Interpolating Gridded Data - MATLAB & Simulink (2024)

Interpolating Gridded Data

Gridded data consists of values or measurements at regularly spaced points that form a grid. Gridded data arises in many areas, such as meteorology, surveying, and medical imaging. In these areas, it is common to take measurements at regular spatial intervals, possibly over time. These ordered grids of data can range from 1-D (for simple time series) to 4-D (for measuring volumes over time) or higher. Some examples of gridded data are:

  • 1-D: Stock prices over time

  • 2-D: Temperature of a surface

  • 3-D: MRI image of a brain

  • 4-D: Ocean measurements in a volume of water over time

In all of these applications, grid-based interpolation efficiently extends the usefulness of the data to points where no measurement was taken. For example, if you have hourly price data for a stock, you can use interpolation to approximate the price every 15 minutes.

MATLAB Gridded Interpolation Functions

MATLAB® provides several tools for grid-based interpolation:

Grid Creation Functions

The meshgrid and ndgrid functions create grids of various dimensionality. meshgrid can create 2-D or 3-D grids, while ndgrid can create grids with any number of dimensions. These functions return grids using different output formats. You can convert between these grid formats using the pagetranspose (as of R2020b) or permute functions to swap the first two dimensions of the grid.

Interpolation Functions

The interp family of functions includes interp1, interp2, interp3, and interpn. Each function is designed to interpolate data with a specific number of dimensions. interp2 and interp3 use grids in meshgrid format, while interpn uses grids in ndgrid format.

Interpolation Objects

griddedInterpolant objects support interpolation in any number of dimensions for data in ndgrid format. These objects also support multivalued interpolation (as of R2021a), where each grid point can have multiple values associated with it.

There are memory and performance benefits to using griddedInterpolant objects over the interp functions. griddedInterpolant offers substantial performance improvements for repeated queries of the interpolant object, whereas the interp functions perform a new calculation each time they are called. Also, griddedInterpolant stores the sample points in a memory-efficient format (as a compact grid) and is multithreaded to take advantage of multicore computer processors.

Grid Representations

MATLAB allows you to represent a grid in one of three representations: full grid, compact grid, or default grid. The default grid and compact grid are used primarily for convenience and improved efficiency, respectively.

Full Grid

A full grid is one in which all points are explicitly defined. The outputs of ndgrid and meshgrid define a full grid. You can create full grids that are uniform, in which points in each dimension have equal spacing, or nonuniform, in which the spacing varies in one or more of the dimensions. Uniform grids can have different spacing in each dimension, as long as the spacing is constant within each dimension.

UniformUniformNonuniform

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An example of a uniform full grid is:

[X,Y] = meshgrid([1 2 3],[3 6 9 12])

Compact Grid

Explicitly defining every point in a grid can consume a lot of memory when you are dealing with large grids. The compact grid representation is a way to dispense with the memory overhead of a full grid. The compact grid representation stores only grid vectors (one for each dimension) instead of the full grid. Together, the grid vectors implicitly define the grid. In fact, the inputs for meshgrid and ndgrid are grid vectors, and these functions replicate the grid vectors to form the full grid. The compact grid representation enables you to bypass grid creation and supply the grid vectors directly to the interpolation function.

For example, consider two vectors, x1 = 1:3 and x2 = 1:5. You can think of these vectors as a set of coordinates in the x1 direction and a set of coordinates in the x2 direction, like so:

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Each arrow points to a location. You can use these two vectors to define a set of grid points, where one set of coordinates is given by x1 and the other set of coordinates is given by x2. When the grid vectors are replicated, they form two coordinate arrays that make up the full grid:

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Your input grid vectors might be monotonic or nonmonotonic. Monotonic vectors contain values that either increase in that dimension or decrease in that dimension. Conversely, nonmonotonic vectors contain values that fluctuate. If the input grid vector is nonmonotonic, such as [2 4 6 3 1], then [X1,X2] = ndgrid([2 4 6 3 1]) outputs a nonmonotonic grid. Your grid vectors should be monotonic if you intend to pass the grid to other MATLAB functions. The sort function is useful to ensure monotonicity.

Default Grid

In some applications, only the values at the grid points are important and not the distances between grid points. For example, most MRI scans gather data that is uniformly spaced in all directions. In cases like this, you can allow the interpolation function to automatically generate a default grid representation to use with the data. To do this, leave out the grid inputs to the interpolation function. When you leave out the grid inputs, the function automatically considers the data to be on a unit-spaced grid. The function creates this unit-spaced grid while it executes, saving you the trouble of creating a grid yourself.

