Estimate Geometric Transformation

Estimate geometric transformation from matching point pairs

• 

Libraries:
Computer Vision Toolbox / Geometric Transformations

Description

Use the Estimate Geometric Transformation block to find the transformation matrix which maps the greatest number of point pairs between two images. A point pair refers to a point in the input image and its related point on the image created using the transformation matrix. You can select to use the RANdom SAmple Consensus (RANSAC) or the Least Median Squares algorithm to exclude outliers and to calculate the transformation matrix. You can also use all input points to calculate the transformation matrix.

Examples

Calculate transformation matrix from largest group of point pairs

1. Examples of input data and application of the Estimate Geometric Transformation block appear in the following figures. Figures (a) and (b) show the point pairs. The points are denoted by stars or circles, and the numbers following them show how they are paired. Some point pairs can be mapped by the same transformation matrix. Other point pairs require a different transformation matrix. One matrix exists that maps the largest number of point pairs, the block calculates and returns this matrix. The block finds the point pairs in the largest group and uses them to calculate the transformation matrix. The point pairs connected by the magenta lines are the largest group.

2. The transformation matrix can then be used to stitch the images as shown in Figure (e).

   

Extended Examples

Apply Affine Transformation to Image

Perform a horizontal shear transformation of an image by calculating an affine transformation matrix.

• Open Script

Video Mosaicking

Create a mosaic from a video sequence. Video mosaicking is the process of stitching video frames together to form a comprehensive view of the scene. The resulting mosaic image is a compact representation of the video data. The Video Mosaicking block is often used in video compression and surveillance applications.

• Open Model

Ports

Input

Pts1 — Point coordinates

M-by-2 matrix

Point coordinates, specified as an M-by-2 matrix of one-based [x y] point coordinates, where M represents the number of points.

The block outputs the same data type for the transformation matrix as the Pts1 and Pts2 image points.

Data Types: single | double | int8 | int16 | int32 | uint8 | uint16 | uint32

Pts2 — Point coordinates

M-by-2 matrix

Point coordinates, specified as an M-by-2 matrix of one-based [x y] point coordinates, where M represents the number of points.

The block outputs the same data type for the transformation matrix as the Pts1 and Pts2 image points.

Data Types: single | double | int8 | int16 | int32 | uint8 | uint16 | uint32

Num — Number of valid points

scalar

Number of valid points to find in Pts1 and Pts2, specified as a scalar. This port appears when you enable the Allow variable-size signal input parameter.

Data Types: int8 | int16 | int32 | uint8 | uint16 | uint32

Output

TForm — Transformation

3-by-2 matrix | 3-by-3 matrix

Transformation, returned as either a 3-by-2 or a 3-by-3 matrix. The block outputs the same data type for the transformation matrix as the Pts1 and Pts2 image points.

Dependencies

• When Pts1 and Pts2 are single or double, the output transformation matrix will also have single or double data type.

• When Pts1 and Pts2 images are built-in integers, the option is available to set the transformation matrix data type to either Single or Double.

Data Types: single | double

Inlier — Points used

M-by-1 vector

Points used to calculate TForm, returned as an M-by-1 vector.

Dependencies

• The Inlier port appears when you enable the Output Boolean signal indicating which point pairs are inliers parameter.

Data Types: Boolean

Parameters

Transformation type — Transformation type

Affine (default) | Nonreflective similarity | Projective

Specify the transformation type as either Nonreflective similarity, Affine, or Projective. See Transformations for a more detailed discussion.

Dependencies

You can set additional parameters depending on the transformation type:

• For Projective transformation, you can specify a scalar algebraic distance threshold for determining inliers.

• For Affine or Nonreflective similarity transformation, you can specify the distance threshold for determining inliers in pixels.

Find and exclude outliers — Find and exclude outliers

on (default) | off

Enable to find and exclude outliers from the input points and use only the inlier points to calculate the transformation matrix. When you turn this parameter off, all input points are used to calculate the transformation matrix.

Method — Method to find outliers

RANdom SAmple Consensus (RANSAC) (default) | Least Median of Squares

Select the method to find outliers as either RANdom SAmple Consensus (RANSAC) or Least Median of Squares See RANSAC and Least Median Squares Algorithms for a more detailed discussion.

