- © 2014 American Society of Plant Biologists. All Rights Reserved.

## Abstract

Leaf vein density (LVD) has garnered considerable attention of late, with numerous studies linking it to the physiology, ecology, and evolution of land plants. Despite this increased attention, little consideration has been given to the effects of measurement methods on estimation of LVD. Here, we focus on the relationship between measurement methods and estimates of LVD. We examine the dependence of LVD on magnification, field of view (FOV), and image resolution. We first show that estimates of LVD increase with increasing image magnification and resolution. We then demonstrate that estimates of LVD are higher with higher variance at small FOV, approaching asymptotic values as the FOV increases. We demonstrate that these effects arise due to three primary factors: (1) the tradeoff between FOV and magnification; (2) geometric effects of lattices at small scales; and; (3) the hierarchical nature of leaf vein networks. Our results help to explain differences in previously published studies and highlight the importance of using consistent magnification and scale, when possible, when comparing LVD and other quantitative measures of venation structure across leaves.

Leaf vein density (LVD), defined as the total length of veins per unit area, has been linked to rates of photosynthesis (Brodribb et al., 2007), plant and leaf hydraulic conductance (Sack and Frole, 2006; Sack and Holbrook, 2006), leaf size and conductance (Scoffoni et al., 2011), and leaf allometry (Price et al., 2012; Sack et al., 2012). Vein density affects the distance that water has to travel through the mesophyll space, thereby providing a mechanism to influence whole-leaf physiological rates (Raven, 1994; Sack and Frole, 2006; Brodribb et al., 2007). Long distances (low vein density) are associated with longer travel times and thus slower physiological rates; conversely, shorter distances (high vein density) are associated with faster rates. It has been suggested that an increase in vein density contributed to the phylogenetic radiation and rise to ecological dominance of the angiosperms. This idea is supported by comparing vein density across basal and more derived angiosperm lineages and also by comparing vein density in fossils spanning the Cretaceous angiosperm radiation (Boyce et al., 2009; Brodribb et al., 2010; Feild et al., 2011). For a recent review of the importance of LVD, see Sack and Scoffoni (2013).

While there has been considerable discussion regarding the physiological, ecological, and evolutionary implications of LVD, there has been almost no discussion of methodological issues associated with its estimation. Most studies use magnified images of cleared leaves to estimate LVD. Image magnification levels reported are variable, from 20× (Boyce et al., 2009) to 25× (Blonder et al., 2011) to 40× (Sack and Frole, 2006), or cover a range of 5× to 40× (Sack et al., 2012). Similarly, the area of each leaf sampled varies, such as 1.5 to 1.9 mm^{2} in Sack et al. (2012) or 5 to 12 mm^{2} in Feild et al. (2011). Variation in sample area and magnification corresponds to measurements that cover a variable total number of areoles and total length of veins. Furthermore, none of the aforementioned studies employing microscopes mentioned, or appeared to consider, the effect of microscope resolving power. The resolving power of a microscope determines the scale at which distinct features within the sample are able to be distinguished in the image. A digital camera used to acquire microscope images should have, at minimum, two pixels spanning the resolving power of the microscope. This will ensure that features able to be resolved through the microscope eyepiece will also be resolvable in the associated digital image.

The length of individual veins in each image is usually determined by tracing the lengths of veins manually via widely used image-analysis programs such as ImageJ (Schneider et al., 2012). Recently, several semiautomated approaches have also been utilized that employ skeletonization on binary representations of cleared images (Blonder et al., 2011; Price et al., 2011, 2012; Dhondt et al., 2012). The use of semiautomated software has enabled estimates of vein density at the scale of entire leaves (Price et al., 2012), leading to questions and criticisms regarding differences between studies (Sack et al., 2012).

