Algorithm for Estimating Length of Bent Objects

ALGORITHM
FOR
ESTIMATING LENGTH
OF
BENT OBJECTS
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By Edmond W. Holroyd III1
ABSTRACT: A prototype algorithm was developed to measure the length of fish larvae, some of which were
bent. The algorithm starts with the longest straight line spanning the object, then repeatedly segments the line
as points are forced to lie along the center line of the object. There is less than 5% length improvement with
the transition from four to eight line segments for objects with a general length/width ratio of about 10. Most
improvements in length were of less than one pixel. Additional segmentation to 16 or more lines may be
appropriate for objects with a greater ratio, but the number of segments should not exceed the length/width
ratio. The algorithm logic can be transferred to other program languages and software packages for routine
image analysis.
INTRODUCTION
1 and the word ‘‘image’’ will denote one of the larvae viewed
in isolation.
Computer automation of labor-intensive measurements of
length, width, and area is a common goal for many applications. Images can be recorded by photography, video, scanners, and digital camera and imported into an image processing system on a computer. Software packages can then
isolate the individual specimens in view and determine their
areas and straight-line dimensions. For example, Jeffries et
al. (1980) determine the greatest straight line lengths between
extreme perimeter pixels of zooplankton. Holroyd (1987) determines snow particle lengths by the extent of the perpendicular projections of perimeter pixels along a linear regression line.
Such straight-line dimensions, however, can be insufficient
for some objects, such as fish larvae. Whether live or preserved, larvae are usually not straight and present challenges
for measuring their total lengths from nose to tail. Software
measurement of the length of a specimen along its spine can
be accomplished manually by following the centerline of the
specimen with a cursor. However, an automated measurement
system is desired to reduce manual labor.
Some software packages offer skeletonization (Russ 1992),
by which the image is eroded by repeatedly removing edge
pixels until only thin lines remain. However, irregularities in
the image edges cause branching in the resulting lines. Special logic is then necessary for selecting those line segments
that actually represent the centerline of the image. Furthermore, skeletonization lines end at the center of curvature of
the ends of the image and do not extend to the image edges.
Therefore, lengths determined from skeletonization are too
short by an amount similar to the widths of the image near
its end points.
Fig. 1 is a view of fish larvae provided by Darrel E. Snyder
(Larval Fish Laboratory, Colorado State University) as a
challenge to develop an algorithm for estimating lengths of
bent objects. The common staple at the top provides a scale
of 12.5 mm length. There is background noise in the image
from debris and from the bottom of the dish as well as faint
shadows of the larvae to their right caused by directed illumination. The toning has been reversed to show opaque objects on a bright background. There are some nearly transparent parts of the larvae that are difficult to detect. In this
report, the word ‘‘scene’’ will denote the entire view of Fig.
1
U.S. Bureau of Reclamation, P.O. Box 25007, D-8260, Denver, CO
80225-0007. E-mail: eholroyd@do.usbr.gov
Note. Discussion open until September 1, 1999. To extend the closing
date one month, a written request must be filed with the ASCE Manager
of Journals. The manuscript for this technical note was submitted for
review and possible publication on December 18, 1998. This technical
note is part of the Journal of Computing in Civil Engineering, Vol. 13,
No. 2, April, 1999. 䉷ASCE, ISSN 0887-3801/99/0002-0130 – 0134/$8.00
⫹ $.50 per page. Technical Note No. 19833.
IMAGE PREPARATION
A threshold was selected to isolate the larvae from the background noise. Two small objects that were not larvae were
allowed to remain in the scene so that they could be excluded
from consideration by having a small area. Holes in the staple
and some images were artificially filled by hand. Some software packages can do such isolation, exclusion, and filling
automatically. A new eight-bit scene was created in which the
background was pure white (gray scale value = 255), and each
larval image, including the two small objects, was filled with
consecutive identification numbers beginning with 2 and numbered from top to bottom in order of encounter. Again, some
image processing software packages (including Optimas, TNTmips, and TNTlite) can automatically isolate such larval images, produce a similar raster (computer picture), and color
code the images by identification number. The algorithm assumes that such automatic preprocessing has already been
done. The result was exported into a file that was a simple
array of bytes for processing by the algorithm. The algorithm
for determining the bent larval lengths was written to examine
that specific file only. It is adaptable for files of other dimensions and for objects of differing length/width ratios.
