# Documentation

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# LocalWeightedMeanTransformation2D

2-D local weighted mean geometric transformation

## Description

A `LocalWeightedMeanTransformation2D` object encapsulates a 2-D local weighted mean geometric transformation.

## Creation

You can create a `LocalWeightedMeanTransformation2D` object using the following methods:

## Properties

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Dimensionality of the geometric transformation for both input and output points, specified as the value 2.

## Object Functions

 `outputLimits` Find output spatial limits given input spatial limits `transformPointsInverse` Apply inverse geometric transformation

## Examples

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Fit a local weighted mean transformation to a set of fixed and moving control points that are actually related by a global second degree polynomial transformation across the entire plane.

Set up variables.

```x = [10, 12, 17, 14, 7, 10]; y = [8, 2, 6, 10, 20, 4]; a = [1 2 3 4 5 6]; b = [2.3 3 4 5 6 7.5]; u = a(1) + a(2).*x + a(3).*y + a(4) .*x.*y + a(5).*x.^2 + a(6).*y.^2; v = b(1) + b(2).*x + b(3).*y + b(4) .*x.*y + b(5).*x.^2 + b(6).*y.^2; movingPoints = [u',v']; fixedPoints = [x',y'];```

Fit local weighted mean transformation to points.

`tformLocalWeightedMean = images.geotrans.LocalWeightedMeanTransformation2D(movingPoints,fixedPoints,6);`

Verify the fit of the `LocalWeightedMeanTransformation2D` object at the control points.

```movingPointsComputed = transformPointsInverse(tformLocalWeightedMean,fixedPoints); errorInFit = hypot(movingPointsComputed(:,1)-movingPoints(:,1),... movingPointsComputed(:,2)-movingPoints(:,2))```

## Algorithms

The local weighted mean transformation infers a polynomial at each control point using neighboring control points. The mapping at any location depends on a weighted average of these polynomials. The `n` closest points are used to infer a second degree polynomial transformation for each control point pair. `n` can be as small as 6, but making it small risks generating ill-conditioned polynomials.

#### Introduced in R2013b

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