The tradeoff between field of view (FOV; i.e. the physical size of the object studied) and the resolution of measurement underlies image analysis in fields ranging from cosmology to biology (Lindeberg, 1998). In the case of microscopes, this tradeoff is expressed fundamentally by the metric known as the space-bandwidth product (Lohmann et al., 1996). In particular, the space-bandwidth product reveals the maximum total number of resolvable pixels that an imaging system can acquire, which is of the same order for nearly all typical light microscopes, such as the one employed in this study. Generally, a microscope will achieve its space-bandwidth product only at its lowest magnification. This is because, for typical microscopes, the FOV is inversely proportional to both magnification and image resolution (i.e. the number of pixels per unit length), yet resolving power is independent of these quantities. Therefore, differences in imaging resolution can lead to differences in the estimation of object dimensions. In a classic example, estimates of the length of the coastline of Britain were found to depend on the scale of measurement, such that finer resolution imaging led to increases in the apparent length of the coastline (Mandelbrot, 1967). One way of overcoming the interdependence of FOV, magnification, and resolution, and to break the space-bandwidth product limit, is to mosaic several microscope images together into a single, large-FOV, high-resolution image (a technique we employ below).

Here, we examine the combined effect of FOV and magnification on the estimation of LVD. We demonstrate that LVD has a strong and systematic dependence on magnification level and FOV, due to both theoretical and empirical considerations. This dependence arises due to three related phenomena arranged roughly in order of decreasing effect: (1) a tradeoff between magnification and resolution in imaging; (2) geometric effects of lattices at small FOVs; and (3) the hierarchical nature of veins, specifically that large veins contribute disproportionally to vein area. The first factor has the potential to influence vein density at all scales of measurement, the second factor will be most pronounced at small FOVs, and the third factor has the strongest effect as vein sample sizes get larger, both within an individual leaf and as leaves themselves get bigger. We show how this scale dependence of LVD has the potential to reconcile estimates of LVD previously reported by groups using magnified (Sack et al., 2012) and unmagnified (Price et al., 2012) images.

## RESULTS

### Tradeoff between Magnification and Resolution

Image magnification affects the measured density and perimeter of veins. In each of the five series we examined, both LVD and vein perimeter density (VPD; see “Materials and Methods”) increased as a function of magnification (Figs. 1 and 2). The fold change (maximum/minimum) in LVD values for *Banksia victoriae*, *Hardenbergia comptoniana*, *Plumeria alba*, *Pittosporum moluccanum*, and *Wisteria floribunda* were 1.81, 1.51, 1.69, 1.38, and 1.34, respectively, with a mean of 1.55. The fold change in VPD values for *B. victoriae*, *H. comptoniana*, *Plumeria alba*, *Pittosporum moluccanum*, and *W. floribunda* were 2.21, 1.6, 1.6, 1.45, and 1.25, respectively, with a mean of 1.62. The fold change between the minimum and maximum measured LVD and VPD values was 2.71 and 2.78, respectively. Images series containing the original image, a binary representation, a skeletonized network superimposed on the binary image, and the binary perimeter superimposed on a grayscale version of the original image are contained in Supplemental Figures S1 to S15.

### Lattice Scale Effect on Entire Leaf Images

The mosaic image utilized to examine scale effects on LVD and VPD measures can be seen in Figure 3. As can be seen in Figure 4, both LVD and VPD depend on the area of the FOV and the number of areoles it contains, particularly at small scales that also exhibit the highest variance. For example, the mean LVD at mean areole numbers of 0.5, 1, 2, 3, 4, and 5 is 5.13, 5.09, 5.05, 5.03, 5.02, and 5.01, respectively, with sd of 0.72, 0.67, 0.61, 0.57, 0.54, and 0.53, respectively. Thus, there is a modest decrease in both mean LVD and its sd with increasing FOV.

The increase in LVD and VPD, and the higher variance at small FOVs, is due to both the lattice effects and the vein hierarchical effects discussed below. For example, because the square regions of interest were chosen at random, some will fall on areas with larger veins, which will thus have both lower LVD and VPD.

