Perspective-n-Point

Key takeaways

Perspective-n-Point belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Perspective-n-Point to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Perspective-n-Point from memory before moving on to harder problems.

Reference excerpt

Perspective-n-Point (PnP) is the problem of estimating the pose of a camera given a set of n 3D points in the world and their corresponding 2D projections in an image. The camera pose consists of 6 degrees of freedom (DOF) which are made up of the rotation (roll, pitch, and yaw) and 3D translation of the camera with respect to an origin point in the world. This problem originates from camera calibration and has many applications in computer vision and other areas, including 3D tracking, robotics and augmented reality. A commonly used solution to the problem exists for n = 3 called P3P, and many solutions are available for the general case of n ≥ 3. A solution for n = 2 exists if feature orientations are available at the two points. Implementations of these solutions are available in open source software.

Problem specification

Definition Given a set of n 3D points in a world reference frame and their corresponding 2D image projections as well as the calibrated intrinsic camera parameters, determine the 6 DOF pose of the camera in the form of its rotation and translation with respect to the world. This follows the perspective projection model for cameras:

s p c = K [ R | T ] p w {\displaystyle s\,p_{c}=K\,[\,R\,|\,T\,]\,p_{w}} . where p w = [ x y z 1 ] T {\displaystyle \textstyle p_{w}={\begin{bmatrix}x&y&z&1\end{bmatrix}}^{T}} is the homogeneous world point, p c = [ u v 1 ] T {\displaystyle \textstyle p_{c}={\begin{bmatrix}u&v&1\end{bmatrix}}^{T}} is the corresponding homogeneous image point, K {\displaystyle \textstyle K} is the matrix of intrinsic camera parameters, (where f x {\displaystyle \textstyle f_{x}} and f y {\displaystyle f_{y}} are the scaled focal lengths, γ {\displaystyle \textstyle \gamma } is the skew parameter which is sometimes assumed to be 0, and ( u 0 , v 0 ) {\displaystyle \textstyle (u_{0},\,v_{0})} is the principal point), s {\displaystyle \textstyle s} is a scale factor for the image point, and R {\displaystyle \textstyle R} and T {\displaystyle \textstyle T} are the desired 3D rotation and 3D translation of the camera (extrinsic parameters) that are being calculated. This leads to the following equation for the model:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Perspective-n-Point

Start with the simplest possible case. Write down what Perspective-n-Point claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Perspective-n-Point before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Perspective-n-Point ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Perspective-n-Point

In research: Perspective-n-Point appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Perspective-n-Point in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Perspective-n-Point is common in secondary-school and first-year university syllabi. It links to neighbouring topics Photogrammetry, Science of photography, so understanding it makes those chapters shorter.
In everyday life: Look for Perspective-n-Point outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Perspective-n-Point in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Perspective-n-Point means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Perspective-n-Point out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Perspective-n-Point in simple terms?

Perspective-n-Point (PnP) is the problem of estimating the pose of a camera given a set of n 3D points in the world and their corresponding 2D projections in an image. The camera pose consists of 6 degrees of freedom (DOF) which are made up of the rotation (roll, pitch, and yaw) and 3D translation…

Why does Perspective-n-Point matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Perspective-n-Point?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Perspective-n-Point.