Measuring the Three-Dimensional Morphology of Crystals Using

Article pubs.acs.org/crystal

Measuring the Three-Dimensional Morphology of Crystals Using Regular Reflection of Light Jayanta Chakraborty, Debasis Sarkar,* Abhishek Singh, and Aman Kumar Bharti Department of Chemical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India ABSTRACT: In this paper we present a new technique for measuring the three-dimensional (3D) morphology of faceted crystals. This new technique is based on the observation that a significant amount of regular reflection of light happens from flat crystal faces. Innovative lighting and camera arrangement is used to obtain the photograph of the crystal where a face is highlighted by huge contrast from the background. Such photographs are used to identify the location of the face corners using an in-house method as well as with the wellknown Harris corner detector. The in-house method is found to be of comparable efficiency as the Harris detector. Using a simple algorithm, the 3D face coordinates are generated from the two-dimensional face coordinates. Finally, the 3D coordinates of the corners are used to reconstruct the crystal. A consistency check is employed during the reconstruction step which corrects for any missing corner or inconsistent data.

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field of robotics.7 Quantifying the exact shape of precious gems like diamond,8 measuring the shape of cavities in intricate machine components, etc. are also important applications of 3D shape measurements. Although there is a large amount of published works on measurement of crystal size using a variety of methods, the literature on measurement of crystal morphology is relatively limited. In the area of crystal research, imaging and image analysis have been the most widely used methods for analyzing crystal morphology. Image analysis captures a two-dimensional (2D) image of the 3D crystal and determines various size and shape parameters from this 2D image. Usually the approximate shape of a crystal is measured using the silhouette of the crystal projected on a horizontal plane.9−11 This approximate method is sufficient to measure crystal shape if the crystals are flat. In such cases, various shape descriptors, for example, Ferret’s diameter, Martin’s diameter, circularity, convexity, elongation, etc. are used to characterize the shape.12,13 However, for a truly faceted crystal (e.g., CuSO4·5H2O) new techniques are needed. Faria et al.14 presented an automated image analysis procedure combined with discriminant factorial analysis to assess the morphology by a set of shape descriptors and then classify agglomerated sucrose crystals according to their shape classes. Li et al.15 presented the “camera model” for integrating crystal morphological modeling with shape measurement techniques by online microscopy. The technique uses morphological modeling to predict 3D shape and the “camera model” to obtain 2D projections of predicted 3D shapes. Then a measure of similarity between such 2D projections and the images obtained online is used for recognition of crystal polymorphic

INTRODUCTION Morphology or shape of a crystal is important for various reasons. For example, the shape of organic and pharmaceutical crystals grown from solution is known to have great influence on their end use properties and downstream processing such as filtration and drying. The crystal morphology may also distort the size obtained in size characterization studies.1 In industrial applications formation of elongated crystals often influences many key properties,2 the shape of the crystal affects dissolution of drugs and their bioavailability,3 and the final feel or appearance of powdered products also depends on the shape. Also, a sudden change in crystal habit may indicate the presence of new polymorph. Hence, understanding the evolution of crystal shape is an interesting and industrially important problem. Although in many practical situations knowledge about approximate shape is sufficient, in some value added products and problems of academic interest the exact shape of a crystal is important.4 For example, during synthesis of catalysts, the total area of crystal faces in a certain crystallographic direction is an important consideration.5 Another familiar example is the modeling of the crystallization process. In the absence of a measurement technique, the crystal is treated as a spherical object of equivalent mass.6 This obscures the fundamental relation between the face growth rates and the rate of overall mass gained by the crystal. If we can measure the exact threedimensional (3D) shape of a crystal by taking samples at intervals, we can measure the growth rate of each individual crystal face, and this should lead to more effective product quality control. Thus, by measuring the true morphology, much more understanding of crystallization process could be achieved. Apart from the field of crystal engineering, measurement of exact 3D shape of a faceted object also finds application in the © XXXX American Chemical Society

Received: August 10, 2012 Revised: November 11, 2012

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dx.doi.org/10.1021/cg301155x | Cryst. Growth Des. XXXX, XXX, XXX−XXX

