Catena 118 (2014) 73–83
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Automatic clod detection and boundary estimation from Digital Elevation Model images using different approaches Olivier Chimi-Chiadjeu a, Sylvie Le Hégarat-Mascle b,⁎, Edwige Vannier a, Odile Taconet a, Richard Dusséaux a a b
LATMOS/IPSL, Université Versailles Saint-Quentin en Yvelines, France IEF, Université Paris-Sud, 91405 Orsay cedex, France
a r t i c l e
i n f o
Article history: Received 30 October 2013 Received in revised form 29 January 2014 Accepted 12 February 2014 Available online xxxx Keywords: Soil micro-topography Aggregates and clod Image processing Contour estimation
a b s t r a c t Soil micro-topography characterization is an important issue for both soil science and remote sensing data interpretation. The objective of present study is to propose and discuss some methods dedicated to the automatic localization of clods (or big aggregates) on Digital Elevation Model images of soil. Two new image processing methods are introduced. The first one deals with the clod detection and the rough estimation of their boundaries. It is based on the adaptation of a famous segmentation algorithm applied to a modified surface enhancing the main features characterizing the clods. The second proposed method deals with the accurate estimation of clod boundaries. Clod boundaries are moved based on dynamic programming. Both proposed methods are validated on laboratory-built surfaces and on an actual surface recorded in an agricultural field. Results show that the proposed methods outperformed previously published methods. The proposed processing of DEM images allows the detection of the aggregates and clods deposited on the soil surface and the accurate estimation of their boundaries. The practice is facilitated by the proposition of default values for the parameters. The implications are the automatic analysis of DEM images that is a step towards micro-topography statistical characterization. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Soil roughness is a key parameter to understand soil properties and physical processes related to energy and mass transfer between the atmosphere and the soil, surface water flow processes, soil erosion and agronomic processes, such as seed germination and seedling emergence. Tillage implements on bare agricultural soils, such as seedbed preparation, lead to low and moderate roughness. The cloddiness can then be quantified in terms of the sizes of the aggregates and clods and their size distribution that are key features of bare soil microrelief to understand two major processes. The first one deals with the choice of optimum tillage to obtain the most favorable plant emergence and the second one is the study of the rainfall impact on microrelief changes. These two processes depend also on other soil physical properties, such as the soil texture, the soil water content and the soil hydraulic properties. Many authors noted that most of the mechanisms of soil fragmentation by tillage implements and of water interaction with the soil surface occur at millimeter scales, (e.g. Arvidsson and Bölenius, 2006; Berntsen and Berre, 2002; Kamphorst et al., 2005; Martin et al., 2008; Roger-Estrade et al., 2004; Rudolph et al., 1997). In the field of remote sensing, soil roughness at small scale occurs for the analysis and interpretation of measurements both in optical domain ⁎ Corresponding author. Tel.: +33 1 69 15 40 36. E-mail address:
[email protected] (S. Le Hégarat-Mascle).
http://dx.doi.org/10.1016/j.catena.2014.02.003 0341-8162/© 2014 Elsevier B.V. All rights reserved.
(visible and near infra-red) and in active microwave one. Indeed, the bidirectional reflectance and the Synthetic Aperture Radar (SAR) signals are very sensitive to soil surface state, especially soil surface irregularities and structures (clod arrangement, furrows). Soil aggregates and clods produced by farming practices affect the bidirectional reflectance by shadowing effect. They have to be separated from soil radiative properties related to biochemical factors, (e.g. Cierniewski et al., 1996; Wang et al., 2012; Wu et al., 2009). The microwave backscattering coefficient also depends on the local surface slope at centimeter scale, which is directly related to clod arrangements. Then, several studies investigate the sensitivity of SAR radars to soil surface parameters, (e.g. Corbane et al., 2011; Holah et al., 2005; Lievens et al., 2011; Verhoest et al., 2008; Zribi et al., 2000). In order to link the remote sensing data to scattering physical models and for modeling purpose, it is important to further characterize key features of the soil micro-relief. Two scales are usually considered for soil roughness modeling. Firstly, the surface is globally characterized by its autocorrelation function model and/or by statistical indices, (e.g. Bertuzzi et al., 1990; Dusséaux et al., 2012; Helming et al., 1993; Römkens and Wang, 1986; Taconet and Ciarletti, 2007). Secondly, a more local and detailed analysis of the surface is performed by focusing on patterns such as aggregates, clods and mound-and-depressions (Ambassa-Kiki and Lal, 1992; Arvidsson and Bölenius, 2006; Borselli and Torri, 2010; Darboux et al., 2001; Kamphorst et al., 2005). In this study we call ‘support surface’ the soil surface model neglecting these aggregates,
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clods and depressions. Several works have already dealt with the problem of clod detection/estimation in the case of an agricultural surface (Jester and Klik, 2005; Vannier et al., 2009). The basic sieving (that allows filtering the clods according to their size) being fastidious, alternative methods have been proposed, such as works based on image processing. Two main kinds of image contents have been considered: visible reflectance images (Sandri et al., 1998; Warner, 1995), providing 2D information, and Digital Elevation Model (DEM) images (Helming et al., 1993), providing 3D information. On the one hand, using visible reflectance images, most classical image processing tools apply. On the other hand, due to the fact that main clod features are geometric, 3D geometric measurements seem much more appropriate than radiometric ones so that several techniques have been developed to acquire DEM images, either based on laser measurements (Bertuzzi et al., 1990; Huang and Bradford, 1990) or on stereovision (Taconet and Ciarletti, 2007). Specifically, stereovision exploits images (photographic or digital) acquired with different view angles to determine the 3D geometry of the scene (in