### Allometry of Vein Area with Leaf Area

We examined if total vein area scales isometrically (slope = 1) or positively allometric (slope > 1) with total leaf area, as predicted by earlier empirical and theoretical work (Niinemets et al., 2007; Price and Enquist, 2007). The slope for the relationship between leaf vein network area and leaf area (Fig. 5) for 350 leaves is 1.084, with 95% confidence intervals from 1.055 to 1.114 (*r*^{2} = 0.932), which do not include 1; thus, the relationship is positively allometric (slope > 1). This suggests that major veins may contribute increasingly to vein area measures as leaves increase in size, although the effect is modest (i.e. the slope is 8% greater than expected under a hypothesis of isometry).

### Vein Size and Density

We report the relationship between the log_{10} value of LVD as a function of leaf area in Figure 6. In doing so, we subdivide the contribution to LVD for each of six vein size classes. The slopes between the log_{10} value of LVD and leaf area in order of decreasing vein size class are −1.0969, −0.2574, −0.3818, −0.3254, −0.4382, and −0.0367, and all slopes significantly differ from zero save that for the smallest vein class (*P* = 0.061). Hence, the negative relationship between LVD and leaf area is significant for all vein size classes, with the exception of the smallest size class.

## DISCUSSION

### Summary of Results

Our results highlight several points with respect to empirical efforts to quantify LVD in cleared leaves. The effect of magnification on LVD measurement is most strongly influenced by the fact that resolution increases with magnification. Our magnified leaf image series all contain the same number of pixels yet span over a 6-fold difference in resolution (pixels mm^{−1}) and a 40-fold difference in FOV. Thus, an individual vein is being represented by an increasing number of pixels within any particular series. The path length increase along any particular vein is due to the fact that at higher magnification it is being measured by what is an effectively finer ruler. This effect has long been known, as exemplified in the aforementioned classic, “How long is the coastline of Britain?” (Mandelbrot, 1967).

We have also shown that vein density measures will be more stable at larger FOV. As seen in Figure 4, as the sample area and/or number of areoles sampled increases, LVD and VPD both flatten considerably with decreasing variance. The variation induced by this estimation procedure is most pronounced when the area sampled is roughly equal to, or smaller than, the size of an individual areole. At sizes smaller than the average areole size, LVD values will be artificially inflated due to the lattice effects we describe above. As an extreme example, consider an image of a single vein where the image encloses the vein but no other part of the leaf. In this extreme case, LVD would be approximately the length of the vein divided by the area of the vein (or its length multiplied by its width). Hence, an upper limit for LVD will be the inverse of vein width.

The decreasing variance with increasing FOV is partially a consequence of the asymptotic behavior of lattices we describe above. As the number of areoles increases, both LVD and VPD measures should stabilize. However, because our leaves are finite in size, the randomly placed subsample windows used to create Figure 4 can overlap, and thus many will have sampled the same regions. Thus, some of the decrease in variance may be due to this factor alone. This is unavoidable in finite-sized leaves using this approach. Additionally, our randomly placed subsample windows may fall on regions with low vein density, such as along a primary vein (midrib), where investigators typically would not sample. We acknowledge that this could also increase the variance at small subsample window sizes. It is difficult, however, to disentangle this effect entirely, because most leaves are hierarchical throughout, and even at small scales, veins within an image can be of different orders and/or diameters, differentially influencing LVD measures. Moreover, an alternative approach, sampling LVD and VPD in nonoverlapping regions, would suffer from a sharp decrease in sample number with area sampled, which would also affect variance measures.

### Explaining Discrepancies between Recently Published Results

In a recent paper, Sack et al. (2012) examined the organization of leaf venation. In doing so, they criticized an earlier analysis of leaf venation networks by Price et al. (2012) and claimed that those authors “did not allow a conclusive test of general scaling of vein traits” (Sack et al., 2012). The results we describe herein suggest that much of the discrepancy between these two studies can be attributed to the scale of measurement; thus, rather than disagreement, we find multiple instances of accord between the two studies.

