m. The first one (upper
formula) calculates coordinates when a dot is in the first
half of pixel (Fig.4a), and the second formula is used to
calculate coordinates when pixel is in the second half of
pixel (Fig. 4b).



















2
,
2
2
,
2
2h
m
Sh
h
h
m
S
mm
x
L
x
L
yx ; (7)
where
BrRh
BB
BB
hS
DL
DP
L










 22 , (8)











DL
DP
BB
BB
BrR . (9)
where BrR – brightness ratio.
Final formula is:






























D
DL
P
D
DL
P
B
BB
B
BrR
h
B
BB
B
BrR
h
m
2
,
2
1
1
2
,
2; (10)
Formulas describe the relation of dots coordinates
and area size that is an S shape curve. In this function
selected region with linearity better than 5%. It is Area (S)
range from 0.1 to 0.9 where subpixel dot coordinates (m)
range varies from 0.15 to 0.85. It means that, when edge
intercepts small triangle that area is less than 0.1 (10%) of
pixel size, calculation is inaccurate. Such pixels can be
ignored or another calculation algorithm must be used.
Edge reconstruction where used both methods shown in
Fig. 6.
Testing and results
Artificial pictures with only one straight-line edge
between dark and light areas were used for testing. There
were used pictures with known different edge angles. This
decision was made to simplify testing and results analysis.
Table 1. Standard deviation cumulative percents
There were tested 14 pictures and calculate 1050 dots.
Each calculated dot position was compared with known
angle straight-line and calculated deviation. Test results
were drawn as histogram of edge reconstruction deviation
shown in Fig. 7. Table 1 shows standard deviation
cumulative percents Method accuracy is six or less percent.
0.01 (1%) 47% 0.02 (2%) 61%
0.03 (3%) 73% 0.04 (4%) 82%
0.05 (5%) 88% 0.06 (6%) 94%
Fig. 5. Area (S) to dot coordinates (m) linearity diagram
Fig. 6. Both methods for edge reconstruction usage
Fig. 7. Edge reconcrtuction deviation histogram

46
Conclusion
Well-known edge detectors cany, sobel, perwit and
other detect edge from blurred images to reduce noise [5,
6]. Trash-hold level is used to extract edge line.
Unfortunately, in practice edge thresholding is often done
intuitively and frequently requiring user tuning of
parameters. Accordingly, edge line width is of one or more
pixels accuracy. These methods are unusable in
applications where high accuracy is need.
Aliased pixel contains information about ratio
between light and dark areas that have covered pixel. This
topic uncovers methods how to get point coordinates with
subpixel precision. Presented point of edge estimation
precision is about 5 percent (Table 1) of the pixel width.
This method does not require Gaussian blur and threshold
turning. Edge detection (restoration) precision is a part of
pixel.
The main uncovered problem of this method is aliased
pixel detection that will be solved in future.
References
1. Russ John C. The image processing handbook, 5th ed. –
CRC Press Taylor & Francis Group. – 2006. – P. 19–25, 135–
145, 292–315.
2. Ramanauskas Nerijus The Investigation of Eye Tracking
Accuracy using Synthetic Images // Elektronika ir
elektrotechnika. – Kaunas: Technologija, 2003. – Nr.4(46). –
P. 17–20.
3. Ritter G. X., Wilson J. N. Handbook of Computer Vision
Algorithms in Image Algebra. – CRC Press. – 1996. – P.
105–121.
4. Ling Guan, Sun-Yuan Kung, Jan Larsen. Multimedia
image and video processing. – CRC Press LLC. – 2001. – P.
83–111.
5. Hansen C., Johnson C. R. The Visualisation Hand Book. –
Elsevier Butterworth–Heinemann. – 2005. – P. 150–162.
6. Nixon M. S., Aguado A. S. Feature Extraction and Image
Processing. – 2002. – P. 99–130.
Received 2008 02 19
V. Vyšniauskas. Subpixel Edge Reconstruction using Aliased Pixel Brightness // Electronics and Electrical Engineering. –
Kaunas: Technologija, 2008. – No. 8(88). – P. 43–46.
One of the most common image features used in machine vision are edges, and there is a substantial body of research on various
techniques for performing edge detection. Edges are useful in many applications as image comparing, recognition and other. Here is
presented edge detection method with subpixel accuracy. Method is based on decision that different intensity and size areas influence
pixel brightness with some relation function. Hear presented functions to calculate one dot of edge going through the pixel. Test results
show that with 0.01 standard deviation is estimated 47% of dots, with 0.05 standard deviation is estimated 88% of dots and 94% with
0.06 standard deviation. Also it is defined, that linearity decrease is more than 5% when edge cut triangle which area is less then 10% of
pixel area. Ill. 7, bibl. 6 (in English; summaries in English, Russian and Lithuanian).
В. Вишняускас.Восстановление контура используя яркость пикселя // Электроника и электротехника. – Каунас:
Технология, 2008. – № 8(88). – С. 43–46.
Контур –одна из самых общих характеристик изображения, используемых в машинном зрении.Существует множество
различных методов для обнаружения контура. Контур полезен во многих применениях таких как сравнивнение, опознавание
изображения и других.Представляется метод обнаружения контура с точностью до доли пикселя. Метод основан на решении,
что зоны разной интенсивности и размера влияют на яркость пикселя и представляют некоторую функцию.Представлены
функции для вычисления точки контура, находящейся на пикселе. Результаты исследования показывают, что с 0.01
стандартными отступлениями определены 47 % точек,с0.05 стандартными отступлениями – 88 % точек и 94 % точек с 0.06
стандартными отступлениями. Также определена нелинейность более 5 %, когда контур отсекает треугольник площадью менее
10 % от площади пикселя.Ил. 7, библ. 6 (на английском языке; рефераты на английском,русском и литовском яз.).
V. Vyšniauskas. Vaizdo kontūrųatkūrimas naudojant persidengusiųtaškųryškumą// Elektronika ir elektrotechnika. –
Kaunas: Technologija, 2008. – Nr. 8(88). – P. 43–46.
Vaizdo kontūro nustatymas yra viena išbendriausiųvaizdųpalyginimo, atpažinimo ir kitokio apdorojimo charakteristikų. Vaizdo
pakeitimas kontūru leidžia gerokai sumažinti kompiuterio skaičiavimųtrukmę. Kontūrams nustatyti naudojami įvairūs metodai
Pristatomas metodas vaizdo kontūrui nustatyti pikselio dalies tikslumu. Metodas paremtas tuo, kad skirtingo ryškio sritys, dengiančios tą
patįpikselįtam tikru proporcingumu, daro įtakąbendram pikselio ryškumui. Pateikiamos funkcijos per pikselįeinančio kontūro taško
koordinatėms rasti. Tyrimais nustatyta, kad su 0,01 neapibrėžtimi nustatomi 47 % taškų, su 0,05 neapibrėžtimi – 88 % taškų, o su 0,06
neapibrėžtimi – 94 % taškų. Taip pat nustatyta, kad netiesiškumas viršija 5 %, kai kontūras atkerta trikampį, kurio plotas sudaro 10 %
pikselio ploto. Tokius pikselius reikia ignoruoti. Il. 7, bibl. 6 (anglųkalba; santraukos anglųrusųir lietuviųk.).