ALGORITHM STEPS
General Overview
The algorithm first scans through the scene for some basic
information. For each larval image, it records the identification
number, the minimum, average, and maximum line and column numbers, the span of the image in the horizontal and
vertical directions, and the total area, all in pixel units. It writes
these numbers to a file for future reference. In the discussion
to follow, variable names will be used to avoid repeated definitions and long descriptions; such names are somewhat arbitrary.
The algorithm was developed specifically for the images in
Fig. 1. They have a length/width ratio of approximately 10.
Objects with a larger ratio (such as worms) can benefit from
a greater number of iteration steps. Objects with a smaller ratio
might use a lesser number of iterations, approaching none at
all for objects (such as eggs) that are unlikely to be bent.
Arrays
Two workspace arrays of an arbitrary 200 by 200 pixels
were created. The dimensions need only be able to contain the
largest image expected. The algorithm only scans the actual
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J. Comput. Civ. Eng., 1999, 13(2): 130-134
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FIG. 1.
Initial Image of Larvae, with 12.5 mm Staple for Scale
span of the object, so excess dimension is harmless in terms
of computational speed. The first array, IM (the image), contains either zeros or the identification number of the image in
the proper pixel locations. The second array, IR (image rim),
contains either zeros or ones at the larval edges.
The algorithm operates by means of ‘‘boxes,’’ ‘‘line segments,’’ and ‘‘anchors.’’ The boxes contain parts of the segmented image. The anchors are the points along the box edges
at which the line segments fold to follow the centerline of the
image. This development extends to the creation of only eight
boxes and lines. Additional segmentation to 16 or more is
possible for long, thin objects, having a greater length/width
ratio. For eight line segments, there must be nine anchors. Four
arrays of dimension 8 keep track of box limits: LT (top line
number), LB (bottom line number), KL (left column number),
and KR (right column number). Two arrays of dimension 9
keep track of the anchor locations: LL (anchor line number)
and KK (anchor column number). One array of dimension 8,
RMAX, records the length of the line segments between the
nine anchor points. The summation of those lengths is the total
length of the larva.
There are some other ‘‘bookkeeping’’ arrays. One pair of
dimension 8 records the line number (LX) or column number
(KX) at which a box is subdivided and along which an anchor
point is located. Such lines or columns are excluded from consideration in a search within the box for the farthest rim location from an anchor point. Another array of dimension 8,
ISBENT, contains a flag to indicate if any part of a line segment falls outside of the larval image. This array is for information only and does not affect the algorithm.
Only the scene between the extreme line and column numbers
for each image is examined. Only those parts of the scene
having the desired identification number are copied into the
upper left corner of the image workspace; the rest remains
zeroed. For illustrative purposes, Fig. 2 shows the algorithm
steps for larva 16 (which is in middle of the left side of Fig.
1 and looks like a question mark), the bent shape of which is
particularly challenging to analyze.
The top and left extremities of the image workspace have
line and column numbers of 1. The bottom and right extremities of the image have line and column numbers equal to the
span of the image in those directions. These are the first box
limits entered into their arrays. The first box is shown in step
0 in the upper left of Fig. 2.
The image workspace, IM, is scanned with a 3 by 3 array.
If the center of this array is nonzero and hence the identification number, then if any of the eight neighbors is a zero, the
center position of the 3 by 3 array is flagged with a 1 as an
edge indicator in the IR rim array workspace.
First Line
The rim image is scanned to identify the pair of pixels with
the greatest separation. The distance between them, illustrated
as the black line in Step 1 in the lower left of Fig. 2, is the
first RMAX. Their coordinates are the first and ninth entries
in the anchor list, with the topmost pixel being listed first. The
line is then examined to see if any part falls outside of the
larval image. If so, the ISBENT flag for the first line is set
to 1.