Example: Temperature Interpolation on 2-D Grid

Consider temperature data collected on a surface at regular 5 cm intervals, extending 20 cm in each direction. Use meshgrid to create the full grid.

[X,Y] = meshgrid(0:5:20)
X = 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20Y = 0 0 0 0 0 5 5 5 5 5 10 10 10 10 10 15 15 15 15 15 20 20 20 20 20

The (x,y) coordinates of each grid point are represented as corresponding elements in the X and Y matrices. The first grid point is given by [X(1) Y(1)], which is [0 0], the next grid point is given by [X(2) Y(2)], which is [0 5], and so on.

Now, create a matrix to represent temperature measurements on the grid and then plot the data as a surface.

T = [1 1 10 1 1; 1 10 10 10 10; 100 100 1000 100 100; 10 10 10 10 1; 1 1 10 1 1];surf(X,Y,T)view(2)

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Although the temperature at the center grid point is large, its location and influence on surrounding grid points is not apparent from the raw data.

To improve the resolution of the data by a factor of 10, use interp2 to interpolate the temperature data onto a finer grid that uses 0.5 cm intervals. Use meshgrid again to create a finer grid represented by the matrices Xq and Yq. Then, use interp2 with the original grid, the temperature data, and the new grid points, and plot the resulting data. By default, interp2 uses linear interpolation in each dimension.

[Xq,Yq] = meshgrid(0:0.5:20);Tq = interp2(X,Y,T,Xq,Yq);surf(Xq,Yq,Tq)view(2)

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Interpolating the temperature data adds detail to the image and greatly improves the usefulness of the data within the area of measurements.

Gridded Interpolation Methods

The grid-based interpolation functions and objects in MATLAB offer several different methods for interpolation. When choosing an interpolation method, keep in mind that some require more memory or longer computation time than others. You may need to trade off these resources to achieve the desired smoothness in the result. The following table gives a preview of each interpolation method applied to the same 1-D data, and also provides an overview of the benefits, trade-offs, and requirements for each method.

MethodDescription

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The interpolated value at a query point is the value at the nearest sample grid point.

  • Discontinuous

  • Modest memory requirements

  • Fastest computation time

  • Requires 2 grid points in each dimension

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The interpolated value at a query point is the value at the next sample grid point.

  • Discontinuous

  • Same memory requirements and computation time as nearest neighbor

  • Available for 1-D interpolation only

  • Requires at least 2 grid points

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The interpolated value at a query point is the value at the previous sample grid point.

  • Discontinuous

  • Same memory requirements and computation time as nearest neighbor

  • Available for 1-D interpolation only

  • Requires at least 2 grid points

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The interpolated value at a query point is based on linear interpolation of the values at neighboring grid points in each respective dimension.

  • C0 continuous

  • Requires more memory and computation time than nearest neighbor

  • Requires at least 2 grid points in each dimension

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The interpolated value at a query point is based on a shape-preserving piece-wise cubic interpolation of the values at neighboring grid points.

  • C1 continuous

  • Requires more memory and computation time than linear

  • Available for 1-D interpolation only

  • Requires at least 4 grid points

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The interpolated value at a query point is based on cubic interpolation of the values at neighboring grid points in each respective dimension.

  • C1 continuous

  • Requires more memory and computation time than linear

  • Grid must have uniform spacing, though the spacing in each dimension does not have to be the same

  • Requires at least 4 points in each dimension

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The interpolated value at a query point is based on a piecewise function of polynomials with degree at most three evaluated using the values of neighboring grid points in each respective dimension. The Akima formula is modified to avoid overshoots.

  • C1 continuous

  • Similar memory requirements as spline

  • Requires more computation time than cubic, but typically less than spline

  • Requires at least 2 grid points in each dimension

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The interpolated value at a query point is based on a cubic interpolation of the values at neighboring grid points in each respective dimension.

  • C2 continuous

  • Requires more memory and computation time than cubic

  • Requires at least 4 points in each dimension

See Also

interp1 | interp2 | interp3 | interpn | griddedInterpolant

Related Topics

  • Resample Image with Gridded Interpolation
  • Interpolation of Multiple 1-D Value Sets
  • Interpolation of 2-D Selections in 3-D Grids

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Interpolating Gridded Data
- MATLAB & Simulink (2024)

FAQs

Interpolating Gridded Data - MATLAB & Simulink? ›

The interpolated value at a query point is the value at the next sample grid point. The interpolated value at a query point is the value at the previous sample grid point. The interpolated value at a query point is based on linear interpolation of the values at neighboring grid points in each respective dimension.

