Answers.Body: <p>I explain in <a href="https://ai.stackexchange.com/a/21048/2444">this answer</a> what a projective transformation (aka projectivity or homography) is. It's a function <span class="math-container">$h$</span> of the form <span class="math-container">$$h: \mathbb{P}^2 \rightarrow \mathbb{P}^2,$$</span> where <span class="math-container">$\mathbb{P}^2$</span> is a <strong>projective space</strong>, so, essentially, a 3-dimensional Euclidean space of <strong>homogenous vectors</strong>.</p>
  <p>You can also represent a homography as a <span class="math-container">$3 \times 3$</span> matrix <span class="math-container">$\mathbf{H}$</span>, so that, when we apply this projective transformation to some input <span class="math-container">$\mathbf{x} \in \mathbb{P}^2$</span>, we get <span class="math-container">$\mathbf{x}' \in \mathbb{P}^2$</span>, so we can represent a projective transformation as follows.</p>
  <p><span class="math-container">$$\mathbf{H}\mathbf{x} = \mathbf{x}'$$</span></p>
  <p>So, basically, a projective transformation is a linear transformation between projective spaces.</p>
  <p>You can generalize these ideas to higher-dimensional projective spaces, i.e. <span class="math-container">$\mathbb{P}^n$</span>.</p>
  <p>Although you can represent a projective transformation as a matrix multiplication, there's more to it. In fact, it's a linear transformation, with 8 degrees of freedom, between projective spaces. You can also view a homography as a generalization of other transformations, like <strong>isometries</strong>, <strong>similarities</strong>, and <strong>affinities</strong>. This is explained more in detail in chapter 2 of the book <a href="http://www.r-5.org/files/books/computers/algo-list/image-processing/vision/Richard_Hartley_Andrew_Zisserman-Multiple_View_Geometry_in_Computer_Vision-EN.pdf#page=42" rel="nofollow noreferrer">Multiple View Geometry in Computer Vision</a> (2nd edition) Richard Hartley and Andrew Zisserman.</p>
  
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Answers.CreationDate: 2021-12-25T09:54:19.807Z
Answers.LastActivityDate: 2021-12-30T10:03:31.427Z
Answers.LastEditDate: 2021-12-30T10:03:31.427Z
Answers.UserDisplayName: nbro
Body: <p>The following are the two types are projections that are generally used in image processing</p>
  <ol>
  <li>Affine transformation</li>
  <li>Projective transformation</li>
  </ol>
  <p>Affine transformation is a backbone operation in neural networks also. It is expressed as</p>
  <p><span class="math-container">$$\mathbf{wx+b}$$</span>
  where <span class="math-container">$\mathbf{w, x, b}$</span> are matrices. In general, <span class="math-container">$\mathbf{x}$</span> is treated as an image in image processing.</p>
  <p>Projective transformation is also a type of transformation on images and it may be different from affine transformation. I want to know whether it can be represented in terms of mathematical expression.</p>
  <p>If yes, what is the expression for projective transformation?</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2021-12-24T22:09:41.427Z
Comments.Text: What do you mean by "projective transformations"? Are you talking about [homographies](https://ai.stackexchange.com/q/21042/2444)? Maybe you could provide a little bit more context. Projective transformations definitely appear in the context of computer vision and computer graphics, but if you provide more context, it may be better. Anyway, maybe later I provide an answer.
Comments.UserDisplayName: nbro
Tags: <computer-vision><image-processing><affine-transformations><image-transformations><projective-transformations>
Title: What is the expression for projective transformation?
UserDisplayName: hanugm
ThreadText: # What is the expression for projective transformation?
  
  (Asked by: username_0 on 2021-12-24)
  
  The following are the two types are projections that are generally used in image processing
  
  1. Affine transformation
  2. Projective transformation
  
  Affine transformation is a backbone operation in neural networks also. It is expressed as
  
  $$\mathbf{wx+b}$$
  where $\mathbf{w, x, b}$ are matrices. In general, $\mathbf{x}$ is treated as an image in image processing.
  
  Projective transformation is also a type of transformation on images and it may be different from affine transformation. I want to know whether it can be represented in terms of mathematical expression.
  
  If yes, what is the expression for projective transformation?
  
  **Tags:** computer-vision, image-processing, affine-transformations, image-transformations, projective-transformations
  
  > **Comments:**
  > **username_1**: What do you mean by "projective transformations"? Are you talking about [homographies](https://ai.stackexchange.com/q/21042/2444)? Maybe you could provide a little bit more context. Projective transformations definitely appear in the context of computer vision and computer graphics, but if you provide more context, it may be better. Anyway, maybe later I provide an answer.  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_1 on 2021-12-25. Score: 1
  
  I explain in [this answer](https://ai.stackexchange.com/a/21048/2444) what a projective transformation (aka projectivity or homography) is. It's a function $h$ of the form $$h: \mathbb{P}^2 \rightarrow \mathbb{P}^2,$$ where $\mathbb{P}^2$ is a **projective space**, so, essentially, a 3-dimensional Euclidean space of **homogenous vectors**.
  
  You can also represent a homography as a $3 \times 3$ matrix $\mathbf{H}$, so that, when we apply this projective transformation to some input $\mathbf{x} \in \mathbb{P}^2$, we get $\mathbf{x}' \in \mathbb{P}^2$, so we can represent a projective transformation as follows.
  
  $$\mathbf{H}\mathbf{x} = \mathbf{x}'$$
  
  So, basically, a projective transformation is a linear transformation between projective spaces.
  
  You can generalize these ideas to higher-dimensional projective spaces, i.e. $\mathbb{P}^n$.
  
  Although you can represent a projective transformation as a matrix multiplication, there's more to it. In fact, it's a linear transformation, with 8 degrees of freedom, between projective spaces. You can also view a homography as a generalization of other transformations, like **isometries**, **similarities**, and **affinities**. This is explained more in detail in chapter 2 of the book [Multiple View Geometry in Computer Vision](http://www.r-5.org/files/books/computers/algo-list/image-processing/vision/Richard_Hartley_Andrew_Zisserman-Multiple_View_Geometry_in_Computer_Vision-EN.pdf#page=42) (2nd edition) Richard Hartley and Andrew Zisserman.
  
  
  ---
  
  