Crystal Growth & Design

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forms. Often, indirect methods are employed to estimate the shape of a powder. Various shape-dependent properties of powdered solid are measured, and an estimate is obtained from such measurements. For example, flow or magnetic properties often depend on shape and those can be measured16 to get an estimate of the particle shape. However, actual shape could not be measured using such techniques. There are a few techniques available in the literature that are capable of measuring exact shape of an object. For example, fringe contrast analysis is routinely used to obtain the depth profile of a surface.17,18 Obtaining multiple silhouettes and fitting those to a synthetic 3D shape is also a popular method.8 Coherent X-ray diffraction,19 stereoscopic imaging,20 and laser defractometry21 have also been used to obtain the 3D shape. However, most of the time these are specially developed for applications other than crystal engineering and need to be modified for current purposes. Hundal et al.22 presented an image analysis method which describes the shape of a convex or concave particle. The authors used Fourier descriptor and a neural network to classify particles according to their shape. Recently, Singh et al.23 used confocal microscopy to extract the exact 3D shape of a crystal. They also performed another remarkable task in addition to measurement of 3D shape. They matched the angular pattern between facets to reveal the miller indices of crystal faces which are otherwise possible only with very sophisticated instruments. However, all available techniques applicable to measure the exact shape of a crystal including that by Singh et al.23 involve costly instrumentation. In this paper, we present a simple and inexpensive technique to measure the exact 3D shape of a crystal from 2D images of its faces, captured using visible light and an ordinary camera. This technique is based on the observation that crystals have plane faces from which regular reflection of light occurs. In the next section we describe the method in detail and then demonstrate it with an example.

Figure 1. Image of a typical copper sulfate pentahydrate crystal produced through seeded growth. Multiple light sources are used to highlight more than one face.

optical zoom and effective 7.2 megapixels resolution is used for capturing images. An ordinary table lamp with a 100 W bulb is used as the light source. Capturing of Images. If parallel rays of light are shone on a crystal, the majority of the light will reflect in a regular fashion as shown in Figure 2. It can be seen that the reflected light is

Figure 2. The pattern of regular reflection of light from a crystal. The positions D1, D2 and D3 are camera positions where the camera receives enough light.

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METHODOLOGY The basic idea of this method is based on the observation that regular reflection of light happens from the flat faces of a crystal. Hence by maintaining proper lighting, we can take images of a crystal in such a way so that only one face of a crystal reflects light; i.e., only one face shines while the others remain dark. Such an image is easy to segment and the geometry of the face can be extracted readily by image analysis. By capturing images in the specified manner, we can thus obtain the geometry of all the individual faces. Because the angles between crystal faces are known constants, dimensions of the face planes can be used to reconstruct the crystal. Hence, the proposed method has three vital components: image capturing, image analysis, and 3D reconstruction. These will be discussed in turn. Materials and Methods. Copper sulfate pentahydrate (S. D. Fine Chemicals, India) was chosen for demonstration purposes because of nice faceted characteristics of its crystals. Such crystals of CuSO4·5H2O were obtained by seeded growth in our laboratory. Water from Milipore system was used in all experiments. For preparation of crystals, first a saturated solution of CuSO4·5H2O is prepared at 10 °C and then a seed crystal of ∼1 mm size (but of irregular shape) is added to this saturated solution. This solution is then cooled to 8 °C by keeping inside a refrigerator. Nice faceted crystal of a couple of millimeter size is formed after four days. A picture of such a crystal is shown in Figure 1. A Sony Cyber-shot camera with 3×

available only at certain directions, and if the camera is held at those directions, it will receive enough light for capturing an image. In other locations it will not receive any light provided the following criteria are met: (i) only regular reflection happens from the faces, (ii) the crystal is not transparent, and (iii) stray lights are blocked. At a given location it will receive light from only one face provided the faces have a reasonable angle with other faces, which is usually the case. While the foregoing conditions are met only at an idealized situation, this technique works very well for practical situations. For example the source of light need not to be parallel and the crystal may be transparent. A certain amount of irregular reflection also happens in all cases. However, the regular reflection from a face always dominates and an ordinary camera with automated settings always adjusts its optics to pick up the shining face. This has been shown in Figure 3. It can be seen that although all the faces including the background reflects a good amount of light, the regular reflection from a single face produces a huge contrast with the background. In fact, it is very close to white. In general, for a reasonably good quality crystal this technique produces images in which a face is highlighted. It can be seen that the edges are very clear and the image can be readily segmented to obtain geometry of a face. It can also be seen that even a very small face can be extracted using this technique. B



