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Plant Physiology 132:1138-1148 (2003) © 2003 American Society of Plant Biologists A New Algorithm for Computational Image Analysis of Deformable Motion at High Spatial and Temporal Resolution Applied to Root Growth. Roughly Uniform Elongation in the Meristem and Also, after an Abrupt Acceleration, in the Elongation Zone1Division of Biological Sciences (C.M.v.d.W., K.K.P., T.I.B.) and Department of Computer Engineering and Computer Science (H.S.J., K.P.), University of Missouri, Columbia, Missouri, 65211; and Institute of Plant Physiology, Russian Academy of Science, Moscow, Russia 127276 (V.B.I.)
A requirement for understanding morphogenesis is being able to quantify expansion at the cellular scale. Here, we present new software (RootflowRT) for measuring the expansion profile of a growing root at high spatial and temporal resolution. The software implements an image processing algorithm using a novel combination of optical flow methods for deformable motion. The algorithm operates on a stack of nine images with a given time interval between each (usually 10 s) and quantifies velocity confidently at most pixels of the image. The root does not need to be marked. The software calculates components of motion parallel and perpendicular to the local tangent of the root's midline. A variation of the software has been developed that reports the overall root growth rate versus time. Using this software, we find that the growth zone of the root can be divided into two distinct regions, an apical region where the rate of motion, i.e. velocity, rises gradually with position and a subapical region where velocity rises steeply with position. In both zones, velocity increases almost linearly with position, and the transition between zones is abrupt. We observed this pattern for roots of Arabidopsis, tomato (Lycopersicon lycopersicum), lettuce (Lactuca sativa), alyssum (Aurinia saxatilis), and timothy (Phleum pratense). These velocity profiles imply that relative elongation rate is regulated in a step-wise fashion, being low but roughly uniform within the meristem and then becoming high, but again roughly uniform, within the zone of elongation. The executable code for RootflowRT is available from the corresponding author on request.
Growth underlies life. Although organisms may be distinguished from crystals by reproduction, there would be nothing to reproduce without growth. In plants, growth is important not only for development of the organism but also for physiology. An animal runs, rolls over, bites, or plays dead; instead, a plant bends away, repositions its leaves, thickens its stem, or makes thorns. All these examples, among many others, involve growth.
The first step to understanding how a plant grows is measurement. Growth
overall can be measured by following the displacement of a terminus, such as
the tip of a blade of grass. By attaching the tip to a position transducer,
the displacement can be measured accurately (e.g.
Hsiao et al., 1970
To study expansion locally throughout a growing organ, one begins by
obtaining the velocity profile (Erickson,
1976
An alternative means to measure the spatial profile of growth is available
in principle from image processing techniques for "image sequence"
analysis (Jähne, 1997
Algorithms based on image sequence analysis work at short time intervals,
do not require marking the plant, and remove the need for tedious and
subjective manual measurement. Although the mathematical principles behind
image sequence analysis were delineated years ago
(Fennema and Thompson, 1979
Although these papers make an important start, they have limitations. The
method for the coleoptile treated the growing organ as a rigid body and found
the velocity of tip movement only, thus functioning in essence like a position
transducer, except that velocity in any direction could be measured
(Barron and Liptay, 1994
Furthermore, we developed the algorithm reported here for our studies of
growth in the root, in which relative elongation rates differ markedly in size
between elongation zone and meristem. The meristem is awkward for marking
methods because it is difficult to apply more than one or two marks within it,
and the displacements are usually too small to measure manually within the
customary time intervals. The previously cited paper using image sequence
analysis on the root (Walter et al.,
2002 Here, we report an algorithm that quantifies velocity confidently at more than 50% of the root pixels, with time steps between images as low as 2 s and an accuracy of better than 1 pixel per nine frames. The velocity profiles we have obtained with the algorithm for the root show velocity increasing more or less linearly through, and perhaps beyond, the meristem, and, surprisingly, also increasing quite linearly, albeit more steeply, through the zone of elongation.
