| OpenCV 5.0.0alpha Open Source Computer Vision

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Geometric Transformations of Images

# Goals

- Learn to apply different geometric transformations to images, like translation, rotation, affine transformation etc.
- You will see these functions: cv.getPerspectiveTransform

# Transformations

OpenCV provides two transformation functions, cv.warpAffine and cv.warpPerspective, with which you can perform all kinds of transformations. cv.warpAffine takes a 2x3 transformation matrix while cv.warpPerspective takes a 3x3 transformation matrix as input.

## Scaling

Scaling is just resizing of the image. OpenCV comes with a function cv.resize() for this purpose. The size of the image can be specified manually, or you can specify the scaling factor. Different interpolation methods are used. Preferable interpolation methods are cv.INTER_AREA for shrinking and cv.INTER_CUBIC (slow) & cv.INTER_LINEAR for zooming. By default, the interpolation method cv.INTER_LINEAR is used for all resizing purposes. You can resize an input image with either of following methods:

import numpy as np
import cv2 as cv
img = cv.imread('messi5.jpg')
assert img is not None, "file could not be read, check with os.path.exists()"
res = cv.resize(img,None,fx=2, fy=2, interpolation = cv.INTER_CUBIC)
#OR
height, width = img.shape[:2]
res = cv.resize(img,(2*width, 2*height), interpolation = cv.INTER_CUBIC)
cv::imread
CV_EXPORTS_W Mat imread(const String &filename, int flags=IMREAD_COLOR_BGR)
Loads an image from a file.
cv::resize
void resize(InputArray src, OutputArray dst, Size dsize, double fx=0, double fy=0, int interpolation=INTER_LINEAR)
Resizes an image.

## Translation

Translation is the shifting of an object's location. If you know the shift in the (x,y) direction and let it be \((t_x,t_y)\), you can create the transformation matrix \(\textbf{M}\) as follows:

\[M = \begin{bmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \end{bmatrix}\]

You can take make it into a Numpy array of type np.float32 and pass it into the cv.warpAffine() function. See the below example for a shift of (100,50):

import numpy as np
import cv2 as cv
img = cv.imread('messi5.jpg', cv.IMREAD_GRAYSCALE)
assert img is not None, "file could not be read, check with os.path.exists()"
rows,cols = img.shape
M = np.float32([[1,0,100],[0,1,50]])
dst = cv.warpAffine(img,M,(cols,rows))
cv.imshow('img',dst)
cv.waitKey(0)
cv.destroyAllWindows()
cv::imshow
void imshow(const String &winname, InputArray mat)
Displays an image in the specified window.
cv::waitKey
int waitKey(int delay=0)
Waits for a pressed key.
cv::destroyAllWindows
void destroyAllWindows()
Destroys all of the HighGUI windows.
cv::warpAffine
void warpAffine(InputArray src, OutputArray dst, InputArray M, Size dsize, int flags=INTER_LINEAR, int borderMode=BORDER_CONSTANT, const Scalar &borderValue=Scalar(), AlgorithmHint hint=cv::ALGO_HINT_DEFAULT)
Applies an affine transformation to an image.

warning

The third argument of the cv.warpAffine() function is the size of the output image, which should be in the form of (width, height). Remember width = number of columns, and height = number of rows.

See the result below:

image

## Rotation

Rotation of an image for an angle \(\theta\) is achieved by the transformation matrix of the form

\[M = \begin{bmatrix} cos\theta & -sin\theta \\ sin\theta & cos\theta \end{bmatrix}\]

But OpenCV provides scaled rotation with adjustable center of rotation so that you can rotate at any location you prefer. The modified transformation matrix is given by

\[\begin{bmatrix} \alpha & \beta & (1- \alpha ) \cdot center.x - \beta \cdot center.y \\ - \beta & \alpha & \beta \cdot center.x + (1- \alpha ) \cdot center.y \end{bmatrix}\]

where:

\[\begin{array}{l} \alpha = scale \cdot \cos \theta , \\ \beta = scale \cdot \sin \theta \end{array}\]

To find this transformation matrix, OpenCV provides a function, cv.getRotationMatrix2D. Check out the below example which rotates the image by 90 degree with respect to center without any scaling.

img = cv.imread('messi5.jpg', cv.IMREAD_GRAYSCALE)
assert img is not None, "file could not be read, check with os.path.exists()"
rows,cols = img.shape
# cols-1 and rows-1 are the coordinate limits.
M = cv.getRotationMatrix2D(((cols-1)/2.0,(rows-1)/2.0),90,1)
dst = cv.warpAffine(img,M,(cols,rows))
cv::getRotationMatrix2D
Mat getRotationMatrix2D(Point2f center, double angle, double scale)
Calculates an affine matrix of 2D rotation.
Definition imgproc.hpp:2629

See the result:

image

## Affine Transformation

In affine transformation, all parallel lines in the original image will still be parallel in the output image. To find the transformation matrix, we need three points from the input image and their corresponding locations in the output image. Then cv.getAffineTransform will create a 2x3 matrix which is to be passed to cv.warpAffine.

Check the below example, and also look at the points I selected (which are marked in green color):

img = cv.imread('drawing.png')
assert img is not None, "file could not be read, check with os.path.exists()"
rows,cols,ch = img.shape
pts1 = np.float32([[50,50],[200,50],[50,200]])
pts2 = np.float32([[10,100],[200,50],[100,250]])
M = cv.getAffineTransform(pts1,pts2)
dst = cv.warpAffine(img,M,(cols,rows))
plt.subplot(121),plt.imshow(img),plt.title('Input')
plt.subplot(122),plt.imshow(dst),plt.title('Output')
plt.show()
cv::getAffineTransform
Mat getAffineTransform(const Point2f src[], const Point2f dst[])
Calculates an affine transform from three pairs of the corresponding points.

See the result:

image

## Perspective Transformation

For perspective transformation, you need a 3x3 transformation matrix. Straight lines will remain straight even after the transformation. To find this transformation matrix, you need 4 points on the input image and corresponding points on the output image. Among these 4 points, 3 of them should not be collinear. Then the transformation matrix can be found by the function cv.getPerspectiveTransform. Then apply cv.warpPerspective with this 3x3 transformation matrix.

See the code below:

img = cv.imread('sudoku.png')
assert img is not None, "file could not be read, check with os.path.exists()"
rows,cols,ch = img.shape
pts1 = np.float32([[56,65],[368,52],[28,387],[389,390]])
pts2 = np.float32([[0,0],[300,0],[0,300],[300,300]])
M = cv.getPerspectiveTransform(pts1,pts2)
dst = cv.warpPerspective(img,M,(300,300))
plt.subplot(121),plt.imshow(img),plt.title('Input')
plt.subplot(122),plt.imshow(dst),plt.title('Output')
plt.show()
cv::warpPerspective
void warpPerspective(InputArray src, OutputArray dst, InputArray M, Size dsize, int flags=INTER_LINEAR, int borderMode=BORDER_CONSTANT, const Scalar &borderValue=Scalar(), AlgorithmHint hint=cv::ALGO_HINT_DEFAULT)
Applies a perspective transformation to an image.
cv::getPerspectiveTransform
Mat getPerspectiveTransform(InputArray src, InputArray dst, int solveMethod=DECOMP_LU)
Calculates a perspective transform from four pairs of the corresponding points.

Result:

image

# Additional Resources

1. "Computer Vision: Algorithms and Applications", Richard Szeliski