In the mathematical field of numerical analysisinterpolation is a type of estimationa method of constructing new data points within the range of a discrete set of known data points.

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In engineering and scienceone often has a number of data points, obtained by sampling or experimentationwhich represent the values of a function for a limited number of values of the independent variable. It is often required to interpolatei. A closely related problem is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original.

The resulting gain in simplicity may outweigh the loss from interpolation error. We describe some methods of interpolation, differing in such properties as: accuracy, cost, number of data points needed, and smoothness of the resulting interpolant function. The simplest interpolation method is to locate the nearest data value, and assign the same value.

In simple problems, this method is unlikely to be used, as linear interpolation see below is almost as easy, but in higher-dimensional multivariate interpolationthis could be a favourable choice for its speed and simplicity. One of the simplest methods is linear interpolation sometimes known as lerp.

Consider the above example of estimating f 2. Since 2.

Generally, linear interpolation takes two data points, say x ay a and x by band the interpolant is given by:. Linear interpolation is quick and easy, but it is not very precise. Another disadvantage is that the interpolant is not differentiable at the point x k. The following error estimate shows that linear interpolation is not very precise.

Denote the function which we want to interpolate by gand suppose that x lies between x a and x b and that g is twice continuously differentiable. Then the linear interpolation error is. In words, the error is proportional to the square of the distance between the data points.

The error in some other methods, including polynomial interpolation and spline interpolation described belowis proportional to higher powers of the distance between the data points.

The dark mode beta is finally here. Change your preferences any time. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. I looked through the internet, and in terms of Bicubic Interpolation, I can't find a simple equation for it. Wikipedia's page on the subject wasn't very helpful, so is there any easy method to learning how Bicubic Interpolation works and how to implement it? I'm using it to generate Perlin Noise, but using bilinear interpolation is way to choppy for my needs I already tried it.

If anyone can point me in the right direction by either a good website or just an answer, I would greatly appreciate it.

Direct Method of Interpolation: Quadratic Interpolation

I'm using C by the way. For those also looking for the answer, here is what I used:. Took Eske Rahn answer and made a single call note, the code below uses matrix dimensions convention of j, i rather than image of x, y but that shouldn't matter for interpolation sake :. Yes it gives the correct values in 0 and 1, but the derivates of neighbouring cells does not fit, as far as I can calculate.

If the grid-data is linear, it does not even return a line The polynomial that fits in 0 and 1 AND also have the same derivates for neighbouring cells, and thus is smooth is almost as easy to calculate. Learn more. Bicubic Interpolation? Ask Question. Asked 6 years, 3 months ago. Active 3 months ago. Viewed 7k times. Nicholas Pipitone.

Nicholas Pipitone Nicholas Pipitone 3, 1 1 gold badge 15 15 silver badges 30 30 bronze badges. Did you checked github about Perlin Noise?

I hope that may be useful: github. Active Oldest Votes. Hello, Nicholas! Could you join me on this question? VenushkaT Hm, I apparently made the correct edits in Sept '16, but didn't ping you about it.

This is a very late ping, but my final comment should be more readable now.Return to GeoComputation 99 Index What's the point? CF37 1DL E-mail: dbkidner glam.

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This paper advocates the use of more sophisticated approaches to the mathematical modelling of elevation samples for applications that rely on interpolation and extrapolation. The computational efficiency of simple, linear algorithms is no longer an excuse to hide behind with today's advances in processor performance.

The limitations of current algorithms are illustrated for a number of applications, ranging from visibility analysis to data compression. A regular grid digital elevation model DEM represents the heights at discrete samples of a continuous surface.