our case, the reconstruction of the surface of the agricultural area in 3D space). Now, when handling images of the surface, the estimation of its physical parameters may be not direct. Then, the first task to study patterns such as clods or aggregates is to delineate them on the image. Indeed from this delineation, some 2D parameters derived from the footprint (such as the mean surface of the aggregates and their spacing) can be computed as well as some 3D parameters (such as the mean volume of the aggregates) if a DEM image is available. For instance, applying their method (Taconet et al., 2010) of automatic clod delineation on DEM images on a large dataset of several hundred of clods, Taconet et al. (2013) shows that these irregular shaped objects can be rather well approximated by simple approached forms (an ellipse for the basis and a half cosine function for the height). In this study we focus on clods or aggregates delineation using DEM images. Since DEM pixel values represent the altitude, some discriminative clod features may be: (i) the clod pixels have a higher value than support surface pixels around the clods, (ii) the clod contours correspond to ‘rapid’ variations of pixel values, and (iii) each clod has one main maximum of values. We use the term ‘contour’ when referring the clod boundary represented in the image domain. Now, in the case of DEM images, classical segmentation methods fail to identify the clods on the support surface, due to the intrinsic non-homogeneity of the clod values (altitudes). Thus, classical image processing tools should be adapted or new ones should be developed. For instance, Taconet et al. (2010) developed a specific level line based approach. The aim of this study is to provide a comparison between different algorithms for clod detection. The first one is (Taconet et al., 2010) mentioned just above. Its main drawback is the under-estimation of the clod size due to a location of the contours inside the clods. The second one is the extension of a preliminary work (Chimi-Chiadjeu et al., 2012), where we had proposed to adapt the so-called ‘watershed’ algorithm to our problem of clod detection. The two last ones deal with the improvement of the clod location rather than with their detection. Both are based on clod contour move. The first one (Chimi-Chiadjeu et al., 2013) aims at moving the initial contour using a simulated annealing optimization. Anyway, because of its computational heaviness, it is not adapted to actual surfaces. Then, in this work, we propose a new approach based on dynamic programming that outperforms previous work. In order to provide some quantitative results, we test the different algorithms on data acquired either on laboratory-built surfaces or on an actual surface obtained on an agricultural soil.
slope features (assuming as in Richard et al. (1999) that clods are roughly comparable to half ellipsoids, boils down to assume a high value for the slope around clod boundaries). The different methods for clod detection from DEM images are thus based on these features (altitude and slope), formalized in different ways, depending on the chosen clod model and on the constraints due to the used estimation algorithm. Let us first define some notations. H is the DEM image, x is the row coordinate, y ∈ {1,⋯, Nr}, and y is the column coordinate, y ∈ {1,⋯, Nc}, so that H(x, y) denotes the altitude value in pixel of image coordinates (x, y). In every pixel, the altitude gradient is a vector with two components respectively along the row and the column directions. Several image processing tools have been proposed to estimate these two components. Simpler techniques are linear filtering, e.g. Sobel operator (Ziou and Tabbone, 1998), whereas ‘optimal filtering’ has also been proposed, (e.g. Canny, 1986; Deriche, 1987; Haralick, 1984). Although in most cases linear estimation is sufficient, in the following we note G(x, y) the estimate of the gradient norm at pixel (x, y), without specifying the way it has been estimated. In this study, we distinguish between methods for clod detection (and rough estimation of their contours) that do not require initialization, and methods for improvement of clod contour estimation that require an initialization. In each case, before presenting the proposed approach, we briefly recall the state of the art for the considered problem in order to further evaluate the interest of the proposed approach relatively to it. 2.1. Clod detection and rough estimation of their contours 2.1.1. Background in clod detection Taconet et al. (2010) proposed an original method mainly based on the analysis of the level lines in the image H. In image processing, a level line is a set of connected pixels having the same value in the image. Thus for DEM images, a closed level line corresponds to an elevation contour. A key point of Taconet's method is the selection of the level lines of interest. Two criteria have been used. First, only the closed level lines of minimum length are considered. In particular, the clods intersecting the image borders (i.e. the clods not completely included in the image domain) are not considered. Second, only the level lines with pixels of ‘highest’ gradient norm values are considered. The word ‘highest’ refers to a percentile p of the pixels presenting the pNr Nc higher values in G image. Then, the definition of a clod is as follows: a clod is a concave 2D function (of altitude). In terms of level lines it means that a clod only includes nested level lines. Practically, from selected level lines (both criteria), a clod center is defined as a level line with no other included level line, and a clod contour is defined as the longest level line having only one clod center included. We refer the reader to Taconet et al. (2010) for more detailed information on the practical implementation of the method. Clod detection proposed by Taconet et al. (2010) is mainly based on altitude information (H image) whereas gradient norm information (G image) is used as a constraint. Altitude information is used without preprocessing (e.g. filtering, contrast enhancement) so that any irregularity in the DEM image has an impact on clod detection. In particular, when a clod has several summits (even very slightly pronounced) it is divided in as many detections. Another drawback of the method is that searching the clod contour among the (selected) level lines, the clod basis can only be horizontal whereas, actually, a clod can stand on a non-horizontal surface (the side of a furrow, for example). To overcome these drawbacks, a new method was investigated.
