Geometry Operations

Types of geometry operations

1.Scaling

2.Translation

3.Rotation

To begin with, this is a basic channel of an image = f(y,x)

y = vertical direction

x = horizontal direction

In geometric transformation, we change the coordinates of the image x and y

f(y,x) => geometric transformation => g(y',x')

g(y',x') = the new image changed by the geometric transformation

Subset of geometric transformation called Affine Transformation

1.Scaling

= reshape the image, we can shrink or expand the image in a horizontal and or vertical direction

Shrinking the image or making the image larger

Scaling: x' = ax

a = scaling factor

ex) scale the image by 2

x' = 2x

y' = y

g(y,2x) = f(y,x)

g(y,2x) will look stretched relative to image f

Ex)

Several column's of g have no corresponding value in f

--> to determine the unknown pixel value, we use interpolation

Interpolation is where we used neighboring pixels to determine the value of an unknown pixel

*In this case, we use the nearest neighbors; the just assigns the value based on the nearest pixel*

Other methods that PIL and open cv use

Image is stretched in the horizontal direction

- use this method to make the image larger

- if the values of a or d are less than zero; the image will shrink

2.Translation

= shift the image

x' = x + tx

We can shift an image horizontally by adding the number of pixels "tx" then by mapping the new location x prime

x' = x + 2 (add two pixels)

Shift the image vertically by adding pixels ty, this shifts the image vertically

y' = y + ty

y' = y + 2

Represent a geometric transformation as a set of equation

x' = ax + tx

y' = dy + ty

Putting the equations in matrix form, we get the Affine Transformation

With open cv, you input this matrix as an array

3.Rotation

We can rotate an image by an angle theta, where the red lines represent the original orientation of the horizontal and vertical access

Use matrix to rotate an image

Rotation matrix will perform a counter-clockwise rotation

*Expression for the matrix*

Simply assuming the isotropic scale factor "r" is 1

PIL and open cv only remove the parameter theta

*PIL, y scale the image by specifying the integer number of pixel's using the method "resize"*

from PIL import image

image = Image.open("lenna.png")

width = 512

height = 512

new_width = 2*width

new_height = height

new_image = image.resize((new_width, new_height)) -----> can disable the width of an image, by

applying the method resize the

image is twice as wide

Rotate the image

image = Image.open("lenna.png")

theta = 45

new_image = image.rotate(theta) ----> the input is the angle we would like to rotate the image by

*open cv*

import cv2

image = cv2.imread("lenna.png")

new_image = cv2.resize(image, None, fx=2, fy=1, interpolation=cv2.INTER_CUBIC)

=> the result will make the image twice as wide

Translation requires the Affine Transformation matrix M

where tx is the number of pixels you shift the location in the horizontal direction

rows,cols,__ = image.shape

tx = 100

ty = 0

M = np.fbat32[[(1,0,tx),(0,1,ty)]]

ty is the number of pixels you shift in the vertical direction

We input the image and matrix into the function warpAffine, we also input the shape of the output image

new_image = cv2.warpAffine(image,M,(cols,rows))

*Bonus*

theta = 45.0

M = cv2.getRotationMatrix2D (center = (cols/2-1,rows/2-1), angle=theta, scale=1)

- obtain the matrix to rotate the image using getRotationMatrix2D, this will make the image by angle theta

- The parameter center is the center of the rotation in the source, scale will be set to one

new_image = cv2.warpAffine(image,M,(cols,rows))

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