Dependencies

This parameter appears when you enable the Find and exclude outliers check box.

Algebraic distance threshold for determining inliers — Algebraic distance threshold for determining inliers

1.5 (default) | scalar

Specify a scalar threshold value for determining inliers. The threshold controls the upper limit used to find the algebraic distance in the RANSAC algorithm.

Dependencies

This parameter appears when you set the Method parameter to Random Sample Consensus (RANSAC) and the Transformation type parameter to Projective.

Distance threshold for determining inliers (in pixels) — Distance threshold for determining inliers (in pixels)

1.5 (default) | scalar

Specify the upper limit distance a point can differ from the projection location of its corresponding point.

Dependencies

This parameter appears when you set the Method parameter to Random Sample Consensus (RANSAC) and you set the value of the Transformation type parameter to Nonreflective similarity or Affine.

Determine number of random samplings using — Determine number of random samplings using

Specified value (default) | Desired confidence

Select Specified value to enter a positive integer value for the number of random samplings. Select Desired confidence to set the number of random samplings as a percentage and a maximum number.

Dependencies

This parameter appears when you select the Find and exclude outliers check box, and you set the value of the Method parameter to Random Sample Consensus (RANSAC).

Number of random samplings — Number of random samplings

500 (default) | scalar

Specify the number of random samplings for the algorithm to perform.

Dependencies

This parameter appears when you set the value of the Determine number of random samplings using parameter to Specified value.

Desired confidence (in %) — Desired confidence (in %)

99 (default) | scalar

Specify a percent desired confidence by entering a number between 0 and 100. The value represents the probability of the algorithm to find the largest group of points that can be mapped by a transformation matrix.

Dependencies

This parameter appears when you set the Determine number of random samplings using parameter to Desired confidence.

Maximum number of random samplings — Maximum number of random samplings

1000 (default) | integer

Specify an integer number for the maximum number of random samplings.

Dependencies

This parameter appears when you set the Method parameter to Random Sample Consensus (RANSAC) and you set the value of the Determine number of random samplings using parameter to Desired confidence.

Stop sampling earlier when a specified percentage of point pairs are determined to be inlier — Stop sampling

off (default) | on

Enable this parameter to stop random sampling when a percentage of input points have been found as inliers.

Dependencies

This parameter appears when you set the Method parameter to Random Sample Consensus (RANSAC).

Perform additional iterative refinement of the transformation matrix — Perform additional iterative refinement

off (default) | on

Specify whether to perform refinement on the transformation matrix.

Dependencies

This parameter appears when you select Find and exclude outliers check box.

Output Boolean signal indicating which point pairs are inliers — Output Boolean signal

off (default) | on

Enable this parameter to output the inlier point pairs that were used to calculate the transformation matrix.

Dependencies

• This parameter appears when you select the Find and exclude outliers check box.

• The block will not use this parameter with signed or double, data type points.

When Pts1 and Pts2 are built-in integers, set transformation matrix date type to — Set transformation matrix date type

Single (default) | Double

Specify transformation matrix data type as Single or Double when the input points are built-in integers.

Dependencies

The block will not use this parameter with signed or double, data type points.

Allow variable-sized signal input — Allow variable-sized signal input

on (default) | off

Enable this parameter to allow variable-sized signal input.

Dependencies

Block Characteristics

Data Types	double | integera | single
Multidimensional Signals	no
Variable-Size Signals	yes
a Generated code will be restricted to MATLAB host computers when you set the FFT implementation parameter to FFTW, or when the transform length is not a power of two.

Tips

The success of estimating the correct geometric transformation depends heavily on the quality of the input point pairs. If you chose the RANSAC or LMS algorithm, the block will randomly select point pairs to compute the transformation matrix and will use the transformation that best fits the input points. There is a chance that all of the randomly selected point pairs may contain outliers despite repeated samplings. In this case, the output transformation matrix, TForm, is invalid, indicated by a matrix of zeros.

To improve your results, try the following:

Increase the percentage of inliers in the input points.

Increase the number for random samplings.

For the RANSAC method, increase the desired confidence.

For the LMS method, make sure the input points have 50% or more inliers.

Use features appropriate for the image contents

Be aware that repeated patterns, for example, windows in office building, will cause false matches when you match the features. This increases the number of outliers.

