a. Basics of the algorithm

The present algorithm classifies the particle habits based on an analysis of dimensionless ratios of simple geometrical measures such as the x and y dimensions, perimeter, and image area. The key to the suggested method lies in considering these ratios for an ensemble of images. It is assumed that each habit category has a unique distribution (or, basis distribution) for a given ratio. For an ensemble of images constructed from a mixture of different habits, the distribution of a particular ratio will be a linear superposition of basis distributions of individual habits. If the basis distributions are known, then the fraction of each habit for an ensemble of 2D images can be found by solving the inverse problem.

In the present analysis, the following measurable parameters of 2D images were used (Fig. 2): N_x, the number of pixels in the x direction (perpendicular to the photodiode array, i.e., along the flight direction); N_y, the number of pixels in the y direction (parallel to the photodiode array); N_total, the total number of shadowed pixels; N_edge, the total number of triggered edge pixels; N_edge1, N_edge2, the number of triggered edge pixels on each side of the particle image; and N_P, the number of perimeter pixels as defined in section 4c(3). The measurable parameters can be converted into their corresponding geometrical parameters as follows: D_x = δN_x the x length, D_y = δN_y the y length, S = δ²N_total the particle shadow image area, and P = δN_P the particle image perimeter. Here δ is the pixel resolution of the OAP-2DC. It is also assumed that the resolution in x and y directions are the same. If this assumption is not valid, an appropriate correction should be introduced in the above equations.

b. Image groups

The classification begins with a break down of 2D images into four groups: “tiny,” “rejected partials,” “complete,” and “accepted partials.”

1. The tiny images are defined as images that trigger less than N^*_total pixels. In the frame of this work the threshold value was set N^*_total = 20 pixels. The images with number of pixels less than N^*_total provide a limited habit resolution and they were removed from analysis. Examples of tiny images are shown in Fig. 3a. Recalling that OAP-2DC does not record the first image slice, the total particle image would contain about 23–25 pixels. Such images correspond somewhat to five pixels or 125 μm in diameter for particles having a quasi-spherical shape. If a particle image has a noncircular shape, the linear size of the image for an equivalent area particle will be larger than that for a circular one. Thus, the particle size D = 125 μm can be considered as a threshold size for the image recognition analysis in the present study. In general case the threshold N^*_total = 20 pixels can be replaced by any larger value. However, N^*_total = 20 pixels should be considered as the lower limit used for habit recognition (section 5a).

2. The group of rejected partials consists of particles where the largest part of image is assumed to be out of the image frame. Such images were defined as N_total ⩽ 180, and N_edge ≥ 4 and N_y < 32. Preliminary analysis showed that the recognition of such images gives poor results. Figure 3b shows an example of rejected partials. Images containing empty slices inside the shadowed zone along the y axes (flight direction) were also rejected (Fig. 3c). Such images usually occur in the case of fogging and in some special cases probe malfunctions. There are a number of other known OAP-2D artifacts caused by electromagnetic noise, shedding, etc. Eliminating these artifacts is a matter of introducing special image filters, which are not discussed in this study.

3. The third group, complete images, includes the images with the number of triggered pixels N_total > N^*_total with the total number of edge pixels N_edge < 4. Examples of complete images are shown in each first two strips in Figs. 1a–c.

4. The accepted partial images were defined as 2D images with N_total > 180 pixels and N_edge ≥ 4 or N_y = 32, which is maximum number of pixels in the y direction. The criterion N_total > 180 was found empirically, and can be changed either side. However, the partial images with lower N_total yield less habit recognition. Examples of accepted partial images are shown in Fig. 1d and in each bottom two strips in Figs. 1a–c.