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reflects too much light from the background making image segmentation difficult. Image Processing. The main tasks during the image processing stage are to extract the dimensions of each face and detect the coordinates of corner points. Figure 5 presents an

Figure 5. Image analysis flow diagram. Figure 3. Images captured with a simple digital camera and ordinary light source. The shining faces are almost white in all cases and the corners are very sharp. This figure also shows that a variety of background surfaces may be used without affecting the contrast significantly.

image analysis flow diagram showing an overview of the steps in processing of each image. The major steps involved in processing of each image are preprocessing, segmentation, and feature extraction. The images obtained by our proposed method of image capturing are of reasonably good quality with highlighted face and very clear edge. Thus, the images can be readily processed using standard image analysis techniques. The raw color image is first converted to grayscale image by calculating the monochrome luminescence from the three color channels. The quality of the image is then further enhanced by increasing the contrast of bright portion by modifying the gamma values of the individual pixels. In the next step, the preprocessed grayscale image is segmented to extract the highlighted face by applying a global threshold, where the elements below and above the threshold values are set to zeros and ones respectively. Segmentation is a critical step in image analysis, and selecting an appropriate threshold value is often a critical issue. However, due to the novel image capturing technique used in this work the selection of threshold value is straightforward here. A high value close to 1 (0.9 used here) works well for all the images analyzed. A global threshold value calculated using Matlab (The MathWorks, Inc.) function graythresh that uses Otsu’s algorithm, also works equally well. Then a morphological operation is performed to remove small objects from the binary image. This is done conveniently by Matlab function bwareaopen by removing all connected objects that have fewer than some specified pixels. Figure 6 shows the results of segmentation of a typical image, and it can be seen that the segmented image is very clear. Next step is to obtain the boundary of the crystal. If the image is very sharp (for example, the left and the top edge of Figure 6), it will produce boundary pixels of an edge

At this point, the capturing of image is manual; i.e., we move the camera in various locations (while keeping the crystal fixed) to capture the images of various faces. The occurrence of a face is interpreted by human intervention. However, with proper instrumentation, this mechanism can be automated. For a crystal lying on a plane, the camera can move around the crystal and perform a search for faces. If it does not encounter a face, it will not receive enough light (absence of a bright spot which can be recognized by standard image processing). As soon as it receives enough light from a bright spot and much less from other locations, signifying a shining face, the camera will pick it up. One point of convenience can be noted at this point: if the camera makes an angle with the face while taking images, the view of the crystal face thus obtained is the perspective view and it is not straightforward to extract the true face dimensions from this view. To make the process simpler, the face search process is modified as follows: the light source is mounted on the camera as shown in Figure 4. Hence, while the camera

Figure 4. Combined arrangement of light and camera for searching of a face.

searches for a face, if a face is shining, it must be parallel (or very close to parallel) to the focal plane of the camera. In that case all dimensions can be converted into real dimension by using a simple image scale factor. It can be noted that even though the lighting system can be mounted like a standard camera flash, use of a normal camera flash is not advisable. In our experience, a camera flash is too strong for the purpose and

Figure 6. Segmentation of a typical image of a highlighted crystal face. C



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Figure 7. The contour of the black and white image before (a) and after (b) fitting convex hull. The quality of edge improves after fitting convex hull.