Do not use this function if the images have significant parallax. You can use the estimateFundamentalMatrix function instead.

Choose the minimum transformation for your problem.

If a projective transformation produces the error message, “A portion of the input image was transformed to the location at infinity. Only transformation matrices that do not transform any part of the image to infinity are supported.”, it is usually caused by a transformation matrix and an image that would result in an output distortion that does not fit physical reality. If the matrix was an output of the Estimate Geometric Transformation block, then most likely it could not find enough inliers.

Algorithms

RANSAC and Least Median Squares Algorithms

The RANSAC algorithm relies on a distance threshold. A pair of points, $$p_i^a$$(image a, Pts1) and $$p_i^b$$(image b, Pts2) is an inlier only when the distance between $$p_i^b$$ and the projection of $$p_i^a$$based on the transformation matrix falls within the specified threshold. The distance metric used in the RANSAC algorithm is as follows:

$$d={\displaystyle \sum _{i=1}^{Num}\min (D(p_i^b,}\psi (p_i^a:H)),t)$$

The Least Median Squares algorithm assumes at least 50% of the point pairs can be mapped by a transformation matrix. The algorithm does not need to explicitly specify the distance threshold. Instead, it uses the median distance between all input point pairs. The distance metric used in the Least Median of Squares algorithm is as follows:

$$d=median(D(p_1^b,\psi (p_1^a:H)),D(p_2^b,\psi (p_2^a:H)),\mathrm{...},D(p_{Num}^b,\psi (p_N^a:H)))$$

For both equations:

$$p_i^a$$ is a point in image a (Pts1)

$$p_i^b$$ is a point in image b (Pts2)

$$\psi (p_i^a:H)$$ is the projection of a point on image a based on transformation matrix H

$$D(p_i^b,p_j^b)$$ is the distance between two point pairs on image b

$$t$$ is the threshold

$$Num$$is the number of points

The smaller the distance metric, the better the transformation matrix and therefore the more accurate the projection image.

Transformations

The Estimate Geometric Transformation block supports Nonreflective similarity, Affine, and Projective transformation types, which are described in this section.

Nonreflective similarity transformation supports translation, rotation, and isotropic scaling. It has four degrees of freedom and requires two pairs of points.

The transformation matrix is: $$H=\left[\begin{array}{cc}{h}_1& -h_2\\ h_2& h_1\\ h_3& h_4\end{array}\right]$$

The projection of a point $${\left[\begin{array}{cc}x& y\end{array}\right]}^{}$$ by $$H$$is: $${\left[\begin{array}{cc}\widehat{x}& \widehat{y}\end{array}\right]}^{}=\left[\begin{array}{ccc}x& y& 1\end{array}\right]H$$

affine transformation supports anisotropic scaling in addition to all transformations that the nonreflective similarity transformation supports. It has six degrees of freedom that can be determined from three pairs of noncollinear points.

The transformation matrix is: $$H=\left[\begin{array}{cc}{h}_1& h_4\\ h_2& h_5\\ h_3& h_6\end{array}\right]$$

The projection of a point $${\left[\begin{array}{cc}x& y\end{array}\right]}^{}$$ by $$H$$is: $${\left[\begin{array}{cc}\widehat{x}& \widehat{y}\end{array}\right]}^{}=\left[\begin{array}{ccc}x& y& 1\end{array}\right]H$$

Projective transformation supports tilting in addition to all transformations that the affine transformation supports.

The transformation matrix is : $$h=\left[\begin{array}{ccc}{h}_1& h_4& h_7\\ h_2& h_5& h_8\\ h_3& h_6& h_9\end{array}\right]$$

The projection of a point $${\left[\begin{array}{cc}x& y\end{array}\right]}^{}$$ by $$H$$is represented by homogeneous coordinates as: $${\left[\begin{array}{ccc}\widehat{u}& \widehat{v}& \widehat{w}\end{array}\right]}^{}=\left[\begin{array}{ccc}x& y& 1\end{array}\right]H$$

Distance Measurement

For computational simplicity and efficiency, this block uses algebraic distance. The algebraic distance for a pair of points, $${\left[\begin{array}{cc}{x}^a& y^a\end{array}\right]}^T$$ on image a, and $$\left[\begin{array}{cc}{x}^b& y^b\end{array}\right]$$ on image b , according to transformation $$H,$$is defined as follows;

