Merge pull request #16243 from collinbrake:grammar_fixes_9

doc/py_tutorials/py_feature2d/py_features_harris/py_features_harris.markdown

@@ -7,25 +7,25 @@ Goal
 In this chapter,
 
 -   We will understand the concepts behind Harris Corner Detection.
--   We will see the functions: **cv.cornerHarris()**, **cv.cornerSubPix()**
+-   We will see the following functions: **cv.cornerHarris()**, **cv.cornerSubPix()**
 
 Theory
 ------
 
-In last chapter, we saw that corners are regions in the image with large variation in intensity in
+In the last chapter, we saw that corners are regions in the image with large variation in intensity in
 all the directions. One early attempt to find these corners was done by **Chris Harris & Mike
 Stephens** in their paper **A Combined Corner and Edge Detector** in 1988, so now it is called
-Harris Corner Detector. He took this simple idea to a mathematical form. It basically finds the
+the Harris Corner Detector. He took this simple idea to a mathematical form. It basically finds the
 difference in intensity for a displacement of \f$(u,v)\f$ in all directions. This is expressed as below:
 
 \f[E(u,v) = \sum_{x,y} \underbrace{w(x,y)}_\text{window function} \, [\underbrace{I(x+u,y+v)}_\text{shifted intensity}-\underbrace{I(x,y)}_\text{intensity}]^2\f]
 
-Window function is either a rectangular window or gaussian window which gives weights to pixels
+The window function is either a rectangular window or a Gaussian window which gives weights to pixels
 underneath.
 
-We have to maximize this function \f$E(u,v)\f$ for corner detection. That means, we have to maximize the
+We have to maximize this function \f$E(u,v)\f$ for corner detection. That means we have to maximize the
-second term. Applying Taylor Expansion to above equation and using some mathematical steps (please
+second term. Applying Taylor Expansion to the above equation and using some mathematical steps (please
-refer any standard text books you like for full derivation), we get the final equation as:
+refer to any standard text books you like for full derivation), we get the final equation as:
 
 \f[E(u,v) \approx \begin{bmatrix} u & v \end{bmatrix} M \begin{bmatrix} u \\ v \end{bmatrix}\f]
 
@@ -34,20 +34,20 @@ where
 \f[M = \sum_{x,y} w(x,y) \begin{bmatrix}I_x I_x & I_x I_y \\
                                      I_x I_y & I_y I_y \end{bmatrix}\f]
 
-Here, \f$I_x\f$ and \f$I_y\f$ are image derivatives in x and y directions respectively. (Can be easily found
+Here, \f$I_x\f$ and \f$I_y\f$ are image derivatives in x and y directions respectively. (These can be easily found
-out using **cv.Sobel()**).
+using **cv.Sobel()**).
 
-Then comes the main part. After this, they created a score, basically an equation, which will
+Then comes the main part. After this, they created a score, basically an equation, which
-determine if a window can contain a corner or not.
+determines if a window can contain a corner or not.
 
 \f[R = det(M) - k(trace(M))^2\f]
 
 where
     -   \f$det(M) = \lambda_1 \lambda_2\f$
     -   \f$trace(M) = \lambda_1 + \lambda_2\f$
-    -   \f$\lambda_1\f$ and \f$\lambda_2\f$ are the eigen values of M
+    -   \f$\lambda_1\f$ and \f$\lambda_2\f$ are the eigenvalues of M
 
-So the values of these eigen values decide whether a region is corner, edge or flat.
+So the magnitudes of these eigenvalues decide whether a region is a corner, an edge, or flat.
 
 -   When \f$|R|\f$ is small, which happens when \f$\lambda_1\f$ and \f$\lambda_2\f$ are small, the region is
     flat.
@@ -60,16 +60,16 @@ It can be represented in a nice picture as follows:
 ![image](images/harris_region.jpg)
 
 So the result of Harris Corner Detection is a grayscale image with these scores. Thresholding for a
-suitable give you the corners in the image. We will do it with a simple image.
+suitable score gives you the corners in the image. We will do it with a simple image.
 
 Harris Corner Detector in OpenCV
 --------------------------------
 
-OpenCV has the function **cv.cornerHarris()** for this purpose. Its arguments are :
+OpenCV has the function **cv.cornerHarris()** for this purpose. Its arguments are:
 
--   **img** - Input image, it should be grayscale and float32 type.
+-   **img** - Input image. It should be grayscale and float32 type.
 -   **blockSize** - It is the size of neighbourhood considered for corner detection
--   **ksize** - Aperture parameter of Sobel derivative used.
+-   **ksize** - Aperture parameter of the Sobel derivative used.
 -   **k** - Harris detector free parameter in the equation.
 
 See the example below:
@@ -103,12 +103,12 @@ Corner with SubPixel Accuracy
 
 Sometimes, you may need to find the corners with maximum accuracy. OpenCV comes with a function
 **cv.cornerSubPix()** which further refines the corners detected with sub-pixel accuracy. Below is
-an example. As usual, we need to find the harris corners first. Then we pass the centroids of these
+an example. As usual, we need to find the Harris corners first. Then we pass the centroids of these
 corners (There may be a bunch of pixels at a corner, we take their centroid) to refine them. Harris
 corners are marked in red pixels and refined corners are marked in green pixels. For this function,
 we have to define the criteria when to stop the iteration. We stop it after a specified number of
-iteration or a certain accuracy is achieved, whichever occurs first. We also need to define the size
+iterations or a certain accuracy is achieved, whichever occurs first. We also need to define the size
-of neighbourhood it would search for corners.
+of the neighbourhood it searches for corners.
 @code{.py}
 import numpy as np
 import cv2 as cv
@@ -139,7 +139,7 @@ img[res[:,3],res[:,2]] = [0,255,0]
 
 cv.imwrite('subpixel5.png',img)
 @endcode
-Below is the result, where some important locations are shown in zoomed window to visualize:
+Below is the result, where some important locations are shown in the zoomed window to visualize:
 
 ![image](images/subpixel3.png)