Researchers extend their previous research on a highly structured and compact algebraic representation of grey-level images which can be viewed as fuzzy sets. Addition and multiplication are defined for the set of all grey-level images, which can then be described as polynomials of two variables. Utilizing this new algebraic structure, researchers devised an innovative, efficient edge detection scheme. An accurate method for deriving gradient component information from this edge detector is presented. Based upon this new edge detection system researchers developed a robust method for linear feature extraction by combining the techniques of a Hough transform and a line follower. The major advantage of this feature extractor is its general, object-independent nature. Target attributes, such as line segment lengths, intersections, angles of intersection, and endpoints are derived by the feature extraction algorithm and employed during model matching. The algebraic operators are global operations which are easily reconfigured to operate on any size or shape region. This provides a natural platform from which to pursue dynamic scene analysis. A method for optimizing the linear feature extractor which capitalizes on the spatially reconfiguration nature of the edge detector/gradient component operator is discussed

Lerner, Bao-Ting

Morelli, Michael V.

English

NASA Technical Reports Server

N90-25577
EXTENSIONS OF ALGEBRAIC IMAGE OPERATORS:
AN APPROACH TO MODEL-BASED VISION1
Bao.Ting Lerner 2
Associate Professor
US Naval Academy
Annapolis, Maryland 21402
andl
NASA/Goddard Space Flight Center
Robotics Laboratory
Greenbelt, Maryland 20771
Michael V. Morelli
Electrical Engineering Department
Fairleigh Dickinson University
Teaneck, New Jersey 07666
and
Jackson and Tull Chartered Engineers
NASA/Goddard Space Flight Center
Robotics Laboratory
Greenbelt, Maryland 20771
ABSTRACT
We extend our previous research on a highly structured and compact
algebraic representation of grey-level images which can be viewed as fuzzy
sets. Addition and multiplication are defined for the set of all grey-level
images, which can then be described as polynomials of two variables.
Utilizing this new algebraic structure, we have devised an innovative,
efficient edge detection scheme. An accurate method for deriving gradient
component information from this edge detector is presented.
Based upon this new edge detection system we have developed a robust method
for linear feature extraction by combining the techniques of a Hough
transform and a line follower. The major advantage of this feature extractor is
its general, object-independent nature. Target attributes, such as line segment
lengths, intersections, angles of intersection, and endpoints are derived by
the feature extraction algorithm and employed during model matching.
The algebraic operators are global operations which are easily reconfigured to
operate on any size or shape region. This provides a natural platform from
which to pursue dynamic scene analysis. We discuss a method for optimizing
the linear feature extractor which capitalizes on the spatially reconfigurable
nature of the edge detector/gradient component operator.
1 This research was supported by the National Aeronautics and Space Administration
(NASA) under grant S-92580-D.
2 Principal Author, to whom correspondence should be addressed.
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INTRODUCTION
There have been various algebraic approaches proposed for "the description
and generation of binary images[2][6]. In this paper we extend our previous
work[Ill3] on a polynomial architecture based on fuzzy sets for the
represention, interpretation and manipulation of grey-scale imagery. An
edge detector and a "compass" gradient component operator are defined for
this algebraic system, and are employed as the front end of a robust linear
feature extractor.
ALGEBRAIC REPRESENTAION OF GREY-LEVEL IMAGES
In a previous paper [1][3] we introduced a polynomial architecture for the
description of grey-level (and binary) images. If Z denotes the integers, then
an image A may be represented as
A = {(x,y) e Z x Z l_tA(x,y) • [0,I]},
where p.A(x,y), denoting the grey-level of pixel (x,y), is a real number
between 0 and 1 with the larger numbers corresponding to the darker grey-
levels. In this way A may be viewed as a fuzzy set.
Image Addition and Multiplication:
Let A and B be grey-level images. We define the addition of A and B as:
A +B = "{(x,y)eAUB [_tA+ B =
min[max()-tA(x,Y),_tB(x,Y)),max(l- _tA(x,y),l%tB(x,y)) ] } (1)
Observe that in the binary case (with tx = either 1 or 0) the addition reduces to
the "exclusive-or" addition defined by Agui, et.al. [2]. That is, in binary the set
theoretic addition of images A and B is
A+B=(AUB)('h (_, t,J -B)
where _, and B are the inverse images of A and B respectively.
We define the multiplication of grey-level images A and B by:
A'B= { l .A..(x,+x.,+r,).
(xl,Yl): A,
(x2,Y2)_ B
rnin [_tA(Xl,Yl),_B(x2,Y2)] }
where :: denotes the addition performed by (1).
(2)
(3)
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Note that the operations of + and * arc commutative and associative, thus
making the set of all grey-level images a commutative semigroup under each
of these operations.
Polynomial
Under the
section, we may represent an image A as the polynomial
Representation of Grey-level Images:
addition and multiplication operations defined in the previous
 nxmy"A ffi2.
m
(m,n)eA (4)
where the coefficient is the grey-level of pixel (m,n) and the summation is
defined by (1).
ALGEBRAIC EDGE DETECTORS
Agui, et.al [2] described the classical differential operator
