Update theory.md

experiment/theory.md

@@ -1,6 +1,6 @@
 <b>1</b> Stereology is an important tool used for the geometrical characterization of microstructures. This microstructural characterization means geometrical characterization such as volume fraction of second phase, particle size distribution, size distribution of grains, density of dislocations. Microstructures evolves due to processing done in the material which in turn governs the properties of the materials and therefore characterization of microstructure is an important aspect of materials science. The term stereology dates to 1960’s and it relates the 2-D images of the microstructures with the 3-D images and structures. In stereology, geometrical features of the structure of the material are quantified using various types of methods. Many mathematical tools are used to transform the 2D measurements made on a planar microstructure to 3D parameters of the structure of the solids. Elements of microstructures:<br>
 
-<b>2 Typical microstructural elements:</b><br><br>
+<b>2 Typical microstructural elements:</b><br>
 <b>2.1</b>	3-D particles<br>
 <b>2.2</b> 2-D surface or interfaces between particles and grains.<br>
 <b>2.3</b>	1-D linear features such as dislocations, fibres, grain edges etc.,.<br>
@@ -10,15 +10,15 @@ In materials science we deal with the opaque materials, and so we take a section
 <b>Figure 1.</b> Section cuts of 3-D images on a 2-D plane.<br><br>
 Another way to represent the images is by producing a projected image and its representation is possible if our material remains transparent to light. Then lines will be projected as lines, surfaces will be projected as area projections and will be forming a projected image. Both 2-D images can be quantified and related to the 3-D structure but stereology behind them functions differently.<br>
 
-<b>3 Tools to observe the microstructures:</b><br><br>
+<b>3 Tools to observe the microstructures:</b><br>
 <b>3.1</b>	 Optical microstructures are studied using the Reflected Light Microscopes (Metallurgical microscopes) for opaque materials.<br><br>
 <b>3.2</b>	 Transmitted light microscopes are used for observing the microstructures of the transparent materials.<br><br> 
 <b>3.3</b>  Electron microscopes are used with higher resolutions to observe the images for obtaining greater details and understand its features in the microstructures. Among these microscopes, most common are Scanning Electron Microscopes (SEM) which shows secondary and back scattered electron images, Transmission Electron Microscopes (TEM) which shows projected images, X-ray map images, and Scanning Tunnelling Microscopes (STM). Stereology has the power to quantify the true interlamellar spacing of the pearlite.<br> <br>
 
-<b>4 Important concepts of Stereology:</b><br><br>
+<b>4 Important concepts of Stereology:</b><br>
 <b>4.1  Sectioning or sampling:</b> Stereology involves sampling in a systematic random way to ensure unbiased representation of the entire structure to be studied. Sectioning involves random sampling of the 3-D block using 2-D plane. Selection of the sample is done randomly and unbiasedly towards any specific regions of interest. <br><br>
 <b>4.2  Sampling probes:</b> These are the basic measurements units used for the microstructural assessments and studies. It can be represented using different shapes such as points, lines, areas, or volumes, depending on the specific stereological method and property of interest. To bring the accuracy to obtained results, proper probe selection is of utmost importance. Section can be done using these lines and probes and it is also known as sampling of materials or sampling by probes of various dimensions. If we section a 2-D system then probe should be 2-D, section with line means a 1-D probe, and sectioning using a point means a 0-D probe.<br> <br>
-<b>5 Structures are sectioned and sampled using different probes.</b><br><br> 
+<b>5 Structures are sectioned and sampled using different probes.</b><br>
 <b>5.1	Sampling using 2-D Probe (Plane probe):</b> Intersection of a plane with the structure. 3-D structure intersects using planes. After sectioning, line appears at the sections of surfaces, points appear at the planes where the lines were present in 3-D structure and area elements and section profiles appears at the places where particles are present in 3-D materials.<br><br> 
 <b>5.2	Sampling by 1-D Probe or Line:</b> If we put grids of lines along horizontal and vertical directions on the 2-D probe then it is known as sampling by 1-D probe or line probe which intersection of a line with the structure. One can check for the intersections line makes with the surface features. <br><br>
 <b>5.3 Sampling using 0-D Probe (Point Probe): </b>Intersection of point with structure. We are interested to look how many points are intersecting with the particles.<br> 
@@ -39,11 +39,11 @@ Use of all the probes mentioned above plays a key role to understand the structu
 Stereology is built on the concept of geometrical probability. So, if there are large number of particles and there is a plane probe which cuts the structure, not all particles will get cut and only some of them will be cut. How many of them will be cut will depend on the probability of the plane probe intersecting the particles.<br> <br>
 <b>6 Estimation:</b> Stereological equations and mathematical models are employed to estimate the properties of interest, such as volume, surface area, length, or number of objects. These estimations are based on the data collected from the samples. <br>
 
