Morphological preprocessing 

A simple thresholding method is rarely sufficient for convective cell identification itself. The cells are often ragged and consist of multiple small pieces. Therefore, a preprocessing method is needed to unify thresholded areas. The GDBSCAN algorithm described in Subsection 5.1.2 is a manner of binding fragmented areas together as clusters. Another way is to use morphological image processing that is applied in this thesis.

In computer vision, morphology is an important method for extracting components that are useful in object description. It is also an important tool in image preprocessing, such as filtering, thinning and pruning. Mathematically morphology consists of set operations that are applied to the image with the user defined structuring element.

Morphology offers a powerful tool to numerous applications. Predominantly it can be used for the following purposes (Sonka 2007):

1. Image pre-processing, such as noise filtering and smoothing

2. Enhancing object structure, such as thinning, thickening and shape simplification. 3. Object segmentation from the background.

4. Calculating descriptive values of objects, such as area and perimeter.

In literature, morphology is presented as a tool of binary image processing, which can be further generalized for gray scale images. After segmenting convective cells from input image, we obtain binary regions that can be processed through morphology. For this reason, general gray scale morphological operations are not covered in here.

Morphological transform is defined by relation of the point set A (e.g. an image) with another small point set B, which is called as a structuring element and expressed with its local origin O (Sonka 2007). Applying morphological operation Ψ(A) to the image A means that the structuring element B is moved systematically along the image. The structuring element B is centered on each pixel corresponding to the O and the operation Ψ(A) is performed for each pixel in the neighborhood defined by B. Therefore, the operation is analogous to the well-known discrete convolution, in which the convolution mask is regarded as a structuring element of the operation. However, the convolution is a linear operation defined by its impulse response whereas morphology is a nonlinear set operation.

Dilation

Dilation is a morphological operation that makes binary objects in an input image thicker

and studier. The structuring element defines the shape and width of the thickening layer. Mathematically the dilation on the image A by the structuring element B is defined as (Gonzalez and Woods 2001)

ˆ

{ | ( )_z },

A B z B   A (5.17)

where ˆB is the reflection of set B

ˆ { | , for }

B w w b b B (5.18)

and ˆ( )B is the translation of set_c ˆB. For a given set S translation is as defined as follows

{ | , for }

c

S  c c s z  s S . (5.19)

Fig. 9 illustrates the dilation. The input image A consists of two rectangular binary sets. A is dilated with a small rectangular structuring element B. In this case, B and its reflection are equal since B is symmetric with respect to its origin (marked with a black dot in Fig. 9). As shown, the two rectangles have enlarged and united together in the operation.

A B

A B

Fig. 9: An example of dilation. a) An input image A and a structuring element B. b) The resulting dilated image.

In the convective cell identification, the dilation can be used for attaching cells together. Through dilation we can attach small distinct pixels to larger entities, eliminate small holes and make ragged areas more uniform. This is also analogous to the GDBSCAN clustering since it can be used for clustering of arbitrary shaped objects such as binary regions.

However, the dilation alone is rarely suitable for the preprocessing of the segmented convective cells. As stated, it increases the size of cells as well as size of small outliers lying outside of larger entities. For this reason, more morphological operations are needed and therefore the concepts of erosion and closing are discussed next.

Erosion

Contrary to the dilation, erosion decreases the size of binary regions in the image. Formally, the erosion of the image A by the structuring element B is defined as follows (Gonzalez and Woods 2001)

ˆ

{ | ( )_z }

A B  z B  A . (5.20)

This means that the erosion extracts the set of points z from A such that B, translated by z, is fully contained in A. In the border region of A, the structuring element B is only partially contained into A and hence the set A decreases.

It can be also shown that the erosion can be defined by means of the dilation as

ˆ

( C )C

A B  A B . (5.21)

Intuitively, the concept of erosion is clear. In the erosion we shrink binary regions in the input set A, which can be also understood as an enlargement of the complement of A. Fig. 10 illustrates the erosion operation. As the figure shows, the operation decreases the size of the two larger objects and even removes the smallest binary object.

B

A

a) b)

A B

Fig. 10: a) The original image A and the structuring element B. b) The image after erosion. Closing

Many morphological operations, such as closing, are based on dilation and erosion. Like dilation, closing tends to fuse narrow breaks and thin gulfs. Moreover, it fills small open holes and gaps but also smoothes the contour. However, it does not thicken the contour like the dilation.

Mathematically closing can be defined as the dilation which is followed by the erosion (Gonzalez and Woods 2001)

( )

A B AB B (5.22)

The interpretation of the closing is simple. In the dilation part, we thicken the edges and fill caps of the contour. The erosion part shrinks and slightly refines the edges. However, since small holes and gaps are filled in the dilation part, erosion is not exposed on these areas anymore. Fig. 11 represents the closing operation applied to the convective binary regions.

Fig. 11: An example of closing. a) Segmented convective regions from input image before morphological preprocessing. The used morphological structuring element is shown in the upper right corner. b) Segmented

regions after dilation. c) Segmented regions after closing, which is obtained by eroding (b).

In the convective cell identification, closing has an important status as it is directly applied to the unprocessed segmented convective regions. The user needs to define the shape and size of the structuring element, which can be done for example by a meteorological expert. Questions arise for example on the minimum distance between two segmented areas that are united together as well as on the desired smoothing effect. As an example, too large structuring element merges individual cells together unnecessarily. In this thesis a round disk having radius of approximately 5 km is used as the structuring element.