DI = (l+x)(l+y) (5)
and applied it to binary images to produce contour-like figures. The action of
D1 on the nine pixel square resulted in a "spotty" edge or contour as depicted
in figure 1.
(a) (b)
Figure 1. (a) Simple binary nine pixel image A, (b) the result
of Dl*A, the differential operator acting on A
We then varied the classical differential operator to the form
D2- l+xy (6)
This new, more compact irreducible polynomial operator gave a substantially
improved contour enhancement capability. Its action on a nine pixel square
appears in figure 2.
Although D2 proved to be a simple, efficient operator for contour
enhancement in images gvith a preponderance of vertical and horizontal lines,
its major drawback was the introduction of a bias :along the -45 degree line, as
can be seen in figure 2. In the next section we shall discuss the derivation of
accurate gradient component information from analysis of this bias.
689
(a) (b)
Figure 2. Operation of D2*A. Figure 3, Operation of D3*A.
To overcome this bias we varied the differentialoperator on an image A to
D3 *A = {(m,n)e (A+Axy)k.J(Ax+Ay) I _l.D3,A(m,n)=
max [_tA +Axy( m,n),_tAx +A y(m,n) ]} (7)
Figure 3 depicts the action of D3 on a nine pixel square. Notice the continuous,
unbroken contour. The definition of the rain-max addition produces a natural
threshold value of 0.5. Unlike classical edge detectors which require the
determination of the optimal threshold from the statisticsof the contoured
image, this natural threshold is a constant for all conditions, thus greatly
simplifying the production of the edge from the contour. The edge produced
by the action of each of the three defined differentialoperators on a
complicated scene is shown in figure 4a 4c.
........ (b) ¸
Figure4, (a)The operationOfDI
on a complex scene.Noticethe
"spotty"edge.(b)D2 appliedtothe
same scene.Thisedgeisbetter,
however thebiasisapparenton
thediagonalstruts.(c)The operation
ofD3 on thescene.Thisoperator
containsno biasand producesa
solidedge.
(c)
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GRADIENT COMPONENT INFORMATION FROM ALGEBRAIC
DIFFERENTIAL OPERATORS ,
We have derived accurate gradient component information by utilizing the
bias (mentioned in previous section) resulting from individual differential
operations and the underlying algebraic structure. We interpret the
differential operation as being a measure of a feature's spatial alignment with
that of the differential operator orientation (the differential vector). A
characteristic of the addition (1) is that as the differential vector converges to
the feature orientation, the number of resultant edge elements (edgels)
produced by that operator decreases. At the point which the differential
operator and feature are aligned, no edgels are produced by the operator.
Conversely, the population of edgels increases as the differential vector
diverges from the feature orientation (Fig 5). There is a strong quantitative as
well as qualitative correspondence between the number of edgels produced by
a differential operator and the orientation of the feature relative to that
differential vector.
The differential operation is symmetric and unsigned, hence four operators
are required to identify all unique (non-redundant) orientations. We define
the four component operators as:
Gx -_ (l+x) (8)
Gy = (l+y) (9)
Gxy = (l+xy) (I0)
C_y'l = (l+xy "1) (11)
Notice that the operators (8), (9), and (10) were used previously to define D2
and D3. The operation of each of these component operators on a simple
binary test image is shown in figure 5a - 5e.
(a) Co)
691
(e)
(d)
Figure 5 (a) A test pattern oriented
15 degrees relative to the abscissa.
This was generated by a Bresenham
algorithm. Co) The operation of Gx,
(c) Gy, (d) Gxy, (e) Gxy "_. The results
of the edgel count for each operation:
Gx : 148
Gy : 516
Gxy : 390
Gxy-1 : 662
The image is operated on by each of the gradient operators and the number of
edgels contributed by each component is compiled. The resultant component
operator with the fewest contributions is the closest to the feature orientation,
and is identified as the primary component (Gp). The component operator
which contributes the next tO the least edgels is identified as the secondary
component (Gs). Gp and Gs bound the sector in which the feature is oriented.
To determine the actual orientation of the feature relative to the primary
component, the inverse trigonometric relationship (12) between the edgel
count of Gp and the edgel count of its orthogonal component operator (Gp.L) is
employed.
orientation = arctan(C,p count/Gpl court0 (12)
As an example, consider the test image data of figure 5. The test line is draw
with a Bresenham algorithm at an orientation of 15 degrees from the
horizontal axis. From the data listed in figure 5 we identify : Gp --- Gx, Gs = Gx y
and Gp.l. = Gy. The relative measure from the Gp by (12) is 16.0 degrees (It
should be pointed out that accuracy of this operation degrades at very low
spatial resolution.) - _- _
Notice that (12) generates no sign information, only a relative measure fro_
the primary component vector. The correct placement of this measure from:
,the primary vector (one side or the other) is fixed by the secondary
component. Hence letting the global orientation of Gx be zero degrees, the
global orientation of the feature in figure 5 is (0+orientation)=16.0 degrees.
692
A major advantage of this edge detector/gradient component system is that it
inherently is a global operation; however reconfiguring it as a spatially
symmetric or asymmetric local operation is a trivial matter. This property