2. Image processing methods for clod estimation In this work we focus on clod detection or estimation using DEM images, that is to say images where the value in every pixel represents the elevation or altitude (z coordinate). Clods are characterized both by altitude features (they are ‘above’ the support surface) and by
2.1.2. Proposed method based on watershed segmentation We propose to split our problem of clod detection (and rough contour estimation) into two sub-problems. The first one deals with the image segmentation into regions so that the clods coincide with some of them. The second one deals with the selection of the
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clods among the obtained regions, since the partition of the image also contains a (some) region(s) representing the support surface between the clods. Besides, for region estimation (segmentation part) we aim at considering both features concerning altitude and gradient norm. Let us first discuss the choice of a segmentation algorithm and the data to which it applies. In image processing, the segmentation is a well-known problem for which numerous approaches have been proposed. It aims at partitioning an image into regions, i.e. clusters of connected pixels, that represent an intermediate level towards the detection of objects. Several criteria have been proposed to perform the segmentation: ‘region’ criteria, i.e. based on the region features, ‘boundary’ criteria, or ‘mixed’ criteria, i.e. involving both region and boundary features. For instance, the ‘region growing’, or ‘split and merge’ (Chen and Pavlidis, 1979) algorithms are examples of ‘region’ based approaches. Basically, an initial image partition is modified by merging or dividing regions, based on the characteristics (area, brightness, color, texture, etc.) of each region. More robust are the approaches based on both ‘boundary’ and ‘region’ criteria. For example, Mumford and Shah (1989) proposed to include constraints on geometric regularity (e.g. minimizing the whole length of the boundaries) in addition to classic region homogeneity criteria, and they gather these different criteria to define a functional1 The parameter weighting the different criteria in the functional acts like a scale parameter by controlling the minimal size of the regions and their final number (Koepfler et al., 1994). All previously cited methods assume that the regions are homogeneous and they fit the region boundaries at local maxima of the gradient norm. Now, considering a DEM image, the clods are not characterized by homogeneous values (since the altitude varies inside the clod), and, in the actual case, the clod boundaries are not defined by local maxima of the gradient norm values (for instance, after rain the clod boundaries become smoother). Apart the mentioned segmentation approaches, the ‘watershed’ transform is derived from mathematical morphology (Haralick et al., 1987; Serra, 1986). While assimilating the image to a topographic surface with pixel values representing the altitudes, from the regional minima, a progressive immersion of the topographic surface is simulated in order to find the watershed boundaries, that is to say the frontiers where the waters of the various watersheds meet. It can be shown (Beucher, 1990; Meyer and Beucher, 1990; Najman and Schmitt, 1996; Vincent and Soille, 1991) that such boundaries are geodesically equidistant from the image minima. Classically, to segment a radiometric image in uniform regions bounded by edges corresponding to high gradient, the watershed segmentation applies to the image of the gradient norm, so that the local minima are located in homogeneous areas and the local maxima on areas presenting rapid variations of gray levels. Coming back to our problem of clod segmentation, we aim at deriving regions such that their boundaries are located on pixels presenting a DEM value as close as possible to the DEM value of the support surface around the clod (first criterion), and a gradient norm value as high as possible (second criterion). We propose to gather both criteria into a functional. Since the watershed segmentation will place the contours at local maxima of the image, we define our functional, I(x, y) such that its local maxima correspond to the clod contours, i.e. such that it decreases with the altitude pixel value and it increases with the gradient norm value. We propose Iðx; yÞ ¼ −H m ðx; yÞ þ γGm ðx; yÞ
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γ is a weight coefficient between the two terms of the functional and G m is the gradient norm image computed on the ‘pseudoaltitude’ image Hm: 8 >
: Hðx; yÞ
if Hðx; yÞNH
where λ is the sigmoid parameter,H is the altitude (DEM value) of the support surface, and α is a coefficient defined so that the function is continuous in H and global contrast of the DEM image (i.e. the difference between the maximum and the minimum values on the DEM image) is H max −H ·H can be estimated as the −1 ½1þ expð−λðHmax −HÞÞ −0:5 mean of the DEM values either globally to the image, or locally to take into account the non-stationarity of the support surface for instance in the case of fields with furrows. Fig. 1 shows some examples of DEM image transformation (Eq. (2)) for different values of λ and H and for H values varying within the [0, 20] interval. We check that H controls the beginning of the sigmoid part of the function, and λ controls the slope of the sigmoid: the higher is λ, the higher is the tangent at the right of the pointH. As said previously, the second sub-problem deals with the selection of the clods among the obtained regions. It can be divided into two steps: the removal of the regions that do not correspond to clods, and the fusion of the regions that correspond to a same clod. In the ‘removal’ step, the regions that do not check the clod features in terms of altitude both ‘significant’ and greater than the support surface altitude are removed. For this, the median values of the altitudes are computed for
preserved: α ¼
∫
k and second on the contour of the each clod k first inside the clod Hmed
∮
k . The proposed condition to ensure that the clod is sufficiently clod Hmed prominent (and thus ‘significant’) is:
∫
∮
k k −H med Nθ H med
ð3Þ ∫
k where θ is a threshold parameter. In addition, the condition Hmed NH implies that the clod is not located in a hole. In the ‘fusion’ step, are merged the regions that represent different ‘summits’ of a same clod. For instance, consider a clod having two altitude maxima spatially close and separated by smooth local minima (e.g. a clod having the shape of the sum of two Gaussian functions with the same standard deviation, σ, and different means μ1 and μ2 so that |μ 1 − μ 2| N 2σ). Then, the functional given by Eq. (1) will present local maxima inducing the dividing of the clod into two regions when using the watershed segmentation algorithm applied to the surface computed by Eq. (1) computed surface. To free from such oversegmentation of clods, we propose a merging criterion consistent with Eq. (1). For any region or union of regions candidate to represent a clod, we define an ‘objective’ function equal to the
ð1Þ
1 In image processing, a functional is a scalar-valued function defined on the space of the 2D functions representing images.
