Goal
----
-In this section,
- - We will learn about distortions in camera, intrinsic and extrinsic parameters of camera etc.
- - We will learn to find these parameters, undistort images etc.
+In this section, we will learn about
+
+* types of distortion caused by cameras
+* how to find the intrinsic and extrinsic properties of a camera
+* how to undistort images based off these properties
Basics
------
-Today's cheap pinhole cameras introduces a lot of distortion to images. Two major distortions are
+Some pinhole cameras introduce significant distortion to images. Two major kinds of distortion are
radial distortion and tangential distortion.
-Due to radial distortion, straight lines will appear curved. Its effect is more as we move away from
-the center of image. For example, one image is shown below, where two edges of a chess board are
-marked with red lines. But you can see that border is not a straight line and doesn't match with the
+Radial distortion causes straight lines to appear curved. Radial distortion becomes larger the farther points are from
+the center of the image. For example, one image is shown below in which two edges of a chess board are
+marked with red lines. But, you can see that the border of the chess board is not a straight line and doesn't match with the
red line. All the expected straight lines are bulged out. Visit [Distortion
(optics)](http://en.wikipedia.org/wiki/Distortion_%28optics%29) for more details.
-This distortion is represented as follows:
+Radial distortion can be represented as follows:
\f[x_{distorted} = x( 1 + k_1 r^2 + k_2 r^4 + k_3 r^6) \\
y_{distorted} = y( 1 + k_1 r^2 + k_2 r^4 + k_3 r^6)\f]
-Similarly, another distortion is the tangential distortion which occurs because image taking lense
-is not aligned perfectly parallel to the imaging plane. So some areas in image may look nearer than
-expected. It is represented as below:
+Similarly, tangential distortion occurs because the image-taking lense
+is not aligned perfectly parallel to the imaging plane. So, some areas in the image may look nearer than
+expected. The amount of tangential distortion can be represented as below:
\f[x_{distorted} = x + [ 2p_1xy + p_2(r^2+2x^2)] \\
y_{distorted} = y + [ p_1(r^2+ 2y^2)+ 2p_2xy]\f]
\f[Distortion \; coefficients=(k_1 \hspace{10pt} k_2 \hspace{10pt} p_1 \hspace{10pt} p_2 \hspace{10pt} k_3)\f]
-In addition to this, we need to find a few more information, like intrinsic and extrinsic parameters
-of a camera. Intrinsic parameters are specific to a camera. It includes information like focal
-length (\f$f_x,f_y\f$), optical centers (\f$c_x, c_y\f$) etc. It is also called camera matrix. It depends on
-the camera only, so once calculated, it can be stored for future purposes. It is expressed as a 3x3
+In addition to this, we need to some other information, like the intrinsic and extrinsic parameters
+of the camera. Intrinsic parameters are specific to a camera. They include information like focal
+length (\f$f_x,f_y\f$) and optical centers (\f$c_x, c_y\f$). The focal length and optical centers can be used to create a camera matrix, which can be used to remove distortion due to the lenses of a specific camera. The camera matrix is unique to a specific camera, so once calculated, it can be reused on other images taken by the same camera. It is expressed as a 3x3
matrix:
\f[camera \; matrix = \left [ \begin{matrix} f_x & 0 & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1 \end{matrix} \right ]\f]
Extrinsic parameters corresponds to rotation and translation vectors which translates a coordinates
of a 3D point to a coordinate system.
-For stereo applications, these distortions need to be corrected first. To find all these parameters,
-what we have to do is to provide some sample images of a well defined pattern (eg, chess board). We
-find some specific points in it ( square corners in chess board). We know its coordinates in real
-world space and we know its coordinates in image. With these data, some mathematical problem is
-solved in background to get the distortion coefficients. That is the summary of the whole story. For
-better results, we need atleast 10 test patterns.
+For stereo applications, these distortions need to be corrected first. To find these parameters,
+we must provide some sample images of a well defined pattern (e.g. a chess board). We
+find some specific points of which we already know the relative positions (e.g. square corners in the chess board). We know the coordinates of these points in real world space and we know the coordinates in the image, so we can solve for the distortion coefficients. For better results, we need at least 10 test patterns.
Code
----
-As mentioned above, we need atleast 10 test patterns for camera calibration. OpenCV comes with some
-images of chess board (see samples/cpp/left01.jpg -- left14.jpg), so we will utilize it. For sake of
-understanding, consider just one image of a chess board. Important input datas needed for camera
-calibration is a set of 3D real world points and its corresponding 2D image points. 2D image points
+As mentioned above, we need at least 10 test patterns for camera calibration. OpenCV comes with some
+images of a chess board (see samples/data/left01.jpg -- left14.jpg), so we will utilize these. Consider an image of a chess board. The important input data needed for calibration of the camera
+is the set of 3D real world points and the corresponding 2D coordinates of these points in the image. 2D image points
are OK which we can easily find from the image. (These image points are locations where two black
squares touch each other in chess boards)
and camera was moved accordingly. This consideration helps us to find only X,Y values. Now for X,Y
values, we can simply pass the points as (0,0), (1,0), (2,0), ... which denotes the location of
points. In this case, the results we get will be in the scale of size of chess board square. But if
-we know the square size, (say 30 mm), and we can pass the values as (0,0),(30,0),(60,0),..., we get
+we know the square size, (say 30 mm), we can pass the values as (0,0), (30,0), (60,0), ... . Thus, we get
the results in mm. (In this case, we don't know square size since we didn't take those images, so we
pass in terms of square size).
