# estimateFundamentalMatrix

Estimate fundamental matrix from corresponding points in stereo images

## Description

example

estimateFundamentalMatrix estimates the fundamental matrix from corresponding points in stereo images. This function can be configured to use all corresponding points or to exclude outliers. You can exclude outliers by using a robust estimation technique such as random-sample consensus (RANSAC). When you use robust estimation, results may not be identical between runs because of the randomized nature of the algorithm.

example

F = estimateFundamentalMatrix(matchedPoints1,matchedPoints2) returns the 3-by-3 fundamental matrix, F, using the least median of squares (LMedS) method. The input points can be M-by-2 matrices of M number of [x y] coordinates, or KAZEPoints, SIFTPoints, SURFPoints, MSERRegions, ORBPoints, or cornerPoints object.

[F,inliersIndex] = estimateFundamentalMatrix(matchedPoints1,matchedPoints2) additionally returns logical indices, inliersIndex, for the inliers used to compute the fundamental matrix. The inliersIndex output is an M-by-1 vector. The function sets the elements of the vector to true when the corresponding point was used to compute the fundamental matrix. The elements are set to false if they are not used.

[F,inliersIndex,status] = estimateFundamentalMatrix(matchedPoints1,matchedPoints2) additionally returns a status code.

[F,inliersIndex,status] = estimateFundamentalMatrix(matchedPoints1,matchedPoints2,Name=Value) specifies options using one or more name-value arguments in addition to any combination of arguments from previous syntaxes. For example, estimateFundamentalMatrix(matchedPoints1,matchedPoints2,Method="MSAC") specifies MSAC as the method to compute the fundamental matrix.

## Examples

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Use the random sample consensus (RANSAC) method to compute the fundamental matrix. The RANSAC method requires that the input points are putatively matched. You can use the matchFeatures function to return these matched points. Outlier points which may still be contained within putatively matched points are further eliminated by using the RANSAC algorithm.

Load stereo points into the workspace.

Estimate the fundamental matrix.

fRANSAC = estimateFundamentalMatrix(matchedPoints1, ...
matchedPoints2,Method="RANSAC", ...
NumTrials=2000,DistanceThreshold=1e-4)
fRANSAC = 3×3

0.0000   -0.0004    0.0348
0.0004    0.0000   -0.0937
-0.0426    0.0993    0.9892

Load the putatively matched points into the workspace.

[fLMedS,inliers] = estimateFundamentalMatrix(matchedPoints1,matchedPoints2,NumTrials=2000)
fLMedS = 3×3

0.0000   -0.0004    0.0349
0.0004    0.0000   -0.0938
-0.0426    0.0994    0.9892

inliers = 18x1 logical array

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Show the putatively matched points.

figure;
showMatchedFeatures(I1,I2,matchedPoints1,matchedPoints2,'montage','PlotOptions',{'ro','go','y--'});
title('Putative Point Matches');

Show the inlier points.

figure;
showMatchedFeatures(I1,I2,matchedPoints1(inliers,:),matchedPoints2(inliers,:),'montage','PlotOptions',{'ro','go','y--'});
title('Point Matches After Outliers Are Removed');

Load the stereo point pairs into the workspace.

Compute the fundamental matrix for input points which do not contain any outliers.

inlierPts1 = matchedPoints1(knownInliers,:);
inlierPts2 = matchedPoints2(knownInliers,:);
fNorm8Point = estimateFundamentalMatrix(inlierPts1,inlierPts2,Method="Norm8Point")
fNorm8Point = 3×3

0.0000   -0.0004    0.0348
0.0004    0.0000   -0.0937
-0.0426    0.0993    0.9892

## Input Arguments

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Coordinates of corresponding points in image one, specified as an M-by-2 matrix of M number of [x y] coordinates, or as a KAZEPoints, SIFTPoints, SURFPoints, MSERRegions, ORBPoints, or cornerPoints object. The matchedPoints1 input must contain points which do not lie on a single planar surface, (e.g., a wall, table, or book) and are putatively matched by using a function such as matchFeatures.

Coordinates of corresponding points in image two, specified as an M-by-2 matrix of M number of [x y] coordinates, or as a KAZEPoints, SIFTPoints, SURFPoints, MSERRegions, ORBPoints, or cornerPoints object. The matchedPoints1 input must contain points which do not lie on a single planar surface, (e.g., a wall, table, or book) and are putatively matched by using a function such as matchFeatures.

### Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: estimateFundamentalMatrix(matchedPoints1,matchedPoints2,Method="MSAC") specifies MSAC as the method to compute the fundamental matrix.

Method used to compute the fundamental matrix, specified as one of the following:

 "LMedS" Least Median of Squares. Use this method if at least 50% of the points in matchedPoints1 and matchedPoints2 are inliers. The function stops the search for inliers and the estimation of the fundamental matrix after 50% of the inliers are found. "MSAC" M-estimator SAmple Consensus. Select the M-estimator SAmple Consensus method if you would like to set the distance threshold for the inliers. Generally, the MSAC method converges more quickly than the RANSAC method. "Norm8Point" Normalized eight-point algorithm [1] . To produce reliable results, the inputs, matchedPoints1 and matchedPoints2 must match precisely.

To produce reliable results using the Norm8Point algorithm, the inputs, matchedPoints1 and matchedPoints2, must match precisely. The other methods can tolerate outliers and therefore only require putatively matched input points. You can obtain putatively matched points by using the matchFeatures function.

Fundamental matrix class, specified as "double" or "single". Use this property to specify the class for the fundamental matrix and for the function's internal computations.

Number of random trials for finding the outliers, specified as an integer. This argument applies when you set the Method to LMedS or MSAC.

• LMedS — The function uses the actual number of trials as the parameter value.