ð2Þ
otherwise;
Fig. 1. Function used to compute ‘pseudo-altitude’ image from DEM image.
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averaged value of I(x, y) over the boundary of the region or of the union of the regions. Then, for any pair of connected regions (i.e. regions presenting a common boundary), we merge them if the objective function is greater for the merged region than when the two regions are considered separately: Let R1 and R2 be two connected regions and R1 ∪ R2 be their union, ∮
∮
∮
I R1 , I R2 and I R1 ∪R2 are the values of the objective function over the ∮R ∪R regions n R1, R2 and o R1 ∪ R2. Then, R1 and R2 are merged if and only if I 1 2 ∮R ∮R N max I 1 ; I 2 . In summary, developing a new method based on the watershed segmentation algorithm, to achieve the clod detection and the rough estimation of their contours, we have to add three processing steps: 1 A transformation of the initial surface so that the local maxima of the transformed surface (given by Eq. (1)) correspond to the clod contours, since the watershed segmentation will place the region boundaries at image local maxima. 2 A selection of the regions corresponding to the clods among the whole set of regions, since the watershed segmentation result is a partition of the whole image, i.e. also including region(s) corresponding to the support surface area. 3 A fusion of some previously selected regions that correspond to a same clod, since the watershed segmentation has the well-known feature to oversegment the image. These steps involved three a priori parameters (γ, λ, θ). In our case, we obtain good results with values equal to (0.5, 0.2, 0.0) for all the considered surfaces (cf. Section 3.2). Thus, we think these values can be used as default values. 2.2. Clod contour accurate estimation Previously presented methods aimed at detecting the clods more or less roughly. Results will show that the watershed segmentation already provides a good estimation for the most of the clods. They use either mainly the altitude information (and the gradient norm only as a hit or miss criterion), or a combination of the altitude and gradient norm information pieces (through Eq. (1)). In this section, we aim at refining the clod contour estimation using more information pieces. 2.2.1. Background in clod contour displacement In Chimi-Chiadjeu et al. (2013), the optimization of a cost function based on a richer modeling of the clods has been proposed. The four criteria characterizing the clod boundaries are: (i) the mean of the gradient norm values on the clod boundary, (ii) the standard deviation of the altitudes on the clod boundary, (iii) the standard deviation of the gradient norm values on the clod boundary, and (iv) the normalized L2-norm of clod boundary altitudes. Thus, in addition to previous criteria on clod boundary altitude and gradient norm values, two additional criteria are introduced, namely the minimal variation (measured in term of standard deviation) of the gradient norm and of the altitude values (over the clod boundary). Note that in Taconet et al. (2010) the standard deviation of the clod contour altitudes was implicitly forced to 0 since only isoaltitude contours are considered as potential clod contours. These four criteria are linearly mixed in a cost function (ChimiChiadjeu et al., 2013), for which the weighting parameters have been learned from the clod reference contours (called further the ‘ground truth’) defined by an expert of soil science. The optimization of the cost function is done in a stochastic way according to a procedure inspired from the simulated annealing approach (Granville et al., 1994; Kirkpatrick and Vecchi, 1983; Le Hégarat-Mascle et al., 1996). Basically, small changes in the clod contour are tested and, according to the result of the test, accepted or rejected. The process is iterative until a stop criterion is verified. The tested changes are the moves of one pixel by one pixel of the clod contour. The induced variation of