### Setup
-So to find pattern in chess board, we use the function, **cv.findChessboardCorners()**. We also
-need to pass what kind of pattern we are looking, like 8x8 grid, 5x5 grid etc. In this example, we
+So to find pattern in chess board, we can use the function, **cv.findChessboardCorners()**. We also
+need to pass what kind of pattern we are looking for, like 8x8 grid, 5x5 grid etc. In this example, we
use 7x6 grid. (Normally a chess board has 8x8 squares and 7x7 internal corners). It returns the
corner points and retval which will be True if pattern is obtained. These corners will be placed in
an order (from left-to-right, top-to-bottom)
-@sa This function may not be able to find the required pattern in all the images. So one good option
+@sa This function may not be able to find the required pattern in all the images. So, one good option
is to write the code such that, it starts the camera and check each frame for required pattern. Once
-pattern is obtained, find the corners and store it in a list. Also provides some interval before
+the pattern is obtained, find the corners and store it in a list. Also, provide some interval before
reading next frame so that we can adjust our chess board in different direction. Continue this
-process until required number of good patterns are obtained. Even in the example provided here, we
-are not sure out of 14 images given, how many are good. So we read all the images and take the good
+process until the required number of good patterns are obtained. Even in the example provided here, we
+are not sure how many images out of the 14 given are good. Thus, we must read all the images and take only the good
ones.
-@sa Instead of chess board, we can use some circular grid, but then use the function
-**cv.findCirclesGrid()** to find the pattern. It is said that less number of images are enough when
-using circular grid.
+@sa Instead of chess board, we can alternatively use a circular grid. In this case, we must use the function
+**cv.findCirclesGrid()** to find the pattern. Fewer images are sufficient to perform camera calibration using a circular grid.
Once we find the corners, we can increase their accuracy using **cv.cornerSubPix()**. We can also
draw the pattern using **cv.drawChessboardCorners()**. All these steps are included in below code:
### Calibration
-So now we have our object points and image points we are ready to go for calibration. For that we
-use the function, **cv.calibrateCamera()**. It returns the camera matrix, distortion coefficients,
+Now that we have our object points and image points, we are ready to go for calibration. We can
+use the function, **cv.calibrateCamera()** which returns the camera matrix, distortion coefficients,
rotation and translation vectors etc.
@code{.py}
ret, mtx, dist, rvecs, tvecs = cv.calibrateCamera(objpoints, imgpoints, gray.shape[::-1], None, None)
@endcode
+
### Undistortion
-We have got what we were trying. Now we can take an image and undistort it. OpenCV comes with two
-methods, we will see both. But before that, we can refine the camera matrix based on a free scaling
+Now, we can take an image and undistort it. OpenCV comes with two
+methods for doing this. However first, we can refine the camera matrix based on a free scaling
parameter using **cv.getOptimalNewCameraMatrix()**. If the scaling parameter alpha=0, it returns
undistorted image with minimum unwanted pixels. So it may even remove some pixels at image corners.
-If alpha=1, all pixels are retained with some extra black images. It also returns an image ROI which
+If alpha=1, all pixels are retained with some extra black images. This function also returns an image ROI which
can be used to crop the result.
-So we take a new image (left12.jpg in this case. That is the first image in this chapter)
+So, we take a new image (left12.jpg in this case. That is the first image in this chapter)
@code{.py}
img = cv.imread('left12.jpg')
h, w = img.shape[:2]
@endcode
#### 1. Using **cv.undistort()**
-This is the shortest path. Just call the function and use ROI obtained above to crop the result.
+This is the easiest way. Just call the function and use ROI obtained above to crop the result.
@code{.py}
# undistort
dst = cv.undistort(img, mtx, dist, None, newcameramtx)
@endcode
#### 2. Using **remapping**
-This is curved path. First find a mapping function from distorted image to undistorted image. Then
+This way is a little bit more difficult. First, find a mapping function from the distorted image to the undistorted image. Then
use the remap function.
@code{.py}
# undistort
dst = dst[y:y+h, x:x+w]
cv.imwrite('calibresult.png', dst)
@endcode
-Both the methods give the same result. See the result below:
+Still, both the methods give the same result. See the result below:
You can see in the result that all the edges are straight.
-Now you can store the camera matrix and distortion coefficients using write functions in Numpy
+Now you can store the camera matrix and distortion coefficients using write functions in NumPy
(np.savez, np.savetxt etc) for future uses.
Re-projection Error
-------------------
-Re-projection error gives a good estimation of just how exact is the found parameters. This should
-be as close to zero as possible. Given the intrinsic, distortion, rotation and translation matrices,
-we first transform the object point to image point using **cv.projectPoints()**. Then we calculate
+Re-projection error gives a good estimation of just how exact the found parameters are. The closer the re-projection error is to zero, the more accurate the parameters we found are. Given the intrinsic, distortion, rotation and translation matrices,
+we must first transform the object point to image point using **cv.projectPoints()**. Then, we can calculate
the absolute norm between what we got with our transformation and the corner finding algorithm. To
-find the average error we calculate the arithmetical mean of the errors calculate for all the
+find the average error, we calculate the arithmetical mean of the errors calculated for all the
calibration images.
@code{.py}
mean_error = 0
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