Sack et al. (2012) report that they find higher mean vein density than our earlier study (Price et al., 2012). We acknowledged (Price et al., 2012) that our measured density values were lower than previous work (Sack and Frole, 2006; Boyce et al., 2009): “We suspect that these differences are due to the fact that our images are unmagnified photographs, which may not resolve all of the very smallest veins, yet which include all major veins.” Our study explicitly considered vein scaling traits at the level of entire leaves utilizing what was, and to our knowledge remains, the best publicly available compilation of entire cleared leaf images (Price et al., 2012). In contrast, Sack et al. (2012) measured vein densities on subsections of leaves of size 1.5 to 1.9 mm^{2} at magnifications from 5× to 40×. Hence, Sack et al. (2012) do not report and/or evaluate the distribution of network properties within a leaf, as done in our work. We offer that the use of different approaches enables different types of conclusions that are in fact complementary to one another.

Sack et al. (2012) report a mean LVD value approximately 3-fold higher than was reported by Price et al. (2012). We note that there is nearly a 3-fold difference across species in the LVD values we found in Figure 2, due entirely to the change in image magnification. Furthermore, Sack et al. (2012) report a higher variance than was found by Price et al. (2012). Increases in variance are expected when imaging small portions of a leaf on the scale of an areole, as demonstrated here (Fig. 4). The increase in variance in estimates of LVD at small scales is likely due to heterogeneity in the structure of leaf venation networks and lattice-type effects at small scales, particularly when vein images do not contain entire areoles, as can occur in leaves with low vein density.

Finally, Sack et al. (2012) found that total leaf vein densities do not vary systematically with leaf size, yet they criticized our study (Price et al., 2012), which also found the same statistically invariant relationship, on the grounds that our conclusion was “not a property of the venation.” In their analysis, Sack et al. (2012) classified leaf veins into corresponding orders and found that vein density declines as a function of leaf area for the midvein and secondary veins but that this relationship flattens and becomes statistically independent of leaf area when examining minor veins (Sack et al., 2012), an important insight. In our prior work (Price et al., 2012), we did not classify veins into corresponding orders. However, it is apparent that measured veins in our paper include veins of all major orders and most, if not all, minor vein orders. We claim that our prior finding of a statistical independence of total vein density with leaf area is a reflection of having measured enough of the leaf venation network to have reached the flattened point of scaling. We evaluate this claim here (Fig. 6) and show that small veins are the largest contributors to total vein length and, moreover, that the total length of small veins is statistically invariant sufficient resolution to capture the invariant relationship between small veins and total leaf area, as we originally claimed (Price et al., 2012). Hence, increases in magnification would, indeed, as found by Sack et al. (2012), increase the baseline expectation for mean leaf vein density (i.e. the *y* intercept; see Fig. 6 in Sack et al., 2012) but not the observation of a flat scaling relationship between LVD and leaf area.

### Recommendations for Future Efforts to Estimate LVD

Ideally, measurement efforts should aim to include larger number of samples, or a larger FOV, at higher resolution and with sufficient magnification to resolve all vein orders. However, for those investigators who rely on vein tracing for measuring vein lengths in cleared leaf images, it may not be realistic to use such approaches, as the time required to trace veins becomes prohibitive. Hand tracing of veins, in some instances, may improve accuracy but may also introduce errors when users differ in the interpretation of what constitutes a vein. In addition, different investigators may use different numbers of straight line segments to approximate a given vein, contributing to variance and potential inconsistency, for curved veins. These challenges highlight the potential benefit of adopting semiautomated approaches (Price et al., 2011). Semiautomated approaches have the advantage of measuring tens of thousands of veins at a rate not achievable by hand-tracing methods. Further improvements in image acquisition, stitching multiple images together to create mosaics (as in Fig. 3), and in the further improvement of thresholding and measurement algorithms are likely to bring such approaches into the mainstream.

