A Research of Boundary Extraction Based on Zero Crossing of Second Directional Derivatives
Qiu Zhicheng
Liu Yutong
(Research Institute of Surveying and Mapping, 16 Beitaipinglu, 100039, Beijing, China)
ABSTRACT:
At present, discrete edge features can be extracted by many edge extraction methods. Because these edge features
are not exact boundary, it is difficult to use in the image analysis and classification.
In this paper, a new boundary extraction approach is introduced based on zero crossing of second directional
derivatives, heuristic searching of artifical intelligence and manual editing.
Using this approach, the boundary on image can be extracted accurately and extraction quality of boundary can be
greatly improved.
KEY WORDS: Edge extraction, Zero crossing, Derivative, Artifical intelligence
1. Introduction
Edge or boundary, generally corresponds to great
change of geometry or physical property of scene, it has
been widely used as the important features in two and
three dimension computer vision. Edge extraction has
become an important research subject in image process-
ing for many years.
Differential operator is a powerful means for ex-
ploring the features of function change, many kinds of
operator have been proposed in recent than 20 years. In
order to improve the accuracy and speed many improve-
ment methods have been proposed also [1][2][3].
The present research results show that most edge
extraction algorithms exist the following problems, for
some algorithms edge feature points along the edge
could be extracted but it is not real edge.
For some algorithms real edge points can be ex-
tracted, but it is discrete. Besides image matching, it is
difficult to use in other area. In order to improve the
quality of edge extraction, a research on boundary ex-
traction based on zero crossing of second derivatives has
been introduced. À test on remote sensing image has
been executed, and the test results indicate that this ap-
proach is successful.
2. The principle of Edge Extraction Using Zero Cross-
ing of Second Directional Derivatives
In digital image. edge generally means that bright-
ness value has great change or the derivatives of bright-
ness value has partical extreme value. More precisely, a
pixel called edge must has the following condition:
within the area around pixel, zero crossing of second di-
rectional derivatives exist on gradient direction.
Digital image grey generally is discrete, for deter-
mining edge accuratelly, the discrete grey value should
be represented by a fitting function. Orthogonal basis
has been selected. There are following relationship for
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discrete orthogonal polynomial.
> Pr (r) (ro + apy" + ... + ar +a,) = 0
r€R
This is a linear equation, after solving the front 4 poly-
nomial function formulas are:
Pic) 1. Pr) =r
P,(r) = 2^ — [734779 P,03) — n — (ml Dr
where: 4k — S SN
SER
For two dimensional discrete orthogonal polynomi-
al, it can be constituted by two one dimension orthogo-
nal polynomial using tensor product.
Suppose R and C are two dimension fitting inter-
val, let {Po(r),
mial on R, {@ (c), … , Que) } is a set of discrete poly-
nomial on C, then, {Po (7) + @o(c), … , Pur) * Qu(c))
is a set of discrete orthogonal polynomial on RXC.
we , Py(r)} is a set of discrete polyno-
Using this relations, a fitting formula of two di-
mensional image can be derived and first derivatives,
second derivatives can be found.
1. Fitting using discrete othogonal polynomial
Suppose R is fitting interval in row direction and
has symmetrical features and n elements, C is in column
direction with fitting interval of symmetrical features
and also has n elements, using tensor product, two di
mension discrete or thogonal Pm (r, ¢) can constituted.
After derivation, coefficient for fitting is:
S SP. (rc) *dí(r,c)
r€R cec
S UL,
SER SEC
Fitting polynomial Q (r, c) can be expressed by the fol-
Os
(D
lowing formula:
K
Que S a, t Pure) (2)
m=0
2. Method of Edge Determination using Directional
Derivatives
According to the definition of edge derivative, it