The main ensemble has now been sorted into four groups. Of these four, a retrieval will be performed on the complete and accepted partial groups. Further for brevity accepted partial images will be referred to as partial images.

c. Basis distributions

The next step of the retrieval is the calculation of the distribution of geometrical ratios for the complete and partial images. Fifteen candidates to ratios to be used in the recognition were considered for both complete and partial images: D_xD_y/S; (D²_x + D²_y)/S; [max(D²_x, D²_y]/S; D_xD_y/P²; (D²_x + D²_y)/P²; [max(D_x, D_y)]/P; D_x/D_y;[min(D_x, D_y)]/[max(D_x, D_y)]; P_circ/P; S_circ/S; S_circ/P²; P²_circ/S; P²/S; [P(D_x + D_y)]/S; and (P_circS_circ)/PS. Here, S_circ and P_circ are expected image area and perimeter, respectively, in assumption that the images are circles [section 4c(2)]. Basis distributions and retrieval matrices (as described in section 4d) were constructed for each of these ratios and tested on special samples of images of particle measured in natural clouds. The ratios that provide the best habit recognition were selected for the analysis. The rest of the ratios were not used.

1) Complete images

For complete images, the ratio

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was found to have the most unique features to characterize each category of habits. Perfect spheres give circular images, R₁ = 1. Due to imperfections of images caused by discretization (Korolev et al. 1998) or the probe electronics, the value of R₁ for real spherical particles may be different from 1. Therefore, the images from an ensemble of spherical particles return a narrow frequency distribution of R₁ centered around R₁ = 1. Needles and columns are implicitly defined as rectangles having a length-to-width ratio (c/a) of between 3 to 8. Therefore, for needles the ratio R₁ ≈ πc/8a ranges from 1 to 3. Due to their unique shape, dendrites and irregular particles will also return a unique distribution for the ratio R₁.

Figure 4a shows three basis distributions of the ratio R₁ for complete spheres, irregulars, and needles for a training set of images like in Fig. 1 measured by OAP-2DC and classified by eyeball. Notice that although all distributions overlap, each particle category leaves a unique distribution signature. Due to rare occurrence of complete dendrites the basis distribution for dendrites is absent (see section 5c).

2) Partial images

The partial images were classified in a different way. The classification begins by assuming that the particle image is a perfect circle. Knowing the number of triggered edge pixels on the image N_edge1 and N_edge2, the perimeter of the circle can be found as

P_circθ₂θ₁r.(2)

Here,

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x₁ = δN_edge1, x₂ = δN_edge2, and y = δN_y. Figure 5 provides explanation for calculation of the perimeter of a partially imaged circle. Note that Eq. (2) is true for all possible positions of a circle in the viewfield of the probe, that is, when x₁ ≠ 0 and x₂ ≠ 0 (case in Fig. 5);x₁ = 0 and x₂ ≠ 0; or x₁ ≠ 0 and x₂ = 0; or x₁ = 0; and x₂ = 0 (complete circle).

The assumed circle image perimeter P_circ is then compared with the measured one P, that is,

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Figure 6a shows four basis distributions of the ratio R₂ for partial spheres, irregulars, needles, and dendrites for a training set of images measured by OAP-2DC in natural clouds.

d. Construction of a retrieval matrix

Let M(R) be the measured distribution of a specific ratio R for an ensemble of measured images such that M(R)dR is the fraction of particles whose ratio lies between R and R + dR. The distribution M(R) can be represented as a linear superposition of basis distributions B_i(R), as

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where f_i is the fraction of particles in the ith habit subject to the normalization constraint Σ_i=1 f_i = 1. Note that M(R) and B_i(R) are normalized to unity. The measured distribution M(R) may differ from the actual distribution due to the Poisson statistics of images in a sample. The shapes of particles in some cloud zones may also not perfectly match those used in the training sample. Therefore, the local basis distributions may be different from that used in the algorithm B_i(R). Due these statistical uncertainties the solution of Eq. (5) may become ill posed for the cases when the distributions M(R) and B_i(R) are continuous or when they are broken into a number of intervals larger than the number habit categories. One way to obtain a unique solution to Eq. (5) for f_i is to discretize the domain of the distributions into a number of intervals [R_j, R_j+1) equal to the number of image categories, thus generating the matrix B with elements

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and the vector M with components

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The boundaries of the intervals [R_j, R_j+1) were choosen by performing an optimization of B that minimized its condition number. The condition number is the ratio of the largest singular value of B to the smallest (Dongarra et al. 1979). The conditioning number measures the sensitivity of the solution of a system of linear equations to errors in the data. Large condition numbers indicate a nearly singular matrix. Figures 4b and 6b shows the result of discretizing of the distributions B_i(R) for complete and partial images. The numerical values of the elements of the matrices B and intervals [R_j, R_j+1) found as a result of the above optimization are shown in appendix A.