the cost function value and a randomly drawn variable determine the decision to accept or reject the tested change. Now, conversely to classical simulated annealing that aims at finding global optimum (through the acceptance of almost any change at the beginning of the convergence), in Chimi-Chiadjeu et al. (2013), the minimization process is local to ensure that the obtained result remains close to the initialization (in the solution space). The method proposed by Chimi-Chiadjeu et al. (2013) has two drawbacks. First, as stochastic optimization processes, it is long and fastidious. Second, as only local optimization is performed, it is highly dependent on the initialization. To overcome these drawbacks, a new method was investigated. 2.2.2. Proposed method based on dynamic programming Dynamic programming (Denardo, 1982; Sniedovich, 2010) is a famous optimization technique. It is based on dividing a complex problem into simpler subproblems. Thus it takes advantage of the resolution of these partial or subproblems to drastically reduce the complexity of the main problem and thus allowing its resolution. Compared to greedy algorithms that do not guarantee the optimal solution, dynamic programming does. It has been applied to numerous optimization problems, including NP-hard problems (such as the knapsack problem or the traveling salesman problem). In addition, some optimization algorithms independently conceived have been found further to be dynamic programming (e.g. Viterbi's algorithm, Dijkstra's algorithm (Sniedovich, 2006)). The idea behind dynamic programming is to recognize that the computation of the main problem solution requires the computation of the solutions of several subproblems, so that the subproblem solutions are computed only once. Either the subproblem solutions are stored so that each time the subproblem is encountered, the solution is directly taken in the memory, or the resolution of a subproblem boils down to only keep a reduced number of potential values for the variable to optimize. In the following, we focus on a particular case of the selection of a subset of the set of the potential solutions thanks to the resolution of a subpart of the whole problem. A trivial example is the minimization in ℝ2 of a separable function, e.g. minimize f(x, y) = A(x) + B(y), that can be divided into two subproblems corresponding to the minimization of each separable part of the function, namely A(x) and B(y) in the example. Another example is the case of a cost function that has the form of a hidden Markov chain: assume the cost at t only depends on the cost at t − 1 and the state at t − 1 and not on the states from instant 1 to t − 1 (the state corresponds to the hidden variable that contains the dependency between instants). Then, the cost minimization at t can be divided into subproblems that are the minimization of the cost at t − 1 given the state at t − 1 and so on. Coming back to our problem (clod estimation), we propose to minimize a functional close to the one used in Chimi-Chiadjeu et al. (2013) but two points. First, in order to avoid some compensation between terms, all terms and weighting coefficients are positive. Thus, since the gradient norm should be maximized, we rather consider its inverse. Second, in order to recover the form of a hidden Markov chain, the criteria on the standard deviation of altitude values and of the gradient norm values are replaced by a sum of (square) difference between successive (altitude or gradient norm) values along the clod contour. Then, the proposed functional writes
J ðC Þ ¼
1 γ4 2 2 ds ∫C γ 1 f 1 ðH ðsÞÞ þ γ2 δH þ γ3 δG þ GðsÞ þ γ 5 jC j
ð4Þ
where C is the clod contour, |C| is the length of C, δH and δG are respectively the local variation of altitude and gradient norm on the contour, γ1, γ2, γ3 and γ4 are some weighting coefficients and γ5 is a constant in particular to avoid the division by zero. f1 is a function of the altitude values in order both to penalize non-linearly the altitude values far from
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the support surface altitude and to take into account the imprecision in the estimated support surface altitude. In our case, we choose a sigmoid based function: f 1 ðxÞ ¼
1 : 1 þ expð−λ2 jx−xm j−λ1 Þ
ð5Þ
This function has the form of a ‘U’ letter centered on x = xm, with λ1 giving the width of the ‘U’, λ2 giving the slope around x − xm = λ1. xm is thus the assumed value of altitude of the support surface, estimated in our case as the median over the surface (or a subpart of the surface in case of non-stationary surface). Assuming a spatial constraint of location of the clod contour, the minimization of J is performed over the region ZR to which it is forced to belong: ^ ¼ argmin J ðC Þ: C ZR
Fig. 2. Potential following nodes versus the geometric configuration of the current path extremity and the preceding node: case of ‘oriented’ connectivity of order 3. Color codes are as follows: ‘preceding node sj(n − 1)’ pixels are in light gray, ‘current extremity sj(n)’ pixels in dark gray, and ‘potential following nodes sj(n + 1)’ pixels in dark.
