It is extended into three functions described as,. Equation 1 is generated by the approximating of columns in the x -direction and approximating the rows in the y -direction. Horizontal details are obtained by approximating columns in the x -direction and detailing rows in the y -direction. Vertical details are computed by detailing columns in the x -direction and approximating rows in the y -direction. The diagonal image is performed by detailing both directions, x , and y -directions Ismail and Khan, ; Salem, In order to create a new image that contains the horizontal, vertical and diagonal edges that had been smoothed in the approximation image, the three detailed images are converted to binary images by a suitable threshold defined for each of them.

Finally, the inverse discrete stationary wavelet transform is performed to obtain the denoised image Raviraj and Sanavullah, Edge detection is one of the most frequently used techniques in image binarization. An edge is characterized by a high local change of the intensity in the image Maini and Aggarwal, Edge detection can be provided by various operators, such as: Sobel, Canny, Roberts, Prewitt, and log.

In this paper we use a mix of Sobel and log operators. The following matrix is called Sobel kernel, and it is applied to the image for extracting horizontal and vertical edges Vincent and Folorunso, The log operator is used to smooth the image with Gaussian filter in order to reduce its sensitivity to noise, and then convolves the smoothed image with a Laplacian filter. Alternatively, the image is convolved with a linear filter - the Laplacian of Gaussian LoG kernel :.

The edges can be detected by finding the zero crossing of the 2 nd derivative of the image intensity. Also, image multi-level thresholding is performed on the image for binarization. This process determines various threshold values to segment the image into several clusters, C. The proposed method applies the convolution process using Sobel kernel, which is favourable for the next operation image multi-level thresholding. Again the edge detection process is performed using log operator for extracting strong edges.

To get a complete contour of segmented objects, dilation, erosion, closing, and region filling operations are applied on the binary image.

Dilation and erosion are the main morphological operations. Morphological image processing is a collection of non-linear operations related to shape or morphology of features in an image. Morphological operations rely only on the relative ordering of pixel values, not on their numerical values, and therefore are especially suited to processing binary images. Suppose A is an image set and B a structuring element.

Dilation and erosion operations are described as in Equations 10 and 11 , respectively. Closing operation is defined as dilation followed by erosion, using the same structuring element for both operations. Region filling is based on set of dilations, complement, and intersections, as in the following equation:. The results of these operations are influenced by the size and shape of the structuring element B. In fact, the choice of the structuring element depends on the shape of the objects in the image.

## Introduction to Particle Technology, 2nd Ed [Martin Rhodes]

To identify the objects, the dilation process is performed, followed by region filling to complete the pores of the objects. Furthermore, a median filter is applied to smooth the contour curve after the erosion operation.

It may happen that two or more objects are connected in their borders - this effect has been neglected by many authors. The problem is that, if the number of connected objects is large with respect to the total number of objects, it may cause significant classification errors Pons and Vivier, Therefore, we shall extract mutually connected objects using a Watershed Transform.

Watershed transformation belongs to a class of methods for region-based segmentation. The basic step of this transformation is to visualize an image grey scale, or binary into a topographic surface, which includes three notions: minima, catchment basins, and watershed lines, as shown in Figure 2. Each regional minimum of this surface is represented as a catchment basin. The idea of watershed transformation is to pierce a hole in each minimum of catchment basin and allowing it to sink in a lake.

The water would be inserted gradually from local minimum firstly until fill up different catchment basins in the image.

When the rising water in different basins would risk merging, dams are built to prevent the merging of water coming from two different basins. These boundary dams correspond to the watershed lines that needed to divide Lu and Ke, There are various methods are used for the watershed transform, such as gradient, marker-controller and distance transform Aayushi, ; Beucher, The gradient magnitude method is applied to gradient images instead of grey level ones Belaid and Mourou, ; Jackway, This method has an over-segmentation problem due to noise in the image.

To solve this problem, the marker-controller watershed approach is introduced. It consists of markers to be defined as new minima of the gradient image objects. After the segmentation process, the watershed line for each object can be separated from its neighbors Aayushi, In the proposed method, the authors have chosen distance transform to perform the segmentation step.

It calculates a distance map from a center point to the edges of the object. The centers are determined by performing multiple successive erosions with a suitable structuring element until all foreground regions of the image are eroded away.

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Then it fills that topological map with imaginary water. Where two watersheds meet a dam is built to separate them Aayushi, ; Lu and Ke, The distance transform provides a metric of the separation of points in the image such as Euclidean, City Block, or Chessboard. The chessboard metric has been used to measure the path between the pixels based on an 8-connected neighborhood. The pixels whose have edges or corners are touched, are one unit apart. The actual structuring element that should be used depends on which distance metric is chosen.

The object is recognized by computing connected component labelling operations after applying the previously described methodologies Stefano and Bulgarelli, Then it is possible to determine the number of cells in each input image and then measure the concentration of cells during growth.

After that, relevant parameters of each cell size can be obtained: area, perimeter, diameter, and radius. The process of classification is unsupervised and based on two fundamental features: cell size, and cell texture. The area of the cell is determined from its radius. Also, cell texture has been chosen to provide a way for accurate classification, ensuring the effectiveness and reliability of the process.

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The texture of an image is represented by the distributions and relationships between the grey levels of the image. The co-occurrence matrix is identified as an estimation matrix for joint probability of two pixels P i, j.

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The dimension of this matrix is equal to the number of grey levels in the image G and its values are associated with the number of occurrences of pairs of grey levels Figure 3. The matrix represents the most popular second-order statistical features for texture analysis Malik et al. Indeed, from this matrix, we can compute the following properties: energy, entropy, contrast, homogeneity and correlation Andrzej and Michal, ; Mihran and Anil, Mostly, energy is the most crucial factor used in our method, while the remaining factors do not properly differentiate between the objects due to the similarity of the particles' values within the image.

The angular second moment energy is computed as:. Two types of datasets are used to evaluate the proposed method.

## AIChE Equipment Testing Procedure - Particle Size Classifiers, 2nd Edition | AIChE

The first data set is a synthetic image, generated by a computer program. The purpose of that is to calibrate and test the reliability of the proposed method. Real images of C. Figure 4 represents the output of each step in the experiment - as described in Figure 1. Figure 4a represents the initial grey image. Figure 4b shows the first step - de-noising using Haar wavelet transform as a pre-processing step for the microalgae microscopic images.

The initial image is decomposed into three levels, and the hard threshold is chosen for all levels. This eliminates noise effectively and also retains the edges information. The next step, image convolution by Sobel kernel, is shown in Figure 4c , and for better visualization, the image contrast is adjusted as shown in Figure 4d. In the resulting image, a clear identification of the cells could not be made. The enhancement process is started by closing and dilating the objects to identify the features of cells, as shown Figure 5b.

Even if the dilation of the cells borders leads to connect previously unconnected cells the watershed technique would be able to overcome this issue. To improve identification, the holes of the particle are filled, as shown in Figure 5c. The resulting image again is not optimal since it contains a large amount of debris.

After that, the problem of connected objects, which are segmented as one component, must be faced.