Here, B⁻¹ is the inverse of B and f is a 4 × 1 vector of fractions of images belonging to different habit categories. For example, if f = (0.5, 0.2, 0.05, 0.25), then the percentage composition of the sample would be 50% for spheres, 20% for irregulars, 5% for needles, and 25% for dendrites.

The final step would be to sum the vectors f_c and f_p for complete and partial images, respectively. This is done by weighting each vector with respect to the number of complete (N_c) and partial (N_p) particles it represents:

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In order for this procedure to be effective, the size of the 2D image sample should be statistically significant. If only a small number of images (less than 100) are used to calculate M, then the errors, related to the Poisson statistics in M, may be 20%, if the fraction of habits are evenly distributed between four categories. The subsequent retrieval procedure may only increase these errors in the final result f. A sample size of five buffers (32 × 1024 pixels) usually represents 200 to 400 particles acceptable for image analysis. Therefore, five buffers is large enough to accurately describe the fractional composition of particle shapes, while at the same time being sufficiently small so as to suitably display the spatial variations of particle habits in a cloud.

e. Some results of image processing

The algorithm was tested on a large dataset collected during four aircraft projects in different weather conditions and climatic zones: the Beaufort Arctic Storm Experiment (Canadian Arctic, September–October 1994); the Canadian Freezing Drizzle Experiment I (Newfoundland, March 1995); the Canadian Freezing Drizzle Experiment 3 (Great Lakes, December 1997–February 1998); and FIRE.ACE (Canadian and U.S. Arctic, April 1998). The results of recognition analysis and habit statistics were summarized in a study by Korolev et al. (2000, hereafter KIH).

Figure 8 shows an example of habit recognition from flight on 23 January 1998 during the Canadian Freezing Drizzle Experiment 3. Figures 8b–e show the fraction of irregulars, dendrites, needles, and spheres, respectively, derived from OAP-2DC data with help of the developed algorithm. It is seen that dendrites and needles occur in localized zones or cells, where the fraction of needles and dendrites may reach up to 90%–100%. These cells are usually surrounded by regions with irregular shaped particles. Figure 9 shows the OAP-2DC images measured in zones with high percentage of (a) spheres, (b) irregulars, (c) dendrites, and (d) needles. A comparison of results of habit recognition by the present algorithm with those obtained by a human classifier shows that on average the error does not exceed 10%–15%.

Quantifying accuracy in any classification scheme for binary images of hydrometeors is rather subjective. Ice particles often come in hybrid forms whose shapes fall in between possible habits. Low resolution and large variety of shapes of cloud particles contribute heavily to uncertainty in classification. Even human classifiers may not agree on the classification of a specific particle. Ice particles are not some absolute object that necessarily fall into a preconceived category. Therefore, there will always be some degree of uncertainty in any classification scheme.

a. Threshold number of pixels for habit analysis

Different habit categories require some minimum number of pixels to be properly classified. Some authors claim to be able to recognize hexagonal plates starting from images having three to five pixels in size (Duroure 1982; Holroyd 1987; Garbrick et al. 1995). However, one may only be able to resolve a hexagonal structure with images having at least 12 to 14 pixels in “diameter.” Plates with smaller sizes would fall into another category other than “plate.” Figure 10 shows two discrete binary images generated from two hexagonal plates having the same orientation but different sizes: 8 pixels (Fig. 10a) and 14 pixels (Fig. 10b) in diameter. It is clearly seen that the discrete image for the smaller plate (Fig. 10a) does not have visual features of hexagonal plate and it will be classified as irregular or sphere. Therefore, the fraction of plates calculated for an ensemble of particles would be a priori biased. For this reason the category of plates was not included into our analysis. Observations of ice particles at the ground level conducted by Zamorsky (1955) showed that the frequency of occurrence of plates is 2.8%. KIH estimated the fraction of plates in polar and midlatitude clouds as 0.4%. This is consistent with a simple visual analysis of OAP-2DC data: on average plates occur rarely in OAP-2DC imagery. Therefore the exclusion of hexagonal plate category from the recognition analysis presumably would not affect the particle habit statistics.