Subsequent Segmentation
Extraction
The algorithm scans through the scene again to analyze each
larval image, following the order of the identification numbers.
The algorithm then performs a number of segmentations to
generate two, four, and eight line segments. A program loop
index, KUT, ranges from 1 to 8, with steps of 8, 4, and 2 with
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J. Comput. Civ. Eng., 1999, 13(2): 130-134
in continuity of the image along the cutting line, but that has
not been a problem with this scene.)
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New Lines
The new boxes are considered in the correct order from the
start to the end of the image. If the box is not at the extreme
end of an image, then it has two anchor points associated with
it, and both are fixed in position along the centerline of the
image. The length of the line segment is the distance between
those anchor points.
If the box is at the start or end of an image, then one anchor
is fixed and the other is free to float to a new position. The
rim image, IR, is scanned within the box to find the farthest
point (new anchor) of the image from the fixed anchor point.
Scanning is not permitted along the last cutting line (a line or
column). The new line length is then recorded. It is this feature
of the algorithm that makes the line segments fold around
major bends of any orientation to find the extremities of the
image. It also lets the line segments extend to the edge of the
image, unlike skeletonization.
The new line is then checked for any points falling outside
the image, indicating that the image is still bent within that
box. New line segments are illustrated along the bottom of
Fig. 2.
Total Length
FIG. 2. Numbered Steps in Algorithm for Image 16, Giving
Type of Product Produced in Each Step
each successive segmentation. KUT is therefore only 1 for the
first segmentation, 1 and 5 for the second, and 1, 3, 5, and 7
for the third. For each KUT the algorithm divides a box into
two, often unequal parts and finds where the new box boundary crosses the centerline of the larval image. That location is
a new anchor point. The algorithm needs to keep careful track
of the order of the new boxes along the image, particularly
for bent images.
Dividing Boxes
The height and width of a box are derived from the limits
stored in the bookkeeping arrays. If they are equal, the algorithm checks whether the anchor points of the last line segment
are on the top and bottom or on the sides of the box. If on
the sides, the box is divided along a column so that the new
anchors are also on the sides. Similarly, a horizontal division
retains the top and bottom pattern.
Otherwise, the algorithm determines which is larger, the
width or the height of the box. The longer dimension is subdivided at the position of the average of the two anchor points,
using integer truncation. This may produce new boxes of unequal dimensions, especially if one anchor was on a side edge
and the other on a top or bottom edge. Successive new boxes
are illustrated along the top of Fig. 2. The extents of the image
within each new box are then determined and carefully stored
at the appropriate locations in the bookkeeping arrays. It is
very important to make sure that the boxes remain in the correct order along the image from start to end.
Interior anchor points are always retained. Those at the start
or end of the series of line segments are movable. New intermediate anchor points are declared to be at the position of the
center of the larval image along the cutting line. All interior
anchor points are illustrated along the middle of Fig. 2. (The
present version of the algorithm fails to protect against a break
Computer determination of the length of a curved line is
frequently by means of a sum of shorter straight line segments.
As the number of segments increases and their lengths decrease, the total length approaches the real length. However,
the practical limit of precision is the size of the pixels comprising the image. Images that are generally straight have no
algorithm challenges. Their initial lengths are similar to final
calculations. Bent images, however, are the reason for the algorithm development. Running the algorithm to 16 line segments might smooth out the upper right in Fig. 2, but there
could be problems from having boxes not extending across the
local width of an image. There would not be much gain in the
length because it is already within a few pixels of the true
length.
Future algorithm development could investigate fitting a
spline to the anchor points to get a curved fit that should follow the centerline of the image better than the straight line
segments.
ILLUSTRATIONS
Fig. 3 shows in a smaller format the progress on all images
in the scene. The images are shaded gray. Black lines are intermediate line segments, and the white lines are the final lines
of eight segments. Images 4 and 11 were not analyzed because
they were too small, as determined by an area threshold specific to the Fig. 1 scene. Images that were nearly straight have
overlapping line segments that may not be resolvable. Bent
images show the progress of the line folding. Sometimes the
floating of the ends of the first and last line segments can be
seen as the folding progresses.
