<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>

CC BY-SA 4.0
2021-12-25T09:54:19.807Z
2021-12-30T10:03:31.427Z
2021-12-30T10:03:31.427Z
nbro
<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>

CC BY-SA 4.0
2021-12-24T22:09:41.427Z
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.
nbro
<computer-vision><image-processing><affine-transformations><image-transformations><projective-transformations>
What is the expression for projective transformation?
hanugm
# 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.  

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## 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.


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