approximately in a line and we can directly proceed to extracting the contour of the white region. However, in many cases, the image may not be very sharp at places and fitting a convex hull to the boundary pixels turns out to be beneficial. The convex hull of the binary image can be easily generated using Matlab function bwconvhull and then the boundary pixels of the binary convex hull image can be readily identified using Matlab function imcontour. The contour of the segmented image before and after fitting convex hull is shown in Figure 7. It can be readily seen that many small uneven areas could be removed by using convex hull making the next part of the algorithm more robust. Detection of Corners. Corners of an image are important 2D features and accurate detection of the coordinates of the corner points of the binary convex hull image is essential for our intended application. Here we present an innovative technique developed by us from scratch for detection of corners. In order to verify the efficiency of our proposed method, we also implement a more established technique such as Harris corner detector24 and compare the results of corner detection by the two methods. If we consider the lines made by a group of subsequent boundary pixels, it will form a consistent variation of slope. The number of boundary pixels taken to form a line should be similar to the number of pixels that forms a corner bend for obvious reasons. The slope of such fitted lines will remain approximately constant as long as we traverse through an edge. This slope will go through a change at a bend and again settle to slope corresponding to the next edge soon after it crossed the bend region. Variation of slope for the image contour shown in Figure 7b is shown in Figure 8. The slope matrix is sorted before plotting such that the lines with the same or very similar slopes are clubbed together which makes the slope identification robust. This becomes a necessary step because in most cases, a crystal face has parallel sides. It can be readily seen that two most prominent steps present corresponds to the sides AB and BC respectively (with slope ∼0.3273 and 4.0 respectively) along with a small step corresponding to side DE. The edges CD and AE have the same slopes as AB and BC, and hence only one slope is detected for one pair of parallel lines. These two flat regions are separated from the slope pool by using a threshold on the first derivative of slope (named rise) and a parameter corresponding to the ‘length’ of the step (named tread). If a preset number of slopes do not differ by a

Figure 8. Variation of slope for line segments drawn with 40 subsequent contour pixels from Figure 7. It shows a rapid change in slope near a bend and a prominent step near an edge.

given tolerance, they are considered as slope corresponding to a side. If only a few subsequent lines are of equal slope they correspond to very small edge and are not considered. The parameters tread and rise used in this method are dependent upon the nature of image and the order of the line lengths. Once the prominent modes of the slopes are found, the intercepts of the lines that are close to the modes (within a specified tolerance) are obtained. It may be noted that there could be (and usually are) multiple lines corresponding to a given slope. To obtain all possible lines, first we isolate the lines that have similar slopes and then separate the intercept modes. The number of segments that fall within the tolerance range of a slope mode is an important quantity and is also calculated. This quantity is named as ‘line prevalence’. The average of the intercepts is found which completes the detection of edges and hence corners. The result of the whole process is shown in Figure 9. The extracted edges along with the original image and the contour are shown in Figure 9. In Figure 9a, lines of all possible prevalence are plotted which includes one less prevalent set of edges. These two could be readily eliminated if we consider only the top four prevalent edges. It can be seen that the corners are detected quite accurately in these cases. This method works well with a variety of images. Even parameter adjustment is not required if a certain kind of face is treated. This generality seems to stem from the linear edges, and this is the only criterion that should be met by the contour. Once all the corners are detected, they are numbered in sequence starting from the top left corner. D



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The Miller indices of faces can be deduced after the morphology is determined by matching the angular pattern as discussed by Singh et al.23 in a recent communication. Now, the next face is to be attached to this face. This attachment is carried out by knowing the edge through which it is attached and its 2D geometry. The angular orientation of this new face is also needed. The connecting edge is identified from the information about the movement of the camera during the image capture as shown in Figure 11. For example, if the Figure 9. (a, b) Detected edges and corners for two different crystals using the slope mode method. Small sides can also be detected if lines of low prevalence are also considered.

Harris corner detector24 is one of the most widely used methods for corner detection, and the algorithm relies on the principle that at a corner point, the image intensity will change greatly in multiple directions. Harris corner detector computes a cornerness measure for each pixel in the image and a cornerness map is thus obtained. The local maxima of this map indicate corners, and these are extracted by nonmaximal suppression and appropriate thresholding of this cornerness map. In order to verify the efficiency of our proposed corner detection technique, we implement the Harris detector using Gaussian function as weighting function and the results are shown in Figure 10 where we mark the corners detected by Harris

Figure 11. Schematic showing the sequence of imaging and the orientation of camera during image capturing. This information is used during reconstruction.

camera moves downward along the xz plane to capture image of face-2 after capturing face-1, the bottom edge of face-1 and top edge of face- 2 will be the same line. Similarly, the positions P1 and P4 are so related that the left edge of face-1 and top edge of face-4 are the same line. We also know the angular orientation of each face from the movement of the camera. It is clear that once the 3D coordinates of the attaching edge, angular orientation of the face and the 2D geometry of a face are specified the face is completely described. Hence, using this information we can obtain the 3D location of the face vertices under question. It can be noted that by specifying the attaching edge (common edge between two faces), we specify two of the coordinates at the beginning. Now, from the angular orientation of the face and using one of the common edge vertices, the equation of the plane can be obtained (eq 1). Next we have to determine the location of other vertices which lie on this plane, at a set distance from the edge vertices ((x1, y1, z1) and (x2, y2, z2)). Hence the distances of any unknown vertex (x, y, z) from the two edge points in 3D can be equated with those distances obtained from 2D face images (d1, d2). These two equations along with equation of the plane (eq 1) can be solved to get the 3D coordinate of the vertex. All other vertices are found in a similar way.

