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Journal of Interdisciplinary Mathematics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tjim20
A computer-assisted mathematical image analysis method for quantifying gravitropic curvature in plant roots a
b
a
Zhong Ma , Bo Forrester , Yu-yu Ren & Todd Hammond a
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Department of Biology
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Department of Mathematics and Computer Science Truman State University , Kirksville, MO , 63501 , USA Published online: 28 May 2013.
To cite this article: Zhong Ma , Bo Forrester , Yu-yu Ren & Todd Hammond (2010) A computer-assisted mathematical image analysis method for quantifying gravitropic curvature in plant roots, Journal of Interdisciplinary Mathematics, 13:1, 1-15, DOI: 10.1080/09720502.2010.10700674 To link to this article: http://dx.doi.org/10.1080/09720502.2010.10700674
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A computer-assisted mathematical image analysis method for quantifying gravitropic curvature in plant roots Zhong Ma
1∗
Bo Forrester Yu-yu Ren
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Todd Hammond 1 Department
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of Biology
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2 Department
of Mathematics and Computer Science Truman State University Kirksville, MO 63501 USA Abstract Gravitropism is the bending growth of plant organs in response to the gravity stimulus. It is essential in the deployment of an underground root system for water and nutrient acquisition, and an aboveground shoot system for photosynthesis. Quantifying such growth phenomena is critical in studying the mechanism of gravitropism. The common method of obtaining the gravitropic bend angle involves manually drawing two tangent lines along the curved organ, and measure the angle between them. The process is usually tedious, subjective, and prone to error. For precise and accurate determination of root gravitropic curvature, we developed a mathematical algorithm that minimizes the ambiguity intrinsic to manual measurement. Combined with edge detection and image segmentation techniques, the algorithm smoothes the identified root medial axis and computes the mathematical curvature at each point along the axis. The gravitropic bend angle is then determined based on the curvature profile. Accurate characterization of gravity response under various conditions is an initial step in investigating changes in gravity perception, and the cellular processes associated with it, thereby advancing our understanding of the gravisensing machinery within the cells. Keywords and phrases : Gravitropic root curvature, medial axis, edge detection, image segmentation, data smoothing.
∗ E-mail:
[email protected]
——————————– Journal of Interdisciplinary Mathematics Vol. 13 (2010), No. 1, pp. 1–15 c Taru Publications °
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Introduction
The successful development of plants depends on the downward growth of the emerging root, and the upward growth of the shoot. Such orientation of plant organs in response to the gravity vector is known as gravitropism. The gravity-induced spatial configuration is essential for plant growth and development, since it establishes a root system for anchorage, water acquisition and mineral uptake from the soil, and a shoot system for the capture of sunlight for photosynthesis aboveground. In roots of higher plants, gravity sensing occurs in specialized sensory cells in the center of the root cap [1-4]. According to the “starchstatolith hypothesis”, gravity sensing is mediated by force-sensitive structures such as dense plastids (amyloplasts) or vesicles (statoliths) [2, 3, 5-9]. With the perception of the orientation of the gravity vector, a biochemical signal, i.e., auxin, is transmitted to the target region, where subsequent response occurs in the form of differential tissue growth [5, 10, 11]. Root gravity sensing and signaling is thought to be regulated by interactions among various cellular structures, such as statoliths, ER, vacuoles, and actin cytoskeleton [5, 12]. It has been suggested that amyloplast sedimentation exerts a pressure on actin filaments connected to the mechanosensitive ion channels in the plasma membrane, leading to channel activation and a subsequent chain of events [13, 14]. However, the exact role of actin cytoskeleton and its interaction with other structures in gravity-sensing remains unclear, and has generated considerable interest and research. To better understand the cellular components and the interactions involved in gravity sensing, it is important to determine the gravitropic response and sensitivity under a variety of physiological and/or environmental conditions. Gravitropic sensitivity can be measured by the “presentation time”, defined as the minimal duration of exposure to the stimulation in a 1-g gravitational field for the induction of graviresponse. It can be calculated by plotting root curvature against the logarithm of the gravistimulation time, followed by extrapolating the regression line to the x-axis at zero curvature ( y = 0◦ ) [3, 4, 7, 15, 16]. The accurate measurement of root curvature is critical in determining gravitropic sensitivity. Methods that depend on visual determination of curvature are unreliable due to human errors in perceiving shape [17]. The need for improved analytical tools that are objective, automated, and accurate is clearly shown in a number of recent reports aimed at extracting numerical
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information through image analysis of plant growth by combined use of software and hardware [18-20]. In this study, we describe an algorithm developed for accurate quantification of root angle from time-lapse images. Mathematical methods used in the root analysis included the Sobel algorithm for edge detection, the Topological Watershed algorithm for image segmentation, and a parametric least-squares algorithm, the latter being used both for data smoothing and in the computation of mathematical curvature at points of the root’s medial axis. Our algorithm provides an important computational and analytical tool for use in future experiments in dissecting the cellular components of the gravisensing structure, and in quantification of complex biological growth patterns. 2.