• MSAC — The function uses the maximum number of trials as the parameter value.

The actual number of trials depends on matchedPoints1, matchedPoints2, and the value of the Confidence parameter. Select the number of random trials to optimize speed and accuracy.

Distance threshold for finding outliers, specified as a positive scalar. This parameter applies when you set the Method to MSAC.

Desired confidence for finding maximum number of inliers, specified as a percentage scalar in the range (0 100). This argument applies when you set Method to MSAC.

Report runtime error, specified as a logical value. Set this parameter to true to report run-time errors when the function cannot compute the fundamental matrix from matchedPoints1 and matchedPoints2. When you set this parameter to false, you can check the status output to verify validity of the fundamental matrix.

## Output Arguments

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Fundamental matrix, returned as a 3-by-3 matrix that is computed from the points in the inputs matchedPoints1 and matchedPoints2.

$\left[\begin{array}{cc}{P}_{2}& 1\end{array}\right]*FundamentalMatrix*{\left[\begin{array}{cc}{P}_{1}& 1\end{array}\right]}^{\text{'}}=0$

P1, the point in matchedPoints1 of image 1 in pixels, corresponds to the point, P2, the point in matchedPoints2 in image 2.

In computer vision, the fundamental matrix is a 3-by-3 matrix which relates corresponding points in stereo images. When two cameras view a 3-D scene from two distinct positions, there are a number of geometric relations between the 3-D points and their projections onto the 2-D images that lead to constraints between the image points. Two images of the same scene are related by epipolar geometry.

Inliers index, returned as an M-by-1 logical index vector. An element set to true means that the corresponding indexed matched points in matchedPoints1 and matchedPoints2 were used to compute the fundamental matrix. An element set to false means the indexed points were not used for the computation.

Data Types: logical

Status code, returned as one of the following possible values:

statusValue
0:No error.
1:matchedPoints1 and matchedPoints2 do not contain enough points. The Norm8Point and MSAC methods require at least 8 points, and the LMedS method requires 16 points.
2:Not enough inliers found.

Data Types: int32

## Tips

Use estimateEssentialMatrix when you know the camera intrinsics. You can obtain the intrinsics using the Camera Calibrator app. Otherwise, you can use the estimateFundamentalMatrix function that does not require camera intrinsics. Note that the fundamental matrix cannot be estimated from coplanar world points.

## Algorithms

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### Computing the Fundamental Matrix

When you choose the Norm8Point method, the function uses all points in matchedPoints1 and matchedPoints2 to compute the fundamental matrix. When you choose any other method, the function uses the following algorithm to exclude outliers and compute the fundamental matrix from inliers:

1. Initialize the fundamental matrix, F, to a 3-by-3 matrix of zeros.

2. Set the loop counter n, to zero, and the number of loops N, to the number of random trials specified.

3. Loop through the following steps while n < N:

1. Randomly select 8 pairs of points from matchedPoints1 and matchedPoints2.

2. Use the selected 8 points to compute a fundamental matrix, f, by using the normalized 8-point algorithm.

3. Compute the fitness of f for all points in matchedPoints1 and matchedPoints2.

4. If the fitness of f is better than F, replace F with f.

For the MSAC method, update N.

5. n = n + 1

### Number of Random Samplings for the MSAC Method

The MSAC method updates the number of random trials N for every iteration in the algorithm loop. The function resets N, according to the following:

 N = min( N, $\frac{\mathrm{log}\left(1-p\right)}{\mathrm{log}\left(1-{r}^{8}\right)}$). Where, p represents the confidence parameter you specified, and r is calculated as follows: $\sum _{i}^{N}\mathrm{sgn}\left(d{u}_{i,}{v}_{i}\right),t\right)/N$, where $\mathrm{sgn}\left(a,b\right)=1$ if $a\le b$ and $0$ otherwise.
When you use the MSAC method, results may not be identical between runs because of the randomized nature of the algorithm.

### Sampson Distance

The function uses the Sampson distance to measure the distance of a pair of points according to a fundamental matrix. The equation below can be used to calculate the Sampson distance. In the equation, u represents matchedPoints1 and v represents matchedPoints2.

$d\left({u}_{i},{v}_{i}\right)={\left({v}_{i}F{u}_{i}^{T}\right)}^{2}\left[\frac{1}{{\left(F{u}_{i}^{T}\right)}_{1}^{2}+{\left(F{u}_{i}^{T}\right)}_{2}^{2}}+\frac{1}{{\left({v}_{i}F\right)}_{1}^{2}+{\left({v}_{i}F\right)}_{2}^{2}}\right]$

where $i$ represents the index of the corresponding points, and ${\left(F{u}_{i}^{T}\right)}_{j}^{2}$, the square of the j-th entry of the vector $F{u}_{i}^{T}$.

### Fitness of Fundamental Matrix for Corresponding Points

The following table summarizes how each method determines the fitness of the computed fundamental matrix:

MethodMeasure of Fitness
LMedS$median\left(d\left({u}_{i,}{v}_{i}\right);i=1:N\right)$, the number of input points. The smaller the value, the better the fitness.
MSAC$\sum _{i}^{N}\mathrm{min}\left(d\left({u}_{i},{v}_{i}\right),t\right)$. The smaller the value, the better the fitness.

## References

[1] Hartley, R., A. Zisserman, Multiple View Geometry in Computer Vision, Cambridge University Press, 2003.

[2] Rousseeuw, P., A. Leroy, Robust Regression and Outlier Detection, John Wiley & Sons, 1987.

[3] Torr, P. H. S., and A. Zisserman, MLESAC: A New Robust Estimator with Application to Estimating Image Geometry, Computer Vision and Image Understanding, 2000.

## Version History

Introduced in R2012b