Overview of the Software
Here, we overview the main features of the software (RootflowRT);
computational details are presented elsewhere
(Jiang et al., 2003
The algorithm in RootflowRT first calculates a velocity field with the tensor method (Fig. 2, A–C). The tensor step is also used to define a mask, segmenting the image into moving and nonmoving components (Fig. 2B). The confident tensor values are then used to constrain and, hence, accelerate a second round where the velocity field is calculated by matching (Fig. 2, D and E). Although confident tensor pixels are few, they tend to be dispersed over the root image (Fig. 2B). The implemented robust matching procedure includes criteria to ensure valid matches, including consistent matching forward and backward in time. The final velocity field is made by combining the confident pixels from both approaches; typically, 5% of the pixels are estimated by the tensor and 50% to 60% estimated by the robust matching. Because confidently estimated pixels appear well spread across the root, the remaining pixels are assigned velocity values based on interpolation (Fig. 2F).
The output thus far is a two-dimensional velocity field. For the purpose of studying elongation, the mask is used to generate a root midline and velocities are computed parallel and perpendicular to this curve, with the parallel component taken to represent elongation. Because the determination of the midline algorithmically is sometimes confounded by root hairs or image irregularities, the software instead can accept midline coordinates entered manually. The velocity profile along the x axis is then determined by averaging the velocities perpendicular to each pixel along the midline.
To obtain the velocity profile for the total growth zone, the root is
imaged in a series of overlapping stacks and profiles are determined for each
stack. Subsequently, the stack profiles are concatenated by the software into
a single profile based on distance from the quiescent center and the movement
of the stage between image stacks, taking into account the movement of the
tip, found by estimating the velocity in the region of the quiescent center.
The approximate quiescent center coordinates are input by the user to improve
precision. The movement of the stage was determined originally by marking the
surface of the agar with graphite particles and capturing a background image
at the same position as each stack (van
der Weele, 2001 For Arabidopsis roots growing about 500 µm h–1, acceptable velocity profiles were computed with time intervals between images ranging between 2 and 20 s (Jiang, 2001), and we selected a 10-s interval as standard. This results in 80 s used per stack and a total imaging time of 5 to 10 min depending on the size of the growth zone. Various magnifications were used, depending on the size of the root, but all in the range of 1 µm pixel–1, which provides an upper limit for spatial resolution. Velocities were measurable from a high of about 0.3 pixels s–1 to a low of approximately 0.01 pixels s–1, defining the temporal resolution.
As a root grows, its tip is propelled through the soil by the cumulative
elongation of all of the cells in the growth zone. An element at the very tip
moves at maximal velocity; elements located at positions progressively distant
from the tip move at progressively lower velocities, until the end of the
growth zone is reached, where the velocity becomes zero. For mathematical
convenience, it is useful to work in a coordinate system where the tip of the
root defines the origin and growth displaces elements away from the tip toward
the base (Erickson, 1976
A representative example of a velocity profile obtained by the algorithm is
shown, with a photograph of the root approximately on the same scale included
for reference (Fig. 3). The
profile has three distinct regions: starting at the quiescent center
(x = 0), velocity increases first gradually with position, then
steeply, and finally it becomes constant, indicating the end of the growth
zone. The regions are separated by fairly abrupt transitions, with the first
transition invariably more abrupt than the second. The increase in velocity in
both the first and second regions appears to have a considerable linear
character. The first region spans the meristem and may extend beyond it
(van der Weele, 2001
The profiles have high-frequency spatial fluctuations, probably indicating
computational noise. Profiles occasionally have larger, irregular, spatial
fluctuations and less pronounced transitions. In approximately 15% of the
profiles, velocity is constant (i.e. has a slope of 0) over an appreciable
region of the meristem, suggesting either that expansion stops transiently in
the meristem or that the algorithm is misled, for example by anomalous
behavior of lateral root cap cells. Similar regions of constant velocity are
present in the maize (Zea mays) meristem in the profiles published by
Erickson and Sax (1956 In view of approximately linear regimes in the velocity profile, the data were fitted to a single model comprising three linear equations joined at two breakpoints. The goodness of fit is evident (Fig. 4A). This impression was confirmed by analysis of residuals for 15 different roots (Fig. 5). For the first line (through the meristem; Fig. 5A), the residuals are small and evenly distributed around the mean, providing no evidence for a systematic departure from linearity. For the second line (through the elongation zone; Fig. 5B), the residuals are again small and randomly distributed, except near the end where they cluster beneath the mean, indicating that the transition to constant (zero) velocity is more gradual than the intersection of two lines.