Figure 10. Captured images of four faces of a crystal. Both the corner detection techniques are displayed here. The Harris corners are shown as filled circles.

⎫ ⎪ ⎪ (x − x1)2 + (y − y1)2 + (z − z1)2 = d12 ⎬ ⎪ (x − x 2)2 + (y − y2 )2 + (z − z 2)2 = d 22 ⎪ ⎭ ax + by + cz = d

detector by filled circles. Figure 10 also shows the corners detected by the new method proposed here. It may be noted all the true corners of the crystal faces are identified correctly by both the methods and no false corner is detected. Thus the performance of our corner detection technique is comparable to the performance of the popular Harris detector. Reconstruction. The reconstruction starts by assigning the 3D coordinates to the corner points of the largest face. The 2D coordinates of the vertices of this face are known from the corner detection algorithm discussed previously, and they are converted into 3D coordinate by assigning the same (but arbitrary) z coordinate to all the vertices. This makes the top face horizontal without any reference to crystallographic axes.

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Once the 3D coordinates of all corners of the crystal are known, the crystal is completely specified and any equivalent description (e.g., in terms of h vector) could be obtained.25 The 3D view of the crystal can also be generated readily by using any standard 3D plotting software. The 3D reconstruction of the crystal shown in Figure 10 is shown in Figure 12. It can be noted that the reconstructed crystal is an exact replica of the original crystal within the experimental accuracy. It can also be noted that not all the faces E



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Figure 12. The original and the reconstructed crystal. The figure shows exact resemblance between the two.

Figure 13. (a) Schematic illustrating the consistency checks. (b) The reconstructed crystal using single edge attachment with one face having a fictitious corner.

of the crystal are observable by this technique. The proposed method can measure the shape of any arbitrary crystal if all the faces are imaged. However, if symmetry is present, the information can be used to advantage and all the faces need not be imaged. Advanced Features of Reconstruction Algorithm: Check for Consistency. The reconstruction algorithm, as described above, is sufficient if all the corners are detected with reasonable accuracy. If the corners are not determined so accurately, the reconstruction algorithm should also check for consistency in addition to the usual task of finding the 3D coordinates. For example, it is possible that one edge of a face (e.g., edge 61 of face 4 in Figure 13a) is not detected if certain corners (1 and 6 in this case) are not detected properly. But the edge exists and hence the other face corresponding to the missing edge (face 5), which will be visible from another perspective, could not be attached to face 4. Such a situation can be handled using multiple edges for attachment rather than a single edge. If the corners are detected exactly, the length of edge 12 will be the same in face 1 and face 4 and a scale factor of about 1 can be used to match the length of the attaching edge to exactness. In this case, all other edge lengths will also show a close match with the corresponding edges of the neighboring faces. However, if the corners are not detected properly, scaling with one edge will lead to a nonunity scale factor and large mismatch in length for other edges. For example, if we could not detect corners 1 and 6 and instead detected point 1′ as a (fictitious) corner, single edge attachment

would lead to a reconstructed crystal as shown in Figure 13b which is clearly inaccurate. Since the attachment using edge 12 fails, as evident from the large mismatch in edge length, any one of the corners corresponding to this edge must be faulty. Next this attachment is tried with edge 23. It may be noted that the information regarding the angular orientation of faces are available from the record of camera movement. For this case all the edges except those involving corner point 1′ match with the attached faces and hence corner 1′ is a fictitious corner. This is also evident otherwise because it is away from the object contour. It is also evident from this check that we have one missing edge for this face. Hence we need to have a pair of true corners instead of the wrong one. This new pair will form the missing edge. The first thing to try is to go down one line in prevalence and include one more line to see if that is the missing line by reiterating the consistency check step. For example, in the current case, the missing edge is the smallest one and hence of least prevalence. While the other edges have a prevalence >10, it has a prevalence of 7. Hence, this strategy would work in this case and revisiting the edge detection step would solve the problem. If this fails, the following algorithm could be used. The missing points can be found by trimming the two edges which are not matching the neighbors. For example, point 1 is placed on the edge 1′2 of face 4 such that the length 12 equals length of that edge in face 1. Point 6 could be found in a similar way provided we have an image of face 5 or face 6. The new points thus found are now considered as true corners. The F