Experiments
We used flax (Linum usitatissimum) seeds as our experimental material for their fast germination. Seeds were germinated in vertically placed cover plate of 9 × 9 cm Petri dishes to avoid capturing the grid at the bottom of the dish in the background of root images to be analyzed. Each Petri dish was filled with about 20 mL of agar. Before the agar solidified, its surface was stamped with three evenly spaced metal bars, each molded to have 12 evenly spaced rectangular-shaped teeth ( L × W = 5 × 2 mm) , resulting in three rows of shallow wells ( L × W × H = 5 × 2 × 1 mm) for accommodating a total of 36 seeds per plate. Seeds were germinated in vertically placed plates in the dark under 25C in a growth chamber, and root emergence generally occurred by 20h since imbibition. At predetermined times, e.g., 22, 26, and 30 hours from imbibition, the Petri dishes were rotated 90 degrees so that the long axes of the newly emerged roots were kept in horizontal orientation. The gravitropic curvature that developed over time was captured through time-lapse images, using a 12.8 Megapixel Canon EOS 5D digital camera with the manufacturer’s 100 mm macro lens. Exposure was in raw mode at ISO 100 with no special lighting equipment. Image-J software (ImageJ 1.34s, http://rsbweb.nih.gov/ij/ ) was used to manually determine root angle values, which were compared to algorithm generated measurements. 3.
Description of image processing and mathematical analysis method Each image was analyzed according to the following series of steps: (i) The user identifies the roots to be analyzed.
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(ii) An image segmentation algorithm is applied to find a rough perimeter for the root. (iii) A smoothing algorithm is applied to the perimeter. (iv) The tip of the root and the basal end near the seed are identified. (v) The medial axis is identified. (vi) A smoothing algorithm is applied to the medial axis. (vii) The curvature of the smoothed medial axis is computed at points along the medial axis.
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(viii) The curvature graph is used to automatically select points at which to measure the bend angle. (ix) Tangents along the smoothed medial axis are used at the selected points to compute the bend angle. We will illustrate these steps using the curved root in Figure 1A, taken with a digital camera (described in Experiments). Artificially generated curved “roots” within the range of 1 to 90 degrees, with increment of 1 degree (see an example at Figure 1I) were used to validate our algorithm. To begin with, the user identifies the roots to be analyzed by marking with a mouse both an area that is on the root and an area outside of the root. We typically identify the latter by circling the root with the mouse. All marked areas are separated from each other later in processing. These marks are shown in Figure 1B in white. If there is a possibility that the image segmentation algorithm may be confused by some artifact in the image such as bubbles in the agar substrate, the user can manually indicate areas of the picture that should be cut from the image. Although the image segmentation algorithm can sometimes manage to segment the seed from the root without help, we typically have found it expedient to separate the seed from the root by hand with a rough drawn line segment (see Figure 1B black line on the root). The marks can be produced either using a Java program called “CreateWatershedMarker” written by Bo Forrester for this project or using common image editing software such as the GIMP or Adobe Photoshop. To help identify the boundaries in the image in a way that is useful for image segmentation, the image is converted to grayscale, and then processed using the Sobel edge detection algorithm [21]. We used the implementation “pgmedge” by Jef Poskanzer et al. (http://netpbm.sourceforge.net/ ). The results of the edge detection algorithm are illustrated in Figure 1C.