Although the velocity profiles are well fitted by lines, linearity is an
oversimplification. Instead of lines, one may fit a set of overlapping
polynomials (Beemster and Baskin,
1998
To determine whether the biphasic profile of velocity found for Arabidopsis typifies other species, we obtained data for alyssum (Aurinia saxatilis), lettuce (Lactuca sativa), tomato (Lycopersicon lycopersicum), and timothy (Phleum pratense). These species have roots of different optical texture. As in Arabidopsis, the velocity profiles have two regions of roughly linear increase separated by an abrupt transition (Fig. 7). As for Arabidopsis, linear regressions fitted to the data from these species conform closely to the curves (data not shown).
Although determining the spatial distribution of growth was the major motivation behind developing the algorithm, the spatial algorithm was modified to measure the velocity of the root tip versus time. In tip-tracking mode, images of the growing root tip are collected at a given frequency for up to 300 frames. Because the root cap may contain tissue fragments moving irregularly as they separate from the root body, fully automatic segmentation of the extreme tip is difficult; therefore, the user initializes the routine by entering the approximate coordinates of the quiescent center. The algorithm then defines a region of interest, with the quiescent center at one edge. The size of the region is defined by the user, and we generally used 200 x 100 pixels. A tensor method alone is used to calculate velocities for the pixels in this box, starting with the first nine frames and moving through the entire sequence by adding the next frame and dropping the last one. All confident velocities are averaged, and this value is reported as tip velocity and used to update quiescent center coordinates.
Results of tip tracking have found that roots grow more or less steadily
during imaging for up to 1 h (data not shown) but that small fluctuations in
tip velocity are usually present (Fig.
8). The fluctuations are irregular, with magnitudes typically less
than 10% of the mean velocity and apparent period ranging from 5 to 20 min.
Satisfactory records could be obtained with 5- or 10-s intervals between
frames. To see if fluctuations could be reduced, we tracked roots illuminated
with yellow light (as used by Beemster and
Baskin, 1998
A New Algorithm for Image Sequence Analysis Applicable to the Plant Root
To measure the velocity field embodying root growth accurately and easily,
we turned to image processing techniques for estimating deformable motion.
Previously, we measured the velocity field manually by marking the plant and
measuring the displacement of the marks
(Beemster and Baskin, 1998
In the most common class of such methods, called "optical
flow," the intensity of a small, moving region of the image is assumed
to be conserved (Fennema and Thompson,
1979
In studies of growth and cell motility in biology, there are a few examples
of the use of each type of algorithm. Tracqui and coworkers have estimated
motion parametrically in crawling and dividing cultured animal cells as well
in as monolayers involved in wound healing
(Germain et al., 1999
We have applied RootflowRT to analyze root elongation under water stress
(van der Weele, 2001
RootflowRT also concatenates the velocity fields from overlapping stacks
spanning the root growth zone, which cannot be imaged at sufficiently high
resolution in a single field of view. Lowering the magnification results in
insufficient gray level texture for reliable feature extraction over the short
time intervals used. Combining the profiles from individual stacks into a
single profile requires knowing the displacement of the stage between stacks
and also the displacement of the tip between successive stacks because the
quiescent center has an x axis coordinate of zero but is moving in
the laboratory frame. To combine the profiles, we assume that the velocity
field is time invariant (i.e. for all x,
dV(x)/dt = 0), an assumption that is also made in
the traditional marking methods (Silk,
1992 Fluctuations can distort the velocity profile in three ways. First, changes in velocity that occur within the 80-s interval of the image stack will lead to an average velocity determination; few fluctuations occur on this time scale. Second, as described above, fluctuations cause a mismatch in the velocity where neighboring stacks overlap. To date, the mismatch amounts to no more than 10% of the velocity itself and is usually much less. Note that the derivative of the velocity profile, which is of chief interest, is not affected by lifting one part of the profile relative to another. Third, the fluctuations in tip velocity mean that the succeeding stacks cannot be placed exactly with respect to the quiescent center. This uncertainty amounts to approximately 1 pixel on the x axis and, thus, is unlikely to distort the final profile appreciably. Given the time required to image the entire growth zone, our algorithmic procedure represents an increase in temporal resolution over previous manual methods by more than an order of magnitude.