How do you interpolate scattered data in MATLAB? ›

Use scatteredInterpolant to perform interpolation on a 2-D or 3-D data set of scattered data. scatteredInterpolant returns the interpolant F for the given data set. You can evaluate F at a set of query points, such as (xq,yq) in 2-D, to produce interpolated values vq = F(xq,yq) .

Which MATLAB function will be useful for interpolating a gridded data on some 3D surface? ›

Vq = interp3( X,Y,Z , V , Xq,Yq,Zq ) returns interpolated values of a function of three variables at specific query points using linear interpolation. The results always pass through the original sampling of the function.

How to implement interpolation in MATLAB? ›

Description
  1. example. ...
  2. Vq = interpn( V , Xq1,Xq2,...,Xqn ) assumes a default grid of sample points. ...
  3. Vq = interpn( V ) returns the interpolated values on a refined grid formed by dividing the interval between sample values once in each dimension.

What is scattered data interpolation? ›

Scattered data consists of a set of points X and corresponding values V , where the points have no structure or order between their relative locations. There are various approaches to interpolating scattered data. One widely used approach uses a Delaunay triangulation of the points.

What are the interpolation techniques used for evenly spaced data? ›

Linear interpolation is often used to regrid evenly-spaced data, such as longitude / latitude gridded data, to a higher or lower resolution.

What is the best way to interpolate data? ›

Linear interpolation is the most straightforward and commonly used interpolation method. It comes naturally when we have two points, we connect them with a straight line to fill out the missing information in between. By doing so, we made our assumption that the points on the line represent the unobserved values.

How to use data from Matlab in Simulink? ›

To load data for a bus, set the Output data type to Bus: <bus object> and specify the name of the Simulink. Bus object that defines the output bus. To load enumerated data, set the Output data type to Enum: <class name> and specify the name of the enumeration class that defines the enumerated data values.

Which method is best for interpolation? ›

In terms of the ability to fit your data and produce a smooth surface, the Multiquadric method is considered by many to be the best. All of the Radial Basis Function methods are exact interpolators, so they attempt to honor your data.

What is the difference between griddedInterpolant and interp2? ›

griddedInterpolant offers substantial performance improvements for repeated queries of the interpolant object, whereas the interp functions perform a new calculation each time they are called.

How to use gridded data? ›

Gridded data can be the output from a numerical model or an objective analysis of observational data. Each gridded data source is comprised of a set of 2- and/or 3-D fields. Each field may have of one or more times associated with it. The IDV supports a variety of ways of visualizing gridded data.

How do you interpolate a surface in MATLAB? ›

The griddata function interpolates the surface at the query points specified by (xq,yq) and returns the interpolated values, vq . The surface always passes through the data points defined by x and y . vq = griddata( x , y , z , v , xq , yq , zq ) fits a hypersurface of the form v = f(x,y,z).

What is linear interpolation in Matlab Simulink? ›

Linear interpolation. This method fits a different linear polynomial between each pair of data points for curves, or between sets of three points for surfaces. Nearest neighbor interpolation. This method sets the value of an interpolated point to the value of the nearest data point.

What is an example of interpolation? ›

Interpolation definition says that interpolation is to estimate the value of a point between two given points in a data set. For example, if a child's height was measured at age 5 and age 6, interpolation could be used to estimate the child's height at age 5.5.

How does meshgrid work in Matlab? ›

[X,Y] = meshgrid(x,y) transforms the domain specified by vectors x and y into arrays X and Y , which can be used to evaluate functions of two variables and three-dimensional mesh/surface plots. The rows of the output array X are copies of the vector x ; columns of the output array Y are copies of the vector y .

What is interpolation in scatter plot? ›

Interpolation is where you use the line of best fit for a value that is within the plotted points. Extrapolation is where you use the line of best fit for a value that is outside the plotted points.

How do you linearly interpolate data? ›

Linear Interpolation Formula
  1. x 1. and.
  2. y 1. are the first coordinates.
  3. x 2. and.
  4. y 2. are the second coordinates. x is the point to perform the interpolation. y is the interpolated value. Solved Examples. ...
  5. y 1. + ( x − x 1 ) ( y 2 − y 1 ) x 2 − x 1. y = 4 + ( 4 − 2 ) ( 7 − 4 ) 6 − 2.

What is the interpolation method for missing data? ›

The process can generally be divided into 2 steps: First, linear interpolation is exploited to process short-term missing data. Then the LightGBM can be used to process long-term missing data. In the process of using LightGBM, linear interpolation is used to interpolate the independent variables of the input model.

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