ð6Þ
The dynamic programming algorithm is divided in two main steps. The first one aims at selecting a subset of the solutions among which there is the optimal solution. The second one aims at determining the optimal solution among the previously mentioned subset of solutions. Then, we define an algorithm in two parts presented in Algorithms 1 and 2 respectively. Let us introduce some notations. Let G be a valued graph whose nodes are the pixels through which the researched clod contour could cross. G is a subset of connected pixels ‘around’ the clod contour initialization, noted Cinit. In G there are two particular nodes that represent respectively the beginning and the end of the clod contour. They are noted sb and se respectively, and a graph path is closed if sb and se are connected nodes. Let Cj be a path graph in G beginning in sb and of length n (equal to |Cj| the cardinality of Cj): Cj is a n-tuple. Then, each pixel s belonging to Cj having an index greater than 1 in the n-tuple (i.e. not equal to sb) has a ‘preceding’ and we can note δH(s) the difference between the altitude values (i.e. pixel values in H image) of s and its preceding and δG(s) the difference between the gradient norm values (i.e. pixel values in G image) of s and its preceding. We define the cost of Cj as: X 2 2 γ1 f 1 ðH ðsÞÞ þ γ2 ½δHðsÞ þ γ3 ½δGðsÞ þ J Cj ¼ s∈C j
γ4 : γ 5 þ GðsÞ
strip, and a set Sb of sb nodes is defined so that it crosses the closed strip from the inside border of the strip to the outside border. For any values sb ∈ Sb, Algorithm 1 is then applied. Additional used notations are: the preceding of sb noted sb(− 1), the set Sn of nodes representing a terminal node of a currently examined graph paths C j and tprec the preceding node of a given node t (if t = s j(n), then tprec = sj(n − 1)). In Algorithm 2, the symbol x‖y denotes the concatenation of x and y tuples. These algorithms involved seven a priori parameters (γ1, γ2, γ3, γ4, γ5, λ1, λ2). In our case, we obtain good results with values equal to 1; 1; 1; 15; G þ 1; :3; 2: , where G is the median of the gradient norm values, for all the considered surfaces (cf. Section 3.3). We think these values can be used as default values. Algorithm 1. Node value computation in dynamic programming algorithm for clod contour accurate estimation
ð7Þ
The value V(s) in a node s is the minimum of J(Cj) (Eq. (7)) among the already examined graph paths from sb to s. Noting sj(n) the last pixel of graph path Cj: V ðsÞ ¼ minC j js jðnÞ ¼s
1 J Cj : n
ð8Þ
According to the dynamic programming principle, one computes the node values V(s) while exploring the different paths to reach a node (pixel) and updating the node values as ‘better’ (i.e. having lower costs) paths are discovered. Then, a key point is the way the different paths are explored. Having constructed a partial path Cj with current extremity sj(n) and preceding node sj(n − 1), Fig. 2 illustrates the potential following nodes sj(n + 1). In Fig. 2, we present a configuration with 3 potential following nodes in the continuity of the direction (sj(n − 1), sj(n)). In Algorithm 1, to simplify the notations, we note Ns,j the set of the potential following nodes sj(n + 1). Algorithm 1 describes the way graph node values are determined as the subset of graph path solutions is also determined. Algorithm 1 assumes known the set of nodes of G. In our case, it is computed using a dilatation process (mathematical morphology operator) of the initial clod contour Cinit by a large structuring element (typically a (2k + 1) × (2k + 1) structuring element is required to displace Cinit of a distance of k pixels). Then the set of G nodes forms a closed
Algorithm 2. Optimal graph path derivation from G
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3. Analysis of performance This section provides some quantitative results obtained in the case of data acquired either on laboratory-built surfaces or on an actual surface obtained on an agricultural soil. Let us first present the used database. 3.1. Database Our approach has been tested on four DEM images of bare agricultural soil surface obtained either by a profile laser scanner (Darboux and Huang, 2003) or by stereo photogrammetry (Helming et al., 1993). These two approaches differ by their acquisition principle. Using the profile laser scanner, each point measure represents the Time of Flight. In order to acquire an image, the scene is scanned varying the incidence angle. In order to overcome the occultation problems, two images are acquired (using incidence angles symmetric relatively to the nadir) that are co-registered using few common reference points. In the case of the stereovision, two images are also acquired from the two cameras. Then, having ‘rectified’ the images from the knowledge or estimation of the geometric acquisition parameters of the camera system, a disparity image can be computed. In this latter, a pixel value represents the difference in column coordinates of homologous pixels, i.e. pixels representing a same 3D point projected in each image of the stereo pair. Disparity is inversely proportional to the depth value of the corresponding 3D point. Now, the final precision of the estimated DEM highly depends on the method used to find the homologous pixels (corresponding to the same 3D point respectively projected in each considered image); e.g. block matching process based on Sum of Absolute Difference, or on correlation (Taconet and Ciarletti, 2007). The three first surfaces, called S1, S2 and S3, are laboratory-built surfaces made in laboratory by sieving a real soil. From S1 to S3, the surface ‘complexity’ increases either due to the non-stationarity of the support surface (S2 exhibiting furrows having the form of a sinusoidal function) or due to the fact that clods are very close and partially overlapping (S3). The spatial sampling is equal to 1 mm on the three axes x, y and z. The size of S1 and S2 is equal to 250 × 250 mm2 and S3 size is 80 × 80 mm2. The fourth surface was acquired on an actual field. Its size is equal to 200 × 200 mm2 with a resolution of 1 mm along the horizontal plane axes and 1.1 mm on the vertical axis. This surface shows a greater range of clod size. Manual delineation of individual clods, what we call reference contours, was completed by a soil scientist on the two last surfaces. Fig. 3 shows the four surfaces in 3D and the corresponding DEM images to be processed. 