For many years, the profile of velocity in the root growth zone has been
estimated by marking experiments and widely accepted to be a smooth curve,
resembling a sigmoid (Goodwin and Stepka,
1945
Marking experiments are intrinsically error prone because they rely on
subjective and manual measurements, and they are limited by the number of
marks that can be applied to the root, typically around 10 per growth zone, in
contrast to the data at every pixel obtained with RootflowRT (thousands of
pixels per growth zone). Most previous publications show velocity profiles
averaged for a group of roots and, therefore, would smooth out abrupt
behavior. To obtain measurable displacements, marking experiments typically
use imaging intervals between 15 and 60 min, compared with the 80 s per stack
used here. Marks moving through an abrupt transition will yield an average
velocity for before and after the transition. Furthermore, if either the
position of a transition or the magnitude of the relative elongation rate
fluctuated during the imaging interval, then the measured displacements would
also reflect average behavior. Fluctuations in overall elongation rate occur
in the Arabidopsis root (Fig.
8) and appear to be widespread among plant organs
(Kristie and Jolliffe, 1986
Our interpretation that the instantaneous velocity field has quite linear
regions is supported directly in the classic paper of Erickson and Sax
(1956
Amazingly, the method used by Erickson and Sax is in essence an analog version of the digital image processing used here. They marked a root densely by dipping it in lampblack and then photographed the longitudinal axis of the root through a narrow rectangular aperture onto film while the film moved continuously. This caused the mark images to leave streaks on the film, with the angle of the streak to the horizontal being proportional to the velocity of movement. The streaks are lines of (roughly) constant intensity and, therefore, are an analog of the structure tensor. In fact, streak photography was designed by Erickson explicitly to reach as "elemental" and "instantaneous" a level as possible, and he appears to have succeeded.
That the root growth zone has a velocity profile with linear phases
separated by an abrupt transition is supported by other data. A pioneering
paper by Brumfield (1942
To the extent that the sigmoid curve reflects averaging out real phenomena
such as growth fluctuations, these curves may be appropriate for understanding
processes taking place in the growth zone over hours to days, such as cell
division (Sacks et al., 1997
The importance of the shape of the velocity curve lies in the implications
for relative expansion rate, which is the derivative of the velocity profile.
Relative elongation rate, technically a strain rate, reflects the deformation
of the underlying cell wall and, thus, is the appropriate parameter for
characterizing the mechanism of growth on the cellular or subcellular scale. A
smooth, sigmoid-like velocity curve gives rise to a profile of relative
elongation rate that is bell shaped, implying that relative elongation rate
changes continuously as a cell traverses the growth zone
(Erickson, 1976
Whatever its character, the profile of relative elongation rate must equal
the profile of cell division rate because congruence of these profiles is
necessary to maintain a constant cell length (per cell file) as observed
(Green, 1976 With our algorithm, we are zeroing in on the truly instantaneous growth behavior of the root: the growth zone comprises two zones, each with more or less uniform relative elongation, separated by a relatively abrupt transition. Future studies can now determine how these zones are maintained and modified in response to the environment and how they correspond to other processes in the root that are delimited spatially, such as cell division and differentiation.
Plant Growth
Seeds of Arabidopsis Columbia, timothy (Phleum pretense), tomato
(Lycopersicon lycopersicum Mill. var. Roma VFN), lettuce (Lactuca
sativa L. var. Black Seeded Simpson), and alyssum (Aurinia
saxatilis L. Desv. var. Gold Dust), the latter three obtained from a
local supermarket, were surface sterilized and germinated on agar-solidified
nutrient solution in 9-cm petri dishes as described previously for Arabidopsis
(Baskin and Wilson, 1997
