\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos(45^\circ) & -\sin(45^\circ) \\ \sin(45^\circ) & \cos(45^\circ) \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}

Rotation is basically the process by which an image is simply rotated around the origin or an image center by a given angle. This one rotates the image or changes the orientation of an image depending on the angle it has been set to.

\begin{bmatrix} \\x' \\y' \\1 \end{bmatrix}=A\begin{bmatrix} \\x \\ y \\1 \end{bmatrix}= \begin{bmatrix} a &b &c \\ d&e &f \\ 0& 0 &1 \end{bmatrix}\begin{bmatrix} \\x \\y \\1 \end{bmatrix}

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\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}

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The interpolation methods most frequently used are nearest neighbor method, bilinear and bicubic interpolation. These methods were further used to obtain quadratic interpolation in order to estimate non-integer pixel values, coming out from the transformation process, to give smooth and exact output images.

\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}

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\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 1 & k_y \\ k_x & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}

Geometric transformations require identification of the type of transformation to be done and its parameters, generation of the required transformation matrix, application of the matrix to the image coordinates, and use of interpolation methods to determine the intensity of the pixel at a new location.

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Thus, geometric transformations expounded upon via the use of mathematical operations through transformation matrices, serve as useful tools for image enhancement, restoration, and analysis and are therefore important in both practical and theoretical applications in the fields of image processing as well as computer vision.

Parallax is very much similar to perspective transformation where lines that are parallel are made to appear as though they are becoming joined at some certain now-where points called vanishes points. Since the perspective transformations do not map parallel lines to parallel lines or distances to distances, they are advantageous for obtaining 3D like effects in the images.

\begin{bmatrix} \\x' \\y' \end{bmatrix}= T\begin{bmatrix} \\x \\y \end{bmatrix}= \begin{bmatrix} \\a &b \\c& d \end{bmatrix}\begin{bmatrix} \\x \\y \end{bmatrix}

Except that Shearing displaces the pixel in one direction and as a result causes an inclined output to be produced. Identical to the rotation transformation, this transformation changes the angles between the axes of the image.

Perspective transformation is some how related to a matrix of order 3*3 where co-ordinate points of the two planes can be transformed. The transformation is defined by:

Scaling enables one to make the image larger or smaller in size or as it is known as scaling it copies or reduces the image proportionately to the original size. Most scaling methods preserve aspect ratio, but the general scaling is achieved by changing the dimension on different axes unlike other methods.

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Translation displaces an image by a certain amount of pixels about the x and y axis. This operation translates the image in a way such that every pixel in the image will be shifted to a new position maintaining the shape and size of the image.

Another important feature of images in Geometric Transformations is that the alteration of such image attributes is significant in image processing as it helps in changing the aspect of the image to fit the analysis or visualization requirements. These operations, changes in image geometry, generate better alignment of images, extraction of features, and improve image data for further process or analysis.

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Geometric transformations are based on the mathematical transformations in which the operations on the coordinates of the points of an image are performed. Such operations are always depicted by transformation matrices. For example:

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Image processing is performed using lots of mathematical tools. One such tool is geometric transformation. Geometric transformation in image transformation consists of two steps, namely spatial transformation of pixels or coordinates and intensity interpolation. Intensity interpolation is used to assign the intensity value of pixels after spatial transformation. The geometric (or spatial) transformation, T of an image is done on a pixel-by-pixel basis. The pixel having coordinate (x, y) will be moved to coordinate (x', y'). That is, the coordinate (x', y') of the output image will have the intensity value of the coordinate (x, y) in the input image. The geometric transformation is given by the equation in matrix form:

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A transformation matrix specifically describes a set of mathematical values that when incorporated with the coordinates of the pixels when implementing an image reshaping process, it achieves the desired result. It is notable that different matrices call for different transformations like translation, rotation, and scale.

Lamidi's and Eakins (2003) and Robinson and Sinton (2004) were identified major issues which includes computational complexity, prediction errors caused by interference, precision issues, which impede the image manipulation and analysis.

Geometric transformation is a mathematical tool used in image processing. Geometric transformation is used in an image to adjust its spatial arrangement of pixel. For example, in operations like scaling, rotation, and translation where pixel coordinates are changed, we use geometric transformations. Geometric transformations in image processing are also called rubber-sheet transformations. This is due to the fact that geometric transformation in image processing is equivalent to the operations performed on a rubber sheet containing the input image. The sheet can be stretched, sheared, rotated, and so on. When these operations are done, the image which is visible on the sheet is the output image.

Parallax shifts the appearing image perspective until the parallel lines appear to be meeting at the vanishing points. Perspective transformation is also known as projective transformation and homograph. It is a geometric transformation where a point from one plane is mapped to another plane. This makes the object appear from different points of views or perspectives. Perspective transformation has application in the field of computer vision as it is involved in tasks like image stitching, camera calibration and 3-D reconstruction.

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Image rotations and transformations are among the core facets of images as they provide generalized methods of managing and analyzing the images. They can also be easily and accurately rotated and scaled and reshaped and translated which make them useful in applications in areas such as medical imaging and remote sensing and computer vision. Although there may be some disadvantages such as computation and interpolation issues, their ability to improve image quality and to help with certain analyses outweighs the pros. Sustaining these transformations provides practitioners with the appropriate technology for formulating radical solutions and improvements in the image processing techniques.

Geometric transformation is the modification of an image by applying geometric transformations that include movement, rotation, scaling, and skewing with intent of changing the place, direction, size or form of the image without any regard on the content of the image.

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To find x' and y', we need to normalize them with respect to w'. Thus the coordinate equations for perspective transformation are:

All the affine transformations except translation, can be represented using a 2 X 2 matrix. But for representing translation, we require a 3 X 3 matrix. Hence, a 3 X 3 matrix, A (affine matrix) is used for affine transformation. An affine matrix has an important characteristic of being invertible. The affine transformation can also be expressed using linear equations of matrix and vectors of order 2. The affine transformation is given by the equations in matrix form:

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Geometric transformations are applied in image registration, object recognition, computer vision, medical imaging, remote sensing, robotics, augmented reality, cartography, digital art and video processing.

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\begin{bmatrix} \\x' \\ y' \\w' \end{bmatrix}=H\begin{bmatrix} \\x \\y \\1 \end{bmatrix}=\begin{bmatrix} a &b &c \\ d &e &f \\ g &h &i \end{bmatrix}\begin{bmatrix} \\x \\y \\1 \end{bmatrix}

Affine transformation can be defined as translation, rotation, scaling, and shearing at once. It maintains the ‘perpendicularity' of the pairs of lines and the ratio between points but not angles or lengths. The commonly used and known geometric transformation is the affine transformation. Affine transformation includes scaling, rotation, translation and shearing. In two dimensions, these transformations preserve points, straight lines and planes. This is the key characteristic of affine transformation. That is, the collinearity between points, parallelism between lines, and convexity of planes are not affected due to affine transformations.

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The perspective transformation is mathematically represented using a 3 X 3 matrix, H (homograph matrix). The affine transformation is given by the equations in matrix form:

\begin{bmatrix} x' \\ y' \\ w' \end{bmatrix} = \begin{bmatrix} h_{11} & h_{12} & h_{13} \\ h_{21} & h_{22} & h_{23} \\ h_{31} & h_{32} & h_{33} \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}