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THE ACCURACY OF THE OBJECTS POSITION MEASURING IN AN IMAGE
V .].Lebedev
Etalon Plant, 2A Novatorskaya Str., 220053 Minsk, Belarus — vil lebedev(@yahoo.com
Commission 1, WG 1/2
KEY WORDS: Vision Sciences, Digital Geometry, Photogrammetry, Application, Scanner, Raster
ABSTRACT:
A discrete location of photo-sensitive elements in CCD and in other matrixes leads to the distortions of shape, reduction of sizes,
shift, etc. of the objects in an image. As a result, the distortions influence on the accuracy of the objects coordinates measuring. In
this paper, we estimate the corner points coordinates measurement errors for arbitrary oriented rectangle and the other objects with
mutually perpendicular boundary segments. We derive the formulas for calculation of the errors depending on the angle orientation
of the figures. Estimations are based on the properties of digital straight segments and geometrical constructions. Some applications
of the results are discussed.
1. INTRODUCTION
In image reading, using the scanners on the basis of CCD and
the other matrixes of photo-sensitive elements, on their object
coordinates measuring accuracy influences to a considerable
extent the discrete location of the elements in a matrix. The
influence of geometrical distortions are less significant even in
low-cost scanners (Cramer, Bill, and Glemser, 1994). On
coordinates measuring accuracy also influences the point spread
function of a digitizer. For example, if binary image is digitized,
the result is a gray scale image, that is its boundary is fuzzy. But
the methods for recovering the binary images solve this problem
(see, for example, Pavlidis and Wolberg, 1986; Li et al. 1986;
Medioni and Yasumoto, 1986). Then the errors remain because
of the finite resolution or super-resolution of the scanner (see
Pratt, 1978; Cohen and Dinstein, 2000). In this paper, we
estimate the corner points coordinates measuring errors of the
rectangular-shaped objects that are arbitrary oriented relative to
the axes OX, QY of the Cartesian coordinate system XOY
superposed with the digitizer coordinate system. Corresponding
estimations we derived (see Lebedev and Zolotoy, 2001;
Lebedev, 2003), but for special angles of orientation and side
lengths of the figures.
2. DIGITIZATION
We assume that an image is read by the receptors field being the
integer valued grid with the constant step equal ] and the
pointed photo-sensitive elements are located in its nodes (see
fig.1). Let the image is quantized in these points on two levels
(0 is a background, 1 is an object), and contour element of the
object is an unit element with four neighbouring elements in
vertical and horizontal directions having at least one zero
element. The shortest closed broken line connecting the contour
elements ( assuming that they are eight-connected) constitutes
the object contour. Let name this contour as the black contour
(according Freeman, 1970). Similarly, introduce the white
contour notation, i.e. the shortest closed line connecting the
eight-connected zero elements bordering in the sense of the
four-connection with the black contour elements. The domain
lying between the black and white contours determines the so-
called zone of non-sensitivity (Pisarevsky et al, 1988). The
expressions for description of such zone has been given (Dorst
and Smeulders, 1984). The width B of this zone influences on
the measuring errors A = (Ax, Ay) of the rectangle corner points
coordinates. In one's turn, the width f depends on the angle of
the rectangle orientation ¢. In next section we will be derive the
equations for this dependence.
o—— 000 0
Figure 1. A digitized rectangle (1 is black contour; 2 is white
contour)
3. THE WIDTH OF NON-SENSITIVITY ZONE
Let a side of a rectangle lies in the first octant of the Cartesian
coordinate system and its slope is tgo = p/q, where p and q are
integers and p/q is irreducible fraction (p S ¢). Then the chain
code of the directions of digital straight segment belonging to
black or white contour and corresponding the side of a rectangle
is the periodic Freeman's sequence L — ],,...,1, (or its reverse
sequence L, 1,1; wherel is OQ or 1, j — 1,2....,q (Pham,
1986). Using the proof of the proposition 3 from the Pham's
work ( see Pham, 1986), we can show that
L-0..0 1.5 lh QOrL79...0,1.. isl) (1)