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Figure 1 (A) Root used to illustrate the algorithm for determining the bend angle. Image resolution is 33 µ m per pixel. The distance from root tip to seed is about 150 pixels, or 5 mm. (B) Marker file and cut file superimposed over root. (C) Root after converting to grayscale and applying the Sobel edge detection algorithm. This image was brightened for easier visibility. (D) Results of applying the watershed algorithm. (E) Smoothed perimeter. (F) Smoothed medial axis. (G) Curvature along root as a function of the distance from the root tip at 0 µ m. (H) Measurement of total bend angle. (I) Example of an artificial “root” with bend angle of 45 degrees used to test performance of algorithms
The image segmentation is then accomplished using the “Topological Watershed Algorithm” of Gilles Bertrand, Michel Couprie, Jean Cousty and Laurent Najman (http://www.esiee.fr/ ∼ info/a2si/tw/ TW programs.html). Figure 1D shows the results after applying the watershed algorithm. The remaining processing is then done by a program written by the authors. This program uses the freely available libraries Imlib 2 (Carsten Haitzler,
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http://freshmeat.net/redir/imlib2/4386/urltgz/imlib2-1.0.6.tar.gz
and http://docs.enlightenment.org/api/imlib2/html/index.html), BLAS (Jack Dongarra et al. http://www.netlib.org/blas/), and the GNU Scientific Library (Free Software Foundation, http://www.gnu.org/software/gsl/). We are making our own software freely available as well, under the terms of the GNU General Public License. Both because of image resolution (in some cases curvatures reached over 1 degree per pixel) and because of irregularities in the roots, it was important to apply smoothing algorithms to the root perimeter. Our smoothing algorithm, which was inspired by the success of Savitzky-Golay filters [22] in smoothing spectroscopic data without greatly diminishing peaks, does a least squares minimization to a best fit to a parametric curve ( x(t), y(t)) with x(t) and y(t) fourth order polynomials. In particular, for any value of t = c where we require an approximation by a smoothed curve, we find the values of a0 , . . . , a4 and b0 , . . . , b4 which minimize the quantity ·µ ¶2 µ ¶2 ¸ k 4 4 j j w x − a ( t − c ) + y − b ( t − c ) , i ∑ i i ∑ j i ∑ j i i =1
j=0
j=0
where ( xi , yi ) (1 ≤ i ≤ k) are the points along the perimeter of the curve, ti is the parameter used for the point ( xi , yi ) and wi ≥ 0 is a weighting (or windowing) function. The parametric curve used near the point t = c is then given by ( xc (t), yc (t)) with 4
xc (t) =
∑ a j (t − c) j ,
j=0
4
yc (t) =
∑ b j (t − c) j .
j=0
As a first stage in obtaining this smoothed curve,, a rough smoothing is done using pixel numbers as the parameter t (more specifically, ti = i with the ( xi , yi ) enumerated going counterclockwise around the root). Then at each i with 1 ≤ i ≤ k we find an approximate arc length from an initial point ( x1 , y1 ) to the point ( xi , yi ) based on the formula Z t q i ( x˙ t (t))2 + ( y˙ t (t))2 dt , si = 1
where x˙ c (t) and y˙ c (t) are the derivatives of xc and yc with respect to the variable t . This approximate arc length along the smoothed curve is then used to re-parameterize the curve and a more accurate smoothing is found using the arc length si as the parameter instead of the original parameter ti = i . The beginning and ending points of the curve are identified with
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each other to avoid artifacts at the endpoints of the parameterization (thus in particular, we take ( x0 , y0 ) = ( xk , yk )) . Rather than smoothing using a moving rectangular window (as in a Savitzky-Golay filter), we used a more computationally intensive cosine window. The window width is based on the width of the root, and is computed automatically. The reason for the cosine window is to minimize changes in the first and second derivatives of the perimeter due to the window shifting to a new pixel. The resulting smoothed perimeter is shown in Figure 1E. The root tip is identified by the fact that it has highest curvature in the sense of angular change per unit arc length, and is shown in the figure in gray. The medial axis is found by identifying points that are equidistant from points both on the upper and lower flanks of the root. The medial axis is then smoothed using methods similar to those used on the perimeter. Figure 1F shows the smoothed medial axis, shown in dashed line with corresponding circles marked in gray. The medial axis is truncated near the seed end by locating what appears to be a sudden change or discontinuity in the two points of contact of the circles of the medial axis. The parametric curves found in smoothing the medial axis are used to get approximate first and second derivative information about the medial axis (here approximate arc length is used as a parameter). Curvature is then calculated using the standard formula
κ=
| x˙ y¨ − y˙ x¨ | . ( x˙ 2 + y˙ 2 )3/2
Because the root width is very small at the tip, a very small change in tip position can correspond to a very large angular change. Similarly, irregularities in the shape of the root near the seed may also affect curvature measurements. To correct for these problems, the measurement of bend angle of the root is taken at the local minima of curvature nearest the endpoints of the curvature graph. These are marked in Figure 1G in dark and light gray at the tip and near the seed. Tangent lines are then used to compute the bend angle θ , as shown in Figure 1H. 4.