-<b>7 Important consequences:</b> <br> <br> 
+<b>7 Important consequences:</b> <br> 
 <b>7.1</b>	Probe is of a lower dimension than the structure and leads to loss of information (e.g., shape). So, during analysis sometimes we use assumed shapes.<br>  
 <b>7.2</b>	Stereological methods are statistical in nature. One obtains average value and are subjected to statistical errors.<br><br>  
 <b>7.3</b>	Stereological methods are based on mathematical rigour and based on probability and statistics.<br>  
-<b>8 Methods of Stereology</b> <br><br>
+<b>8 Methods of Stereology</b> <br>
 <b>8.1	Point Counting:</b> In point counting, a grid with regularly spaced points is superimposed on a sample microstructure, and the number of points falling within the region of interest is counted. This method is commonly used for the estimation of volume fractions and particle densities.<br> <br>
 <b>8.2	Line Intercept Method:</b> The line intercept method involves the placement of lines randomly onto the sample microstructure followed by counting the total number of times line intersects the structure of interest. This is the most common method used to estimate the length and surface area.<br> <br>
 <b>8.3	Area Fraction Method:</b> This method estimates the volume fraction of the structures within a 2-D plane. It involves the count of total intersection points of the structures with a grid of points or lines.<br> <br>
@@ -52,7 +52,7 @@ Stereology is built on the concept of geometrical probability. So, if there are
 <b>Figure 2.</b> Microstructures sectioned with both horizontal and vertical grids.<br>
 
 Structures coming from spherical particles in 3-D. Structures are coming from the spherical particles in 3-D. If we want to calculate area fraction of the particles. i.e., total area fraction of the particles in a unit area of image. Since microstructures vary in statistical manner so we want to use stereology for this calculation. <br><br>
-<b>Concept:</b><br><br> 
+<b>Concept:</b><br> 
 <b>9.1</b>	Put grid of points on the given microstructure. Now, count the total number of points and then count the total number of points lying inside the circular particles. <br><br>
 <b>9.2</b>	Now, measure the point fraction (P<sub>P</sub>)<br>
 <image src="images/Picture3.png"><br>
@@ -71,7 +71,7 @@ where, n = Total number of points intersecting the particles<br>
 <image src="images/Picture5.png"><br>
 n = number of intersections<br>
 L = Total length of all the horizontal lines<br><br>
-<b>11 Applications of Stereology</b><br><br>
+<b>11 Applications of Stereology</b><br>
 <b>11.1	Biological Sciences:</b> Cells, tissues, organs, and neural networks are studied with greater details with the help of stereology.<br><br>
 <b>11.2	Materials Science:</b> Microstructural analysis, grain size calculations, fiber orientations, porosity, are studied using the concepts of stereology for different metallic and materials systems and extremely helpful in the quality control of the material design.<br><br> 
 <b>11.3	Geology:</b> Geological sample properties such as rock microstructures, textures, grain sizes and porosities are studied using the stereology.<br><br>