While results at high magnification have higher variance than those at low magnification due to lattice and vein hierarchy effects, results within the same level of magnification remain comparable. For example, investigators comparing vein density across a series of leaves all measured at 10× will not have their results influenced by the issues we raise here, save the higher variance observed at small scales. However, as the range of areas and magnifications sampled increases within a study, unintended biases may be introduced, particularly at small sample sizes. For example, a sample measured at 5× may have a lower LVD than one measured at 40× with a smaller FOV, due simply to the phenomena we describe herein and not to developmental, ecological, or evolutionary drivers.

We have also addressed how to obtain magnified images of entire leaves using image-stitching techniques (Fig. 3). Measuring properties of leaf venation networks in magnified images of whole leaves is rarely done except perhaps for species whose mature leaves are sufficiently small and/or during early ontogeny for other species (Donner et al., 2009). Imaging entire leaves at higher magnification requires stitching image subsections together as we have done here, which is time consuming and computationally intensive even for modestly sized leaves. There remain both technological and computational challenges involved in assembling and processing large-FOV images of plant leaves at high magnification.

Finally, we address future efforts to characterize the effect of hierarchical network structure on estimates of LVD. Leaf vein networks contain veins of different orders serving both resource delivery and support roles. It is generally thought that the support functions in leaves are borne primarily by lower order veins (i.e. first and second order veins; Niklas, 1999), with higher order veins functioning primarily in resource delivery (Sack and Holbrook, 2006; Sack et al., 2008; Scoffoni et al., 2011). Due to both biomechanical and hydraulic considerations, the size of lower order veins must increase disproportionately with leaf size (Sack and Holbrook, 2006; Price and Enquist, 2007; Price et al., 2012; Sack et al., 2012). For example, previous analyses have demonstrated that petiole radius is positively allometric (slope > 1) with leaf area, with a mean interspecific exponent of approximately 2.64 (Price et al., 2009). Similarly, the mass fraction of primary veins is positively allometric with leaf size (Niinemets et al., 2006, 2007). This positive allometry implies that the fraction of the vein area occupied by larger, lower order veins increases as leaves increase in size, just as the area of veins increases faster than the area of leaves (Fig. 5). Hence, we recommend that efforts to link LVD with leaf size also specify the extent to which they address major and minor veins.

## CONCLUSION

Our results demonstrate that LVD exhibits a strong and systematic dependence on the scale of inquiry due to magnification effects, geometric lattice effects, and vein hierarchical effects. Our findings caution against using data obtained from multiple scales to make comparative inferences about physiological, ecological, or evolutionary mechanisms driving changes in LVD. We have also shown that seemingly contradictory results from previously published reports can be attributed in part to these scale effects. Advances in image acquisition, and in the development and adoption of algorithmic approaches to quantifying leaf vein measurements, have the potential to increase the scale at which investigators collect data, increasing confidence and facilitating comparisons between measured results.

## MATERIALS AND METHODS

### Definition of Terms

FOV is the physical size of an object or part of an object able to be imaged by a microscope, reported in dimensional units, such as cm^{2}. Image size is the total number of pixels in the image (equivalently, the total number of bytes required for uncompressed storage). Magnification is the ratio of an image’s apparent size to its true size (e.g. objects in a 5× image are five times their actual size). Resolving power is the minimum distance between two distinct features in the sample, such that they can still be visually resolved (reported in dimensional units, such as cm). Resolution is the equivalence between pixels and a dimensional length (e.g. reported in pixels cm^{−1}).

### Biological Specimens

We cleared and stained one leaf from each of the following five species, *Banksia victoriae*, *Hardenbergia comptoniana*, *Plumeria alba*, *Pittosporum moluccanum*, and *Wisteria floribunda*, following standard protocols (Gardner, 1975). Specimens were collected locally on the campus of the University of Western Australia (latitude −31.9 and longitude 115.8). All species have hierarchical veins, as commonly found in many angiosperm taxa.