allows for the compilation of the gradient components over any size and shape
regions. In the last section we shall discuss how this reconfiguration is
perfectly suited for dynamic scene processing.
LINEAR FEATURE EXTRACTOR
The algebraic edge detectoffgradient component operator has been employed
as the front end of a linear feature extraction system which is based on the
Hough transform. Our primary goal is to establish and maintain object
independence and system invariance to object rotation, translation and
scaling in the image. The system takes as input the edge and gradient
component information (as discussed in the next section) and produces a list of
line segments.
The Hough transform is a technique for parameterizing binary image points
into a family of curves representing the analytical equations of image
features [4][5]. We employed a linear transform which is based upon the form
p = x*cos(0)+y*sin(0), (13)
where x and y are the coordinates of the edgel, 0 is the evaluated orientation
relative to the x axis (evaluated in the range of -_/2 to n/2), and p is the normal
vector to the line of that orientation. The evaluation of (13) produces the
transform parameter space.
We have developed an adaptable and reliable method for processing the Hough
transform parameter space to extract unique image feature equations[3]. The
analytical feature representations are then utilized to locate actual feature
segment endpoints via a line following/segment locating mechanism.
The processing sequence of the linear feature extraction system is shown in
figure 6a - 6e. Details of this system can be referenced in [3].
(a) (b)
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(c)
(e)
(de
Figure 6. (a) The operation of D3
on an ORU handle scone. Co) Hough
parameter space based on the normal
equation of a line. Rho is the
ordinate, theta the abscissa. (c) Peak
detected parameter space. (d) The
analytically derived lines super=
imposed on the edged image. Note the
accuracy. (e) The line segments
determined by the line follower/line
locater. This infomation is passed to a
model marcher.
DYNAMIC SCENE ANALYSIS
The major drawback of the Hough transform is the large amount of
computation that it requires. It can be seen from (13) that the number of
iterations will equal the number of edgels multiplied by the theta resolution.
It is apparent that the optimization of the transform requires the reduction of
the parameter space theta resolution as well as the area of the image which is
transformed. The intent is to perform this optimization in a way which does
not violate the goals of object independence and system invari_ance to r6_tation,
translation and scaling. This can be accomplished by evaluating the image
only in desired areas and at high theta resolution at selected orientations. This
optimizing information can be derived, in a dynamic sence, from previous
processing. In previous work [3] we discuss the optimization of the Hough
parameter space via the utilization of information from a geometric world
model.
We have recently developed a mutually optimizing system in which spatial and
gradient information from the algebraic edge/gradient component operator
optimizes the Hough transform, and the confirmed object feature information
is used to redefine the size and shape of the algebraic operator. In this way
the highest degree of accuracy, optimization and confirmation is achieved.
694
The unique ability to redefine the size and shape of the algebraic edge
detector/gradient component operator provides a natural environment to
perform this optimization.
CONCLUSION
The algebraic approach provides an innovative methodology as well as a sound
mathematical framework for image representation and manipulation. It is
significant that this algebraic system provides insight into the morphological
structure of the image itself, as well as the operators on the image. Real-time
implementation of these operations can be easily realized due to the efficient
min-max nature of our algebraic system. The simple extraction of the gradient
component information from the polynomial structure demonstrates the
strong potential of this approach.
ACKNOWLEDGEMENTS
The authors would like to thank David Provost, Danny Dalton, and Stanford
Ollendorf of NASA/Goddard Space Flight Center Flight Telcrobotics Laboratory
for their encouragement and support of this research. This research was
primarily funded under NASA grant S-92580-D. The original research leading
to this paper was performed by Professor Lemer under NSF grant IST-$409621.
REFERENCES
[1]
[2]
[3]
[6]
B. Lemer and H. Thomas, "Polynomial Representation of Grey-level
Images", SPIE Applications of Artificial Intelligence VI,
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T. Agui, M. Nakajima, and Y. Arai, "An Algebraic Approach to the
Generation and Description of Binary Pictures", IEEE Trans.on PAMI,
Vol. PAMI-4, No. 6, Nov. 1982, pp. 635-641.
B. Lerner, M. Morelli, and H. Thomas, "Applications of Algebraic Image
Operators to Model-Based Vision", SPIE Applications of Artificial
Intelligence VII, Vol. 1095, March 1989, pp. 36-47.
D. H. Ballard and C. M. Brown, Computer Vision, Englewood Cliffs, N.J. :
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J. Illingworth and J. Kittler, "A Survey of the Hough Transform",
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G.X. Ritter and P.D. Gader, "Image Algebra Techniques for Parallel Image
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1987.
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Extensions of algebraic image operators: An approach to model-based vision

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