Only image 13 appears to have a strange behavior on the
last folding. As a result of the particular shape of the broad
head of a bent larva, the last line segment folds back rather
than proceeding to a rim position on the extreme right. The
total length of the image is not significantly affected by this
reversing fold. However, extending the algorithm to 16 line
segments will probably produce a significant error for this particular image. It appears that the number of segments should
not greatly exceed the length/width ratio of an image that is
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FIG. 3.
Line Fitting Progress on All Images in Scene
that there is not much improvement with the increased segmentation. Bent images plot to the right. There are sometimes
significant jumps in total length as the line segments fold
around a bend in the image. The greatest improvement going
from four to eight segments is about 5% for image 16. For
two-thirds of the images, the last improvement was less than
one pixel, and so further improvement cannot be expected for
them. It appears that further division of the images to 16 line
segments would generally produce an improvement of less
than 2%.
DISCUSSION
FIG. 4. Length measurements Get Closer to Eight-Line Segment Value As Number of Segments Increases
not bent because of the possibility that a final box may not
contain opposite edges across an image.
FINAL LENGTHS
As the algorithm progresses from one to eight line segments, the total length of each image usually increases. Fig. 4
shows a graph indicating the convergence toward the final (unknown) value. The vertical axis consists of three values of the
number of line segments, 1, 2, and 4. The horizontal axis gives
on a logarithmic scale the difference in length from the eightsegment value, expressed as a fraction of the eight-segment
length.
Straight images have points plotting on the left, indicating
An algorithm has been developed that will approximate the
length of larvae even if they are bent and in random orientations. It operates by folding line segments so that anchor points
fall along the centerline of the image. The concept appears
simple, but the program logic requires careful bookkeeping of
box and line segment coordinates in the proper order. Lengths
converge rapidly to a ‘‘final’’ value with eight line segments
for these particular images.
It appears that the number of segments should not exceed
the general length/width ratio of the unbent objects. Neither
should the number approach the pixel length of the object,
because pixels cannot be subdivided. It is suspected that excess
segmentation will start to produce irregularities of measurement of images with unusual shapes. Boxes may no longer
extend across the local width of the images. Side bulges, like
those in image 15, may fold the lines away from the general
center. The reverse folding of image 13 should be taken as a
warning against further segmentation of images of this particular length-to-width ratio.
This algorithm appears general in nature and could be used
for other natural and artificial objects that are long but possibly
bent. Some diverse possibilities are plankton, worms, algal
strands, wood pulp fibers, elbow macaroni, and oxbow lakes.
The algorithm is not a full package and needs to be integrated
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into processes already available in commercial software systems.
The algorithm coding was in FORTRAN, but the logic
should transfer to the more popular varieties of C. The image
processing routines in MIPS (Map and Image Processing System, from MicroImages, Inc., Lincoln, Nebr.) were used to
examine the algorithm progress and prepare the resulting figures. Their more modern TNTmips and free TNTlite multipurpose software could incorporate this algorithm.
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ACKNOWLEDGMENTS
Encouragement was received from Straton Spyropoulos, formerly of
Optimas Corp., and from Darrel E. Snyder of the Larval Fish Laboratory,
Colorado State University, to develop the algorithm for incorporation into
the Optimas software. Steve Hiebert of the Bureau of Reclamation provided review and further encouragement.
APPENDIX.
REFERENCES
Holroyd, E. W. III. (1987). ‘‘Some techniques and uses of 2D-C habit
classification software for snow particles.’’ J. Atmospheric and Oceanic
Technol., 4(3), 498 – 511.
Jeffries, H. P., Sherman, K., Maurer, R., and Katsinis, C. (1980). ‘‘Computer-processing of zooplankton samples.’’ Estuarine perspectives,
Victor S. Kennedy, ed., Academic, San Diego, 303 – 316.
Russ, J. C. (1992). The image processing handbook. CRC Boca Raton,
Fla., 315 – 318.
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