A petri dish was placed on the stage of a horizontal compound microscope so the plants remained vertical. Images were taken through the lid of the petri dish to prevent evaporation of the water film around the root and consequent movement of the root. To accommodate for the focal length of the objective, a dimple lid was constructed. Light from the built-in microscope lamp (12-V halogen bulb) was passed through either yellow acrylic (Plexiglas J2208, Cope Plastic, St. Louis) for broad-band yellow light or glass (Schott RG-9, Bes Optics, West Warwick, RI) for infrared light. Roots were imaged with a 10x objective and a CCD camera (C2400, Hamamatsu Co., Hamamatsu, Japan) with the infrared cutoff filter removed and coupled to the microscope with either a 5x or 2.5x intermediate tube lens. A time stamp was placed on the image with a time date generator. Images were captured on an Apple Macintosh G3 (Apple Computer, Cupertino, CA) equipped with a frame grabber board (Scion LG-3) and the image analysis program Scion Image (www.Scioncorp.com, Scion Image, Frederick, MD). Each frame is 640 x 480 pixels in size, with 0.8 to 1.5 µm pixel–1, depending on the tube lens. For tip tracking, the tip was placed in the field of view and images taken every 5 or 10 s for a total of up to 200 images. For spatial analysis, a series of stacks were obtained, spanning the growth zone and including nongrowing regions of the root. Each stack has nine images, with the time interval chosen by the user (usually 10 s). The algorithm references the calculated velocity to the time of frame five, being at the center of the stack. To concatenate the velocity output from the single stacks into a single profile, one must determine the amount of movement of the stage between the positions used to obtain each stack. Initially, this movement was determined by collecting, for each stack, a background image of the agar surface that had been marked to enable the backgrounds to be registered. As marks, good results were obtained with graphite particles or cornstarch, although care had to be taken to avoid disturbing the root. Disturbing the root could be avoided by incorporating inert, latex beads into the agar, but this required focusing into the agar and resulted in successive background images being captured at different focal planes. Because capturing a background image lengthened the time required to image the growth zone, we obtained an electro-optical position transducer (Inchworm; Burleigh Instruments, Fishers, NY), and built a cradle for the microscope stand that allowed the inchworm to engage the stage controller in the vertical direction. The inchworm moves the stage in preset increments, accurate to ±1 µm, and in this way the stage movement between stacks could be accounted for absolutely without recourse to background images. The software accepts either method for concatenation. In addition to accounting for the movement of the stage, one must also account for the movement of the tip. For frame five of the tip stack, the user enters the coordinates of the quiescent center (in Arabidopsis, the quiescent center contains four cells, and its position is determined with reference to the columella, detectable because its abundant amyloplasts scatter light and give rise to dark bands running across the tissue—the absolute tip of the root could also be entered). The position of the quiescent center is taken as x = 0 for the velocity profile. The velocity at this region is multiplied by the time between center frames to obtain the tip displacement between a pair of stacks, and the output from each is adjusted accordingly. Source code for RootflowRT is available for downloading from the corresponding author. Both spatial and tip-tracking versions are available. At present, the code compiles and runs under Unix and Windows operating systems with development done primarily on SGI MIPS Irix (Silicon Graphics, Mountain View, CA) and HP Intel P4 computers (Hewlett Packard, Palo Alto, CA).
The velocity profiles were fitted either with overlapping second degree
polynomials as described by Beemster and Baskin
(1998
We gratefully acknowledge Dr. Michael Keller for his help on statistics, and we thank Jan Judy-March for flawless technical assistance and Mayandi Sivaguru for the data shown in Figure 8. K.K.P. worked on this project as part of his senior year curriculum at David Hickman High School (Columbia, MO). Received January 30, 2003; returned for revision February 25, 2003; accepted March 23, 2003.
Article, publication date, and citation information can be found at www.plantphysiol.org/cgi/doi/10.1104/pp.103.021345.
1 This paper is dedicated to Ralph O. Erickson on the occasion of his 89th
birthday. This work was supported by the U.S. National Science Foundation
(award no. IBN 9817132 to T.I.B.) and by the U.S. Department of Energy (award
no. 94ER20146 to T.I.B.), which does not constitute endorsement by that
Department of views expressed herein.
2 Present address: Department of Cell Biology and Molecular Genetics,
University of Maryland, College Park, MD 20742.
3 Present address: Department of Biology, Carnegie Mellon University, P.O.
Box 3320, Pittsburgh, PA 15230. * Corresponding author; e-mail BaskinT{at}Missouri.edu; fax 573–882–0123.