For classification of needles, depending on the a/c ratio, an image having three to four pixels all in a line would be enough for proper classification. Therefore, the images of needles with 20 pixels in total would more than adequately describe the needle/column habit. An image having 8–10 pixels in linear size and about 20–25 pixels in total may adequately represent the stellar habit. The operational definition of a spheres is not trivial. On one hand, it depends on uncertainty related to discretization of the image (Korolev et al. 1998). On the other hand, it depends on how accurate we want to describe a deviation of the image from circular in a recognition analysis. The irregularities on the surface of the sphere should be about pixel size to generate a deviation of the image from circular, and thus generate a reason to put it in the “irregular” category. In this study we deliberately decreased the threshold N^*_total down to 20 pixels in order to keep the number of accepted images large. KIH analyzed about 3 × 10⁷ particle images measured by OAP-2DC. Approximately 73% of the images were rejected as tiny with N_total < 20 pixels. A decrease of N^*_total may dramatically decrease the number of accepted particles that may make the recognition analysis meaningless. Using the threshold N^*_total = 20 allows RABIT to distinguish needles and dendrites, though it leaves some uncertainties in the separation of irregulars and spheres for small images (20 < N_total < 60). From a retrieval matrix B for complete images (appendix A) the value of this uncertainty may be estimated as 20%–25%. For larger images this uncertainty decreases to few percent or less. So, the threshold N^*_total = 20 gives reasonably accurate results for recognition analysis of the above four habit categories.

b. The effect of particle orientation on habit statistics

Image orientation can be a severe impediment to image classification. The illuminating laser beam of the OAP-2DC installed on the NRC Convair-580 was purposely oriented vertically to combat orientational effects. Since an ice particle with the Reynolds number Re > 10 (List and Schemenauer 1971) falls with its largest cross section plane oriented horizontally, it is expected that the OAP-2DC measures the largest projection of the particle, that is, when the particle image exhibits its best orientation for habit classification. From laboratory studies (List and Schemenauer 1971; Hallett et al. 2000), columns fall with their c axis being horizontal, while for plates and dendrites the c axis is vertical. As mentioned above, the planar ice particles, like plates and dendrites, may give a rectangular projection, which may be recognized as a needle or a column. In the case of aircraft measurements this may occur if the ice particle changes its natural orientation due to wind shear caused by the aircraft and the local flow around the instrument. If this were the case, we would observe on a regular basis in zones of dendrites a constant fraction of images classified as needles due to reorientation of dendrites. Similarly in the zones of needles, a certain fraction of circular would be observed all the time. However, such a coexistence of shapes is not observed (e.g., see Figs. 8c,d), and the effect of reorientation of planar and columnar ice particles on the habit statistics is estimated as no larger than a few percent.

c. Observation of “complete” dendrites

The growth of plates along the a axis is changed to dendritic growth when the terminal fall velocity exceeds a certain threshold (Re > 0.5) (Alena et al. 1990). Since the terminal fall velocity is related to a plate size, there is a threshold plate size for dendritic growth. For typical atmosphere conditions it is about 200 μm. A particle starts to exhibit stellar or dendritic features only when it reaches some 400–600 μm in size. This explains why in OAP imagery the frequency of occurrence of complete dendrites is significantly less than that of partial dendrites. For this reason we were unable to obtain statistically significant training set of complete images of dendrites, and, thus they were not included in the retrieval matrix B.

d. Dependence of basis distribution on particle size

The dependence of particle habits on size is discussed in detail in the recent study by KIH. In this work the particle habit were classified for three size groups: D > 125, D > 250, and D > 500 μm. This classification was done using three different retrieval matrixes derived for each size range. The comparison of basis distributions showed that the differences between habit bins do not exceed 5%. This allows one to make a conclusion about the generality of the retrieval matrix for the particle size range D > 125 μm (or N_total > 20 pixels).