3.2. Validation of the watershed segmentation approach In this section, we aim at validating the proposed approach (watershed segmentation based) for clod detection and rough estimation, either in absolute terms or in comparison with the cited background method (Taconet et al., 2010). Fig. 4 shows the results obtained using the watershed segmentation adaptation on the four studied surfaces. The obtained clod contours are plotted in red. For the two first surfaces (S1 and S2), the obtained clod contours are highly consistent with visual interpretation of the surface (that is easy in the case of these two surfaces rather simple). For S1, the result of manual clod delineation by photo-interpretation is plotted in dashed green. In the case of S 3 and S 4, the reference contours are plotted in dashed green. Reference contours are only defined when the clod is completely included in the DEM image. Watershed segmentation based approach provides the clods once the DEM image includes a significant part of them. This is clearly visible on Fig. 4c where the high density of clods leads to several overlapping with image borders. Qualitative analysis of the obtained results on this surface leads to the conclusion that most clods are well detected: 5 ‘non-detections’ (small clods with center coordinates about (15,10), (70,10),
(20,60), (10,65), (20,75)) out of 29 right detections (in agreement either with reference contours or with visual interpretation for clods overlapping image borders), and 1 ‘false detection’ (small clod about in (57,62)). In addition, we note that the estimated clod contours are close to the reference contours (when available) and, for the clods not completely included in the image, the obtained contours seem consistent with visual interpretation. Fig. 4d presents the results obtained for S4 surface. Even if the surface is much more complex (for instance visual interpretation becomes very tricky), the conclusions about the qualitative performance of the method remain valid. In S4 case, we note 5 ‘non-detections’ (small clods about in (80,40), (140,30), (20,85), (100,80), (175,75)) out of 24 right detections, and no ‘false detection’. In order both to measure the precision of the estimated clod contours and to provide a quantitative indicator of the performance of the method(s), we base on the performance indicator proposed in Chimi-Chiadjeu et al. (2013): Let τj be the variable representing the overlapping of the computed clod j (by any method) and the corresponding reference clod defined from the reference contours. τj can be simply computed as follows. Let B e be the binary image of the estimated clods (pixel value is 1 if it Cj
belongs to an estimated clod, 0 otherwise), and BC^ j be the binary image of the reference clods. The cardinality of a binary image B, noted |B| is the number of pixels equal to 1 in B. Binary AND, noted ∧, and OR, noted ∨, operators are defined between images applying the classical binary operators on every pixel. Then, τ j ¼ Be ∧BC^ Cj
j
=B ∨B ^ e C Cj
: j
ð9Þ
The Cumulative Distribution Function (CDF) of τ represents, for each possible value of τj (τj ∈ [0,1] by definition), the probability of clods having a τj value lower than τ. The probability is simply estimated as a ratio between the number of clods having τj = x b τ and the total number of clods. Fig. 5 shows the quantitative evaluation of the results in the case of S 3 and S4 (reference contours are required for quantitative evaluation). The ordinate at the origin (τ = 0) is the proportion of the 0-overlapping clods, that is to say either ‘non-detections’ or ‘false detections’. In case of perfect result the CDF of τ would be such that ∀ τ ∈ [0, 1), CDF(τ) = 0 and CDF(1) = 1. Thus, the closer to the x-axis the curve, the higher the achieved performance. We see that for S3 the median (CDF (τ) = 0.5) overlapping rate τj is 0.72 (half clods have an overlapping rate greater than 0.72), and 25% of the clods has an overlapping rate greater than 0.82 (CDF(τ) = 0.75 ⇒ τ ≈ 0.82). The CDF integral (that ideally should be null) is equal to 0.61 for Taconet et al. (2010) method and is equal to 0.42 using the proposed method, inducing thus an improvement of about 20%. For S4, the median overlapping rate τj is equal to 0.5 and 25% of the clods has an overlapping rate greater than 0.69 (CDF(τ) = 0.75 ⇒ τ ≈ 0.69. For S 4 , the CDF integral decreases from 0.62 (Taconet et al. (2010) method) to 0.52 using the proposed method (improvement about equal to 10%). For comparison, we have considered the method previously proposed by Taconet et al. (2010). From Fig. 5 also showing the performance obtained using Taconet et al. (2010) method, we measure the improvement brought by the proposed method in the case of S3 and S4 surfaces, by the distance between the two curves. Results using Taconet et al. (2010) method have also been reported Fig. 4a, b, c and d. In the case of Fig. 4a, the difference is noticeable only on the clod located about at (225,225). In the case of Fig. 4b, it is visible on the clod located about at (225,50) and, to a lesser extent, the clod located about at (30,160). The surface S2 represents a freshly plowed surface: the clods are objects lying on the support surface, with a strong gradient at the clod boundary, so that the altitude at the clod boundary varies
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(a) S1
(b) S2
(c) S3
(d) S4 Fig. 3. 1st column: 3D representation and 2nd column: DEM image of the 4 studied surfaces Si, i ∈ {1,…,4}. The x, y and z units are in mm, with 3D origin corresponding to the upper left pixel for (x, y) and to the mean of the DEM values for z axis. S1, S2 and S3 are laboratory-built surfaces whereas S4 is an actual agricultural field surface.
faster than the support surface altitude. Thus, the domination of the altitude variations at clod boundary allows getting good results even using initial method (Taconet et al., 2010) (despite the assumption of isoaltitude of the clod contour, assumption that is not checked in case of non stationarity of the support surface). In the cases of Fig. 4c and d, from the qualitative observation of the relative location of the clod contours provided by the two methods (in red and in yellow) relatively to the reference (in green), we note that the better performance of the proposed approach is due to a strong underestimation of some clod
contours by Taconet et al. (2010) method, whereas for other clods results obtained the two methods may be close. 3.3. Validation of the dynamic programming approach In this section, we aim at validating the proposed approach (dynamic programming based) for clod contour accurate estimation, either in absolute terms or in comparison with the cited background method (Chimi-Chiadjeu et al., 2013).