Barron JL, Liptay A (1994) Optical flow to measure minute increments in plant growth. Bioimaging 2: 57–61 Barron JL, Liptay A (1997) Measuring 3-D plant growth using optical flow. Bioimaging 5: 82–86[CrossRef] Baskin TI (2000) On the constancy of cell division rate in the root meristem. Plant Mol Biol 43: 545–554[CrossRef][Web of Science][Medline] Baskin TI, Wilson JE (1997) Inhibitors of protein kinases and phosphatases alter root morphology and disorganize cortical microtubules. Plant Physiol 113: 493–502[Abstract] Beauchemin SS, Barron JL (1995) The computation of optical flow. ACM Comput Surv 27: 433–467[CrossRef]
Beemster GTS, Baskin TI (1998) Analysis of cell
division and elongation underlying the developmental acceleration of root
growth in Arabidopsis thaliana. Plant Physiol
116:
1515–1526 Black MJ, Anandan P (1996) The robust estimation of multiple motions: parametric and piece-wise smooth flow fields. Comput Vision Image Understanding 63: 75–104[CrossRef] Black M, Jepson AD (1996) Estimating optical flow in segmented images using variable-order parametric models with local deformations. IEEE Trans Pattern Anal Machine Intell 18: 972–986[CrossRef] Black MJ, Rangarajan A (1996) On the unification of line processes, outlier rejection, and robust statistics with applications in early vision. Int J Computer Vision 19: 57–91 Brumfield RT (1942) Cell growth and division in living root meristems. Am J Bot 29: 533–543 Degli Agosti R, Jouve L, Greppin H (1997) Computer-assisted measurements of plant growth with linear variable differential transformer (LVDT) sensors. Arch Sci Genève 50: 233–244 Dormann D, Siegert F, Weijer CJ (1996) Analysis of cell movement during the culmination phase of dictyostelium movement. Development 122: 761–769[Abstract] Dormann D, Weijer C, Siegert F (1997) Twisted scroll waves organize Dictyostelium mucoroides slugs. J Cell Sci 110: 1831–1837[Abstract] Erickson RO (1976) Modeling of plant growth. Ann Rev Plant Physiol 27: 407–434 Erickson RO, Sax K (1956) Elemental growth rate of the primary root of Zea mays. Proc Am Philos Soc 100: 487–498 Farnebäck G (2000) Fast and accurate motion estimation using orientation tensors and parametric motion models. In A Sanfelieu, ed, Proceedings of the 15th International Conference on Pattern Recognition, Barcelona, Spain, Vol 1. IEEE Computational Society, Los Alamitos, CA, pp 135–139 Fennema C, Thompson W (1979) Velocity determination in scenes containing several moving objects. Comput Graphics Image Proc 9: 301–315 Fox MD, Puffer (1976) Analysis of transient plant movements by holographic interferometry. Nature 261: 488–490 Frensch J (1997) Primary responses of root and leaf elongation to water deficits in the atmosphere and soil solution. J Exp Bot 48: 985–999 Germain F, Doisy A, Ronot X, Tracqui P (1999) Characterization of cell deformation and migration using a parametric estimation of image motion. IEEE Trans Biomed Eng 46: 584–599[CrossRef][Medline] Gibson J (1966) The Senses Considered as Perceptual Systems. Houghton Mifflin, Boston Goodwin RH, Stepka W (1945) Growth and differentiation in the root tip of Phleum pratense. Am J Bot 32: 36–46[CrossRef] Green PB (1976) Growth and cell pattern formation on an axis: critique of concepts, terminology and modes of study. Bot Gaz 137: 187–202[CrossRef]
Hau Hejnowicz Z (1959) Growth and cell division in the apical meristem of wheat roots. Physiol Plant 12: 124–138 Horn BKP, Schunck BG (1981) Determining optical flow. Art Intel 17: 185–204[CrossRef]
Hsiao TC, Acevedo E, Henderson DW (1970). Maize
leaf elongation: continuous measurements and the dependence on plant water
status. Science 168:
590–591 Ishikawa H, Hasenstein KH, Evans ML (1991) Computer-based video digitizer analysis of surface extension in maize roots. Planta 183: 381–390[Web of Science][Medline] Ivanov VB, Dobroahaev AE, Baskin TI (2002) What the distribution of cell lengths in the root meristem does, and does not, reveal about cell division. J Plant Growth Regul 21: 60–67[Medline] Ivanov VB, Maximov VN (1999) The change in the relative rate of cell elongation along the root meristem and the apical region of the elongation zone. Russ J Plant Physiol 46: 73–82 Jähne B (1997) Motion. In Digital Image Processing, Ed 4. Springer, Berlin, pp 395–450 Jiang H, Palaniappan K, Baskin TI (2003) A combined matching and tensor method to obtain high fidelity velocity fields from image sequences of the non-rigid motion of plant root growth. In MH Hamza, ed, IASTED International Conference on Biomedical Engineering, BioMED 2003, Acta Press, Calgary, Canada (in press)