For projective transformation:

$$D(p_i^b,\psi (p_i^a:H))={({({\widehat{u}}^a-{\widehat{w}}^a{x}^b)}^2+{({\widehat{v}}^a-{\widehat{w}}^a{y}^b)}^2)}^{\frac{1}{2}}$$, where $$\left[\begin{array}{ccc}{\widehat{u}}^a& {\widehat{v}}^a& {\widehat{w}}^a\end{array}\right]=\left[\begin{array}{ccc}{x}^a& y^a& 1\end{array}\right]H$$

For Nonreflective similarity or affine transformation: $$D(p_i^b,\psi (p_i^a:H))={({({\widehat{x}}^a-x^b)}^2+{({\widehat{y}}^a-{\widehat{y}}^b)}^2)}^{\frac{1}{2}}$$,

where $${\left[\begin{array}{cc}{\widehat{x}}^a& {\widehat{y}}^a\end{array}\right]}^{}=\left[\begin{array}{ccc}{x}^a& y^a& 1\end{array}\right]H$$

Algorithm

The block performs a comparison and repeats it K number of times between successive transformation matrices. If you select the Find and exclude outliers option, the RANSAC and Least Median Squares (LMS) algorithms become available. These algorithms calculate and compare a distance metric. The transformation matrix that produces the smaller distance metric becomes the new transformation matrix that the next comparison uses. A final transformation matrix is resolved when either:

• K number of random samplings is performed

• The RANSAC algorithm, when enough number of inlier point pairs can be mapped, (dynamically updating K)

The Estimate Geometric Transformation algorithm follows these steps:

1. A transformation matrix $$H$$ is initialized to zeros

2. Set count = 0 (Randomly sampling).

3. While count < K , where K is total number of random samplings to perform, perform the following;

   1. Increment the count; count = count + 1.

   2. Randomly select pair of points from images a and b, (2 pairs for Nonreflective similarity, 3 pairs for affine, or 4 pairs for projective).

   3. Calculate a transformation matrix $$H$$, from the selected points.

   4. If $$H$$has a distance metric less than that of $$H$$, then replace $$H$$ with $$H$$.

      (Optional for RANSAC algorithm only)

      1. Update K dynamically.

      2. Exit out of sampling loop if enough number of point pairs can be mapped by $$H$$.

4. Use all point pairs in images a and b that can be mapped by $$H$$ to calculate a refined transformation matrix $$H$$

5. Iterative Refinement, (Optional for RANSAC and LMS algorithms)

   1. Denote all point pairs that can be mapped by $$H$$ as inliers.

   2. Use inlier point pairs to calculate a transformation matrix $$H$$.

   3. If $$H$$has a distance metric less than that of $$H$$, then replace $$H$$ with $$H$$, otherwise exit the loop.

Number of Random Samplings

The number of random samplings can be specified by the user for the RANSAC and Least Median Squares algorithms. You can use an additional option with the RANSAC algorithm, which calculates this number based on an accuracy requirement. The Desired Confidence level drives the accuracy.

The calculated number of random samplings, K used with the RANSAC algorithm, is as follows:

$$K=\frac{\log (1-p)}{\log (1-q^s)}$$

where

• p is the probability of independent point pairs belonging to the largest group that can be mapped by the same transformation. The probability is dynamically calculated based on the number of inliers found versus the total number of points. As the probability increases, the number of samplings, K, decreases.

• q is the probability of finding the largest group that can be mapped by the same transformation.

• s is equal to the value 2, 3, or 4 for Nonreflective similarity, affine, and projective transformation, respectively.

Iterative Refinement of Transformation Matrix

The transformation matrix calculated from all inliers can be used to calculate a refined transformation matrix. The refined transformation matrix is then used to find a new set of inliers. This procedure can be repeated until the transformation matrix cannot be further improved. This iterative refinement is optional.

References

[1] R. Hartley and A. Ziserman, “Multiple View Geometry in Computer Vision,” Second edition, Cambridge University Press, 2003

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Fixed-Point Conversion
Design and simulate fixed-point systems using Fixed-Point Designer™.

Version History

Introduced in R2008a