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三维重构

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opencv3中关于三维重建的经典教程。Packt.OpenCV.3.Computer.Vision.Application.Programming.Cookbook.3rd.Edition.2017.对其单独截取部分

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Chapter 11. Reconstructing 3D

Scenes

In this chapter, we will cover the following recipes:

Calibrating a camera

Recovering camera pose

Reconstructing a 3D scene from calibrated cameras

Computing depth from stereo image

Introduction

We learned in the previous chapter how a camera captures a 3D scene

by projecting light rays on a 2D sensor plane. The image produced is an

accurate representation of what the scene looks like from a particular

point of view, at the instant the image was captured. However, by its

nature, the process of image formation eliminates all information

concerning the depth of the represented scene elements. This chapter

will teach how, under specific conditions, the 3D structure of the scene

and the 3D pose of the cameras that captured it, can be recovered. We

will see how a good understanding of projective geometry concepts

allows us to devise methods that enable 3D reconstruction. We will

therefore revisit the principle of image formation introduced in the

previous chapter; in particular, we will now take into consideration that

our image is composed of pixels.

Digital image formation

Let's now redraw a new version of the figure shown in Chapter 10 ,

Estimating Projective Relations in Images, describing the pin-hole

camera model. More specifically, we want to demonstrate the relation

between a point in 3D at position (X,Y,Z) and its image (x,y) on a

camera specified in pixel coordinates:

Notice the changes that have been made to the original figure. First, we

added a reference frame that we positioned at the center of the

projection. Second, we have the Y-axis pointing downward to get a

coordinate system compatible with the usual convention that places the

image origin in the upper-left corner of the image. Finally, we also

identified a special point on the image plane: considering the line coming

from the focal point that is orthogonal to the image plane, the point

(u0,v0) is the pixel position at which this line pierces the image plane.

This point is called the principal point. It could be logical to assume

that this principal point is at the center of the image plane, but in

practice, this one might be off by a few pixels, depending on the

precision with which the camera has been manufactured.

In the previous chapter, we learned that the essential parameters of a

camera in the pin-hole model are its focal length and the size of the

image plane (which defines the field of view of the camera). In addition,

since we are dealing with digital images, the number of pixels on the

image plane (its resolution) is another important characteristic of a

camera. We also learned previously that a 3D point (X,Y,Z) will be

projected onto the image plane at (fX/Z,fY/Z).

Now, if we want to translate this coordinate into pixels, we need to

divide the 2D image position by the pixel width (px) and height (py),

respectively. We notice that by dividing the focal length given in world

units (generally given in millimeters) by px, we obtain the focal length

expressed in (horizontal) pixels. Let's define this term, then, as fx.

Similarly, fy =f/py is defined as the focal length expressed in vertical

pixel units. The complete projective equation is therefore as follows:

Recall that (u0,v0) is the principal point that is added to the result in

order to move the origin to the upper-left corner of the image. Note also

that the physical size of a pixel can be obtained by dividing the size of

the image sensor (generally in millimeters) by the number of pixels

(horizontally or vertically). In modern sensors, pixels are generally

square, that is, they have the same horizontal and vertical size.

The preceding equations can be rewritten in matrix form as we did in

Chapter 10 , Estimating Projective Relations in Images. Here is the

complete projective equation in its most general form:

Calibrating a camera

Camera calibration is the process by which the different camera

parameters (that is, the ones appearing in the projective equation) are

obtained. One can obviously use the specifications provided by the

camera manufacturer, but for some tasks, such as 3D reconstruction,

these specifications are not accurate enough. By undertaking an

appropriate camera calibration step, accurate calibration information

can be obtained.

An active camera calibration procedure will proceed by showing known

patterns to the camera and analyzing the obtained images. An

optimization process will then determine the optimal parameter values

that explain the observations. This is a complex process that has been

made easy by the availability of OpenCV calibration functions.

How to do it...

To calibrate a camera, the idea is to show it a set of scene points for

which their 3D positions are known. Then, you need to observe where

these points project on the image. With the knowledge of a sufficient

number of 3D points and associated 2D image points, the exact camera

parameters can be inferred from the projective equation. Obviously, for

accurate results, we need to observe as many points as possible. One

way to achieve this would be to take one picture of a scene with many

known 3D points, but in practice, this is rarely feasible. A more

convenient way is to take several images of a set of some 3D points

from different viewpoints. This approach is simpler but requires you to

compute the position of each camera view in addition to the

computation of the internal camera parameters, which, fortunately, is

feasible.

OpenCV proposes that you use a chessboard pattern to generate the set

of 3D scene points required for calibration. This pattern creates points at

the corners of each square, and since this pattern is flat, we can freely

assume that the board is located at Z=0, with the X and Y axes well-

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