Jiang Z, Staude W (1989) An interferometric
method for plant growth measurements. J Exp Bot
40:
1169–1173 Kristie DN, Jolliffe PA (1986) High resolution studies of growth oscillations during stem elongation. Can J Bot 64: 2399–2405
Liptay A, Barron JL, Jewett T, van Wesenbeeck I
(1995) Oscillations in corn seedling growth as measured by
optical flow. J Am Soc Hortic Sci
120:
379–385
Ma Z, Baskin TI, Brown KM, Lynch JP (2003)
Regulation of root elongation under phosphorus stress involves changes in
ethylene responsiveness. Plant Physiol
131:
1381–1390 Metaxas D, Terzopoulos D (1993) Shape and nonrigid motion estimation through physics-based synthesis. IEEE Trans Pattern Anal Mach Intel 15: 580–591[CrossRef] Mullen JL, Ishikawa H, Evans ML (1998) Analysis of changes in relative elemental growth rate patterns in the elongation zone of Arabidopsis roots upon gravistimulation. Planta 206: 598–603[CrossRef][Web of Science][Medline] Muller B, Stosser M, Tardieu F (1998) Spatial distribution of tissue expansion and cell division rates are related to irradiance and to sugar content in the growing zone of maize roots. Plant Cell Environ 21: 149–158[CrossRef] Nagel HH (2000) Image sequence evaluation: 30 years and still going strong. In 15th IEEE International Conference on Pattern Recognition, Vol 1. Computer Vision and Image Analysis. IEEE, Los Alamitos, CA, pp 149–158 Odobez JM, Bouthemy P (1995) Robust multiresolution estimation of parametric motion models. J Visual Commun Image Represent 6: 348–365 Ronot X, Doisy A, Tracqui P (2000) Quantitative study of dynamic behavior of cell monolayers during in vitro wound healing by optical flow analysis. Cytometry 41: 19–30[CrossRef][Web of Science][Medline] Sacks MM, Silk WK, Burman P (1997) Effect of water stress on cortical cell division rates within the apical meristem of primary roots of maize. Plant Physiol 114: 519–527[Abstract] Schmundt D, Stitt M, Jähne B, Schurr U (1998) Quantitative analysis of the local rates of growth of dicot leaves at high temporal and spatial resolution, using image sequence analysis. Plant J 16: 505–514[CrossRef][Web of Science] Shapiro L, Stockman GC (2001) Computer Vision. Prentice-Hall, Engle-wood Cliffs, NJ
Sharp RE, Hsiao TC, Silk WK (1990) Growth of
the maize primary root at low water potentials: II. Role of growth and
deposition of hexose and potassium in osmotic adjustment. Plant
Physiol 93:
1337–1346
Sharp RE, Silk WK, Hsiao TC (1988) Growth of
maize primary root at low water potentials: I. Spatial distribution of
expansive growth. Plant Physiol
87:
50–57 Siegert F, Weijer CJ, Nomura A, Miike H (1994) A gradient method for the quantitative analysis of cell movement and tissue flow and its application to the analysis of multicellular dictyostelium development. J Cell Sci 107: 97–104[Abstract] Silk WK (1992) Steady form from changing cells. Int J Plant Sci 153: S49–S58[CrossRef] van der Weele CM (2001) Cell production, expansion, and the role of auxin in the response of the root of Arabidopsis thaliana exposed to water deficit. PhD thesis. University of Missouri, Columbia Vedula S, Baker S, Seitz S, Kanade T (2000) Shape and motion carving in 6D. Proc IEEE Computer Vision Pattern Recognition 2: 592–598
Walter A, Spies H, Terjung S, Küsters R,
Kirchge Zhou L, Kambhamettu C, Goldgof DB, Palaniappan K, Hasler AF (2001) Tracking nonrigid motion and structure from 2D satellite cloud images without correspondences. IEEE Trans Pattern Anal Machine Intel 23: 1330–1336[CrossRef] Zhuang X, Palaniappan K, Haralick RM (1999) Highly robust statistical methods based on minimum-error Bayesian classification. In CW Chen, Y-Q Zhang, eds, Visual Information Representation, Communication and Image Processing, Optical Engineering Series 64. Marcel-Dekker, New York, pp 415–430 Zoccolan D, Giachetti A, Torre V (2001) The use of optical flow to characterize muscle contraction. J Neurosci Methods 110: 65–80[CrossRef][Web of Science][Medline] This article has been cited by other articles:
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RESULTS
ecker and
Jähne, 1997