Results
4.1
Variations in manual determination of gravitropic curvature using Image-J
We assessed variations in manual measurements of root angle values among individuals having varied experience with the Image-J software. A total of six people, including one experienced user and five first
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time users of Image-J, measured the same set of 14 curved roots using the angle tool in Image-J. Measurements were made by drawing two intersecting tangent lines along the curved roots. Each person determined the orientation of the two tangent lines relative to each other based on the curvature of the root surface. For each curved root, bend angle was measured with the pair of tangent lines placed at three locations along the root, i.e., the top and bottom flanks, and the mid-line of the root. A correlation analysis of data obtained by inexperienced and experienced users indicates that perception and measurement of root angles varied with individuals and the locations where measurements were made (Table 1). Table 1 Correlation of root angle measurements obtained from experienced and inexperienced users of the Image-J software. Root angle values were obtained manually in Image-J by each user through drawing two tangent lines along curved roots. Each set of the two tangent lines were made at three different locations, i.e., top and bottom flanks, and mid-line of each root R2 values
Experienced user of Image-J vs.
Top flank
Mid-line
Bottom flank
Inexperienced user 1
0.3829
0.8677
0.7046
Inexperienced user 2
0.5849
0.8407
0.7117
Inexperienced user 3
0.7142
0.8362
0.9004
Inexperienced user 4
0.9088
0.7824
0.9173
Inexperienced user 5
0.8251
0.8724
0.9023
The numbers of individuals who obtained highly correlated results to those by the experienced user ( R2 > 0.9) were one and three out of five when measurement was made from the top and bottom flanks of the roots, respectively. None of the first time users obtained highly correlated results with those by the experienced user when measurements were made from the mid-line of the roots, although the correlations were at a consistently similar level overall ( R2 ≈ 0.84 ; Table 1). Paired t -tests indicate that in general, results from all users differed significantly between the locations on the root where measurements were made ( P values less than 0.0001 for all location comparisons).
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Algorithm validation via artificial roots
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Artificial roots with bend angles ranging from 1 to 90 degrees at one degree increment were created to validate the accuracy of the algorithm measurements (Figure 1-I and 2). The angle measurements by the algorithm were highly correlated with the expected values, with an R2 = 1 , and a slope of 1.0335, a 3.4% deviation from 1. Overall, the deviation from expected angle values is about 4%.
Figure 2 Validation of the algorithms. Artificial roots with bend angles ranging from 1 to 90 degrees at one degree increment were analyzed by the algorithms. High correlation was found between measured and expected angle values
4.3
A comparison of algorithm computed vs. manually determined root angles
The same set of the 14 roots measured manually by experienced and inexperienced users of Image-J as described above (see section 4.1) was analyzed by the algorithm we developed, and results were compared to those from the manual measurements. Regardless of the experiences with Image-J or the locations on the roots for manual determination of root angles, the correlation between manually obtained and algorithm generated results was invariably low (Table 2). In addition, manual determination in Image-J generally overestimated root angle values by about 15% when measured from the top flank of roots, whereas underestimated root angle
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values by about 9% and 25% respectively when measured from the midline and bottom flank of roots (Figure 3). Paired t -tests indicate that on average, results from manual measurement by six people made at all three locations on the root differed significantly from those obtained by the algorithm (P values were 0.0005 for top flank measurement, 0.00001 for bottom flank measurement, and 0.0104 for mid-line measurement). Table 2 Correlation between root angle values obtained from the algorithm we developed and Image-J. Measurements from Image-J were made as described in Table 1 R2 values
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Algorithm vs. Image-J measurement by
Top flank
Mid-line
Bottom flank
Experienced user
0.5381
0.4859
0.5129
Inexperienced user 1
0.5414
0.5953
0.6079
Inexperienced user 2
0.4550
0.4538
0.4860
Inexperienced user 3
0.6127
0.6126
0.3950
Inexperienced user 4
0.5864
0.3878
0.4071
Inexperienced user 5
0.7541
0.5007
0.4130
Figure 3 A comparison of root angle values determined manually via Image-J by six people and via the algorithm. In manual quantification, bend angle was measured with tangent lines placed at three locations along the root, i.e., the top and bottom flanks, and the mid-line of the root
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Test of algorithm robustness
The robustness of the algorithm was tested by analyzing images of different tilting angles and resolutions, in order to accommodate various purposes of study and conditions of image acquisition.
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4.4.1 Effect of image tilting We captured images of curved roots in a vertical petri dish rotated at several tilting angles from zero degree, as well as at horizontal orientation (zero tilting), and calculated root angles with the algorithm. In general, the algorithm produced highly correlated results between tilted and nontilted root images (Figure 4A; R2 = 0.9884 and 0.9776, slope = 1.0235 and 1.0299 for clockwise tilting of 7 and 18.5 degrees, and R2 = 0.9976 and 0.987, slope = 1.0006 and 0.9261 for counterclockwise tilting of 8 and 20 degrees), and the overall variation is less than 1% compared to the nontilted image. 4.4.2 Effect of image resolution We tested the sensitivity of the algorithm to the resolution of root images. Again, highly correlated results were obtained when image resolution was increased by 2-fold from 33 to 16 um/pixel (Figure 4B; R2 = 0.9717 , slope = 0.9736 ), although on average this gain in resolution increased root angle measurements by 7%. 5.
Discussion and conclusion
The main axis formed by plant root and shoot is typically aligned in parallel to the direction of the gravity vector, and when this orientation deviates from the vertical direction, plants are capable of sensing the change and responding accordingly by curved growth. Such growth adjustment is due to gravity, and is manifested by a bend angle that results in the realignment of the main plant axis with gravity. Quantifying gravitropic response of plant organs is important for understanding the mechanism of gravity sensing in plant cells. Methods that involve visual determination of curvature are both tedious and subject to human errors in perceiving shape, and are therefore unreliable [17]. This is also evidenced by our assessment of manual estimation of root angles from time-lapse images using the angle tool in Image-J by individuals of varied experience with the software. It can be seen that the degree of correlations in angle measurements between experienced and inexperienced users varied with
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Figure 4 Robustness test of the algorithms: (A) Effect of image tilting, (B) Effect of changing image resolution
users as well as with the locations on the root where angles were estimated (Table 1). In addition, manual results from Image-J by even the experienced user correlated poorly with angle values determined by the algorithm regardless of which part of the root was used for manual measurement (Table 2). Compared to algorithm generated measurements, manual quantification either underestimated or overestimated root angle values significantly, depending on where the tangent lines were drawn
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(Figure 3). The inconsistency and unreliability in manual quantification demonstrates the importance of employing a mathematical algorithm that eliminates the subjectivity and ambiguity intrinsic to the manual measurement, and achieves quantification of root angles precisely and accurately. The method and algorithm we developed is especially suited for quantifying large numbers of curved roots from time-lapse images captured with a digital camera of reasonable resolution. It is simple to use, and can determine root angle values with greater accuracy and precision (Figure 2). It is also robust enough to generate highly correlated results from images of both high and low resolution as well as tilted images (Figure 4A and 4B). However, to minimize variations due to image resolution or tilting, it is possible to maintain consistent camera settings for image acquisition, and align the camera with the object so as to avoid a tilting angle of the non-curved main axis. Although the main application of the algorithm is to compute gravitropic bend angles, it is interesting to note that the curvature profile generated by the algorithm may reveal subtle features of root curving growth that are neither detectable by the human eyes from the root image, nor reflected by a single angle value output, such as the double peaks shown on the curvature profile of Figure 1G, which may bear biological significance. In summary, we believe that our method of image analysis and algorithm will provide a robust, reliable, and useful computational and analytical tool in various gravitropism studies. Acknowledgements. The authors gratefully acknowledge support received from the National Science Foundation’s Interdisciplinary Training for Undergraduates in Biology and Mathematics program under Grant No. 0436348, “Research-focused Learning Communities in Mathematical Biology,” and Grant No. 0337769, “Mathematical Biology Initiative”. We would also like to thank T.W. Sorrell for technical assistance, and Truman State University undergraduates Allison Schafers, Jenna Landwehr, James Franklin, and Jeremy Oliver for helping with the Image-J analysis. References [1] P. Basu, A. Pal, J. P. Lynch and K. M. Brown (2007), A novel imageanalysis technique for kinematic study of growth and curvature, Plant Physiol., Vol. 145, pp. 305–316. [2] T. Bjorkman (1988), Perception of gravity by plants, in Advances in Botanical Research (Vol. 15), J. A. Callow (editor), Academic Press, San Diego, CA, pp. 1–41.
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[19] W. K. Silk (1984), Quantitative descriptions of development, Annu. Rev. Plant Physiol., Vol. 35, pp. 479–518. [20] I. Sobel and J. Feldman (1968), A 3 × 3 isotropic gradient operator for image processing, presented as a talk within the Stanford Artificial Intelligence Project, (unpublished). [21] D. Volkmann and A. Sievers (1979), Graviperception in multicellular organs, in Encyclopedia of Plant Physiology (Vol. 7), W. Haupt and M. Feinleib (editor), Springer-Verlag, Berlin, pp. 573–600. [22] H. Q. Zheng and L. A. Staehelin (2001), Nodal endoplasmic reticulum, a specialized form of endoplasmic reticulum found in gravitysensing root tip columella cells, Plant Physiol., Vol. 125, pp. 252–265. Received May, 2009