### Imaging Procedure

We photographed each cleared leaf at 5×, 20×, and 60× magnification using a Nikon SMZ 800 stereomicroscope with a numerical aperture of 0.09 and stored leaf images as .jpg files. The resolving power is normally expressed as *d*_{min} = λ_{0}/NA, where λ_{0} is the mean wavelength of light, assumed to be 0.53 μm for white light, and NA is the numerical aperture of the objective lens. When a digital camera is employed to acquire images, the image size must be sufficiently large such that there are at least two pixels in the image spanning a distance equal to the magnified resolving power of the microscope. In particular, the image resolution must be at least 2/(M*d*_{min}), where M is the magnification; if this is not satisfied, then the resolving power of the microscope in combination with the digital camera is reduced below *d*_{min}. From the specifications of the Nikon SMZ 800, we were able to determine that the camera does not reduce the resolving power, which is approximately equal to 3.6 μm.

Multiple images were taken of each individual leaf at increasingly higher magnifications. Leaves were not moved, so that each higher magnification image in the series represents a subarea of the leaf within the same region (Supplemental Figs. S1–S15). The focal region(s) of all leaves was selected approximately halfway between the point of petiole attachment and the leaf tip, and between the midrib and the leaf margin, to minimize the effects of larger veins whenever possible.

### Vein Estimation Procedure

Following image capture, the LEAF GUI software (Price et al., 2011) was used to identify the venation network. All magnified leaf images were originally 2,560 × 1,920 pixels but were cropped to remove a scale bar and down sampled by a factor of 1:6 to facilitate vein measurement in LEAF GUI, resulting in images that were 297 × 297 pixels. The down-sampling factor was chosen so that the minor-most veins in the 5× images were approximately several pixels in diameter. Down sampling was done in the Matlab software environment and employs bicubic interpolation, in which the output pixel value is the weighted average of pixels in the nearest four-by-four neighborhood. The resulting resolution for each image series was 34.3, 104.2, and 217.4 pixels mm^{−1}, with FOV of 74.7, 8.13, and 1.87 mm^{2}, for images at 5×, 20×, and 60×, respectively.

Binary representations of the leaf veins were then created using a combination of local and global thresholding approaches. Isolated pixel regions that were not connected to the network were then removed via size filtering. When necessary, isolated single-pixel-wide spurs were selectively removed. Detailed descriptions of the approaches utilized in the LEAF GUI software are available at www.leafgui.org and in Price et al. (2011).

Once the venation network was identified, we utilized standard methods to contract the vein network to a single-pixel-wide “skeleton.” LEAF GUI uses the skeleton to estimate the total length of the vein network within any image. We also identified a single-pixel-wide “perimeter” to the vein network and used this perimeter to estimate the total vein perimeter within any image. In high-resolution images, skeletonization can introduce spurious vein tips. To confirm that the scale dependence in LVD we observe was not overly influenced by errors introduced in skeletonization, we introduce a second parameter, VPD, such that VPD = *L*_{p}/*A*, where *L*_{p} is the total length of the perimeter of the identified vein network and *A* is the area of the venation network (Fig. 1; Supplemental Figs. S1–S15). VPD does not rely on skeletonization and thus provides a complementary measure of network length. Detailed descriptions of the approaches utilized in the LEAF GUI software, including a user manual and instructional videos, are available online (www.leafgui.org) and in Price et al. (2011), with several worked examples described in Price (2012).

### Modeling the Effect of Scale on LVD

#### FOV Effects of Regular Lattices

In an earlier report, we laid out predictions for the number of edges, nodes, length, and density per unit area for three regular lattice types, triangle, square, and hexagonal, ignoring finite size effects (Price et al., 2012). Here, we consider explicitly the effects of finite size on measures of theoretical vein density. For example, consider a square lattice composed of cells whose sides have unit length and whose sides are proxies for veins. The area of a lattice of size *n* × *n* is simply *n*^{2}. The total length of edges (i.e. veins) is equal to the total length of the vertical and horizontal edges, which, including the perimeter, is *n*(*n* + 1) + *n*(*n* + 1) = 2*n*^{2} *+* 2*n* in an area of size *n*^{2}. Thus, the theoretical analog to LVD is 2 + 2/*n*. However, if the outer edges are not included, then LVD should be equal to 2 − 2/*n*. One can see that for small *n* (i.e. small sample area with few or no complete areoles), vein density is sensitive to the area sampled and may overestimate or underestimate the value of LVD (even in the absence of noise), which approaches a constant value as *n* becomes large. Similar results are easily obtained for hexagonal and triangular lattices.