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(a) S1
(b) S2
(c) S3
(d) S4
Fig. 4. Results of the proposed watershed segmentation adaptation on the 4 studied surfaces: S1, S2, S3 and S4. The contours obtained using the proposed method are plotted in red, the reference contours are plotted in green and the contours obtained by Taconet et al. (2010) method are plotted in yellow. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
First we check the relative robustness of the proposed approach relatively to the clod contour initialization. Fig. 6a shows, in the case of S1, the results obtained using either the watershed segmentation based method or the Taconet et al. (2010) method, as initialization of the dynamic programming approach. Using these two initializations
(a) S3
(visible on Fig. 4a), the clod contour accurate estimation results are plotted respectively in magenta and in cyan (on Fig. 6a). We note that for this simple surface the results are robust to the initialization. For comparison, the results obtained after moving the clod contours using Chimi-Chiadjeu et al. (2013) method from the same two initialization
(b) S4
Fig. 5. Performance of the proposed watershed segmentation adaptation (in black) on the 2 surfaces S3 and S4 and comparison with Taconet et al. (2010) method (in gray).
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(a) Dynamic programming
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(b) (Chimi-Chiadjeu et al., 2013) method
Fig. 6. Robustness of clod contour accurate estimation on the surfaces S1 versus clod initialization, either using the proposed dynamic programming adaptation (left) or using Chimi-Chiadjeu et al. (2013) method (right). The color refers to the used initialization: either the watershed segmentation based method in magenta or Taconet et al. (2010) method in cyan. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
cases (and same surface S1) are presented on Fig. 6b. It clearly appears that the proposed method is more robust to the initialization than the previous method (Chimi-Chiadjeu et al., 2013). Note that another advantage of the proposed method is the reduced computation time (from few hours for Chimi-Chiadjeu et al. (2013)
method to few minutes with dynamic programming). It is due to the determinist feature of the new algorithm. For quantitative performance evaluation, let us focus on surfaces S3 and S4. The considered initialization is provided by Taconet et al. (2010) method rather than by the watershed segmentation based
Fig. 7. Results (1st row) and performance (2nd row) of the proposed dynamic programming method for clod contour accurate estimation in the case of actual surface S3 (1st column) and S4 (2nd column). Contour initialization is plotted in yellow, dynamic programming result is plotted in cyan, reference contour is plotted in green, and Chimi-Chiadjeu et al. (2013) result (computed on S4 for comparison) is in magenta. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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method. First the results obtained by watershed segmentation based method are already good. Second the clod contour accurate estimation is easier starting from an initialization ‘inside’ the clod rather than ‘outside’ the clod. Indeed since there are several clods on the surface, initializing a given clod contour outside of the actual clod, there is a risk to drift towards another clod. Fig. 7 shows the results obtained on S 3 (1st column) and S 4 (2nd column). Concerning S3, we note that only 5 clod contours have been moved in a noticeable way, namely the clods about centered in (25,20), (35,55), (50,55), (15,10), and (10,65). Indeed, these clods were among the most poorly estimated. This very timely contour displacement is confirmed by Fig. 7c showing the CDF improvement due to the dynamic programming based algorithm. In this section, CDF are computed only considering a subset of the clods for which initialization is available (‘false detections' due to the initialization are removed) so that the CDF curves start from the origin point (0,0). Concerning S4, we also note that the clod contour displacement is variable from one clod to the other and seems relevant. Comparing our results with those obtained in Chimi-Chiadjeu et al. (2013), we check that the proposed method provides better results in terms of CDF criterion (Fig. 7d). 4. Conclusion This paper proposed two new methods for the automatic estimation of clod contours. The definition of efficient image processing algorithms dealing with the soil surface automatic analysis is an important issue for both communities of soil science (e.g. agronomy, hydrology) and of remote sensing interpretation. The two proposed methods are complementary. The first one allows detecting the clods, so that the algorithm outputs are the number of clods, the rough estimation of their location and their contour. The second one allows refining the estimation of the clod contour, based on an initialization of these latter. Both methods are computationally efficient. The obtained results on laboratory-built surfaces as well as on the considered actual surface (agricultural field) are very promising. In particular, they outperformed previously published methods, and they are free from several constraints (e.g. with the proposed methods, contours are not necessarily isolines). The efficient obtained results in automatic clod delineation offer then the opportunity in deriving some inherent 2D parameters computed from the footprint (surface of the clod basis) and 3D parameters (volume, mean height). Then, this clod characterization will be realized on large dataset of several hundred of clods detected in field conditions (such as the freshly tilled seedbed from which surface S4 was extracted). Refinement of image processing for clod detection allows us to study more complex targets as on fields with furrows. Besides, considering a surface subject to successive rainfalls, we could analyze the evolution of clod contours and the merging of the clods or even their disappearing. Such studies will be done in collaboration with soil scientists or experts. In order to study larger scale, the link with radar backscattering coefficient will be investigated. From DEM image at millimeter resolution (such as those considered in this study) and their processing, some empirical statistical models will be developed to simulate realistic field surfaces. Then, the development and use of direct electromagnetic models will allow connecting the surface parameters to the backscattering coefficient measured at meter resolution. Acknowledgments The authors thank F. Darboux for his expertise and his help during the acquisition of the surfaces, and his comments on the paper. References Ambassa-Kiki, R., Lal, R., 1992. Surface clod size distribution as a factor influencing soil erosion under simulated rain. Soil Tillage Res. 22, 311–322.
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