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known orientation is not so straightforward, and we are exploring use of electric field to accomplish this task with precision for small particles. For industrial powders there are many additional challenges that should be met, for example, rough faces, for which the image processing should be more robust. However, this technique has the potential to extract an enormous amount of information with a cheap attachment to an ordinary microscope. Even a limited amount of information, e.g., area of a given type of face (more active in terms of catalytic or dissolution property), could be of enormous practical value.

reconstructed crystal is now identical to the original crystal as shown previously in Figure 12 and has not been shown again. Adaptation of the Proposed Technique for Practical Use. Although the technique discussed here is demonstrated for a single large size (∼0.5 cm) crystal, which is moved manually, it can be applied for many practical purposes with suitable modifications. In this section we will discuss some of those modifications briefly, details of which with demonstration will follow in a separate communication. The first issue we address is the applicability of the technique to small particles of ∼100 μm size. Figure 14 shows an image of

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CONCLUSIONS In this work, we have demonstrated a simple, inexpensive yet effective approach to construct 3D crystal shape from 2D images of its faces using an ordinary digital camera and lighting. We have shown that the face geometry of a crystal with sharp features can be extracted readily by using our new image capturing technique based on regular reflection of light from a face. The key information regarding a face is the location of its corners. The corner coordinates are determined using two techniques: one is the well established Harris detector and the other is a new method developed by us. Both the methods yield good results. The 2D geometry of the faces is used in an innovative way to reconstruct the crystal. The proposed method does not assume any shape of the crystal a priori and hence can be used to measure any arbitrary shape if all faces can be imaged. However, if symmetry is present, the information can be used to advantage and all the faces need not be imaged. The reconstructed crystal is found to be the exact replica of the original crystal.

Figure 14. An image of micrometer size crystals captured under microscope with reflectance mode (Leica DM 2500) without any special arrangement or setting. This image clearly shows that face highlight is possible for small crystals.

such a small particle taken by a Leica microscope camera using reflectance mode. Here again the face that is parallel to the focal plane of camera (i.e., horizontal plane) only shines. It can be readily seen that the shining face makes very good contrast with the rest of the crystal. However, as the crystal is placed on an ordinary glass slide, the background reflects light with equal intensity. So, special treatment of the background (e.g., coating with carbon black) is required in this case. The second limitation of the technique is that it is demonstrated for a single isolated crystal. So, before it can be applied for powders or slurry, we need to have a sound strategy to isolate each crystal for observation. Currently we are exploring the following methods: (i) mounting individual crystal on a Teflon coated wire using natural stickiness of crystals due to electric charge on crystals, and (ii) filtering a dilute slurry on a small drum filter to form an array of well separated crystals. For powders, a small drum (diameter ∼ 500 μm) rolls on the powder and picks up powders on the drum. This wire/drum then rotates under the observation field of the microscope to expose various faces of the crystal. Third and the most important issue is the presence or absence of symmetry of the crystal with respect to a horizontal plane: in the demonstration, we assume that the crystal is symmetric such that if we can image four faces of the crystal, it is completely described. However, in general the crystal may not pose such symmetry. Even in the demonstration, the symmetrically opposite face of the smallest face was absent on the crystal, although this face was determined automatically as it was surrounded by other faces. For such cases, we need to picture all distinct faces of a crystal. This calls for precise movement of the crystal, e.g., rotating in three independent axes and flipping it over. Rotation is easily achieved mechanically. Flipping over a particle with

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AUTHOR INFORMATION

Corresponding Author

*Tel: +91-3222-283920. E-mail: [email protected]. Fax: +91-3222-282250. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the anonymous reviewers for giving some useful insights that significantly improved the manuscript. REFERENCES

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Measuring the Three-Dimensional Morphology of Crystals Using

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