#### Lattice Scale Effect on Entire Leaf Images

We hypothesize that vein and perimeter density estimation are scale dependent in the case of idealized lattices (as explained above) and also in the case of veins surrounding irregularly shaped areoles in leaves. Due to the tradeoff between magnification and FOV, magnified images of entire leaves are rare, save but for the smallest leaves. To enable the quantification of lattice effects on a magnified leaf image, we photographed subregions of a single *H. comptoniana* leaflet at 5× to obtain images of the minor veins. To cover the entire leaf took a total of 27 individual subregion images that were 1.15 × 0.84 mm. These subregions were then stitched together using the MosaicJ stitching plugin in ImageJ. The entire mosaic image was 22,942 × 4,848 pixels and was downsized by a factor of 1:4, resulting in an image that was 5,736 × 1,212 pixels. Following image processing in LEAF GUI, we subsampled randomly placed square regions within the leaf ranging in size from 0.5 mm on a side (0.25 mm^{2}) to a square with a side length equal to the maximum distance from any point within the leaf to a point outside of the leaf divided by 2.2, chosen so that the side of the sampling square is less than half the maximum distance. For example, in an idealized circular leaf, the maximum distance would be the circle radius, so the largest square would have a side length of radius/2.2. Dividing the maximum distance by a number greater than 2 allows multiple subsampling squares to be placed within the leaf (we note that other choices would yield similar results). The maximum square size for the *H. comptoniana* leaf was 3.61 mm a side or 13 mm^{2}. At each square size, 10,000 subsampling squares were randomly placed within the leaf without overlapping leaf borders, and both LVD and VPD were calculated. This allows us to examine the mean and sd for a given magnification and, thus, the effect of FOV on LPD and VPD measures.

#### Allometry of Vein Area with Leaf Area

Veins of different orders contribute equally to vein density per unit vein length but unequally to vein area. Using a previously described database of 350 unmagnified cleared leaf images (Price et al., 2012), we utilize LEAF GUI to estimate the amount of area occupied by veins as a function of increasing total leaf area.

#### Vein Size and Density

We investigated the influence of both leaf size and vein size on LVD estimates, using vein width as a proxy for overall vein size. For each leaf of the 350 unmagnified leaf images utilized in an earlier study (Price et al., 2012), we binned vein widths into six linearly spaced bins from the narrowest to the widest vein. We chose six bins for our analysis but note that the results are robust to variation in bin number. We then calculated the corresponding vein density for each width class.

### Supplemental Data

The following materials are available in the online version of this article.

**Supplemental Figures S1–S15.**Image series corresponding to the data presented in Figure 2. For each species and magnification we show the cleared leaf image, binary representation, a blue version of the binary image with vein skeleton (LVD) superimposed, and a gray-scale version of the cleared leaf image with the vein perimeter (VPD) superimposed.

## Footnotes

The author responsible for distribution of materials integral to the findings presented in this article in accordance with the policy described in the Instructions for Authors (www.plantphysiol.org) is: Charles A. Price (charles.price{at}uwa.edu.au).

↵1 This work was supported by Discovery Early Career Research Awards from the Australian Research Council (to C.A.P. and P.R.T.M.) and by a Career Award at the Scientific Interface from the Burroughs Wellcome Fund (to J.S.W.).

↵[C] Some figures in this article are displayed in color online but in black and white in the print edition.

↵[OPEN] Articles can be viewed online without a subscription.

↵[W] The online version of this article contains Web-only data.

### Glossary

- LVD
- leaf vein density
- FOV
- field of view
- VPD
- vein perimeter density

- Received July 5, 2013.
- Accepted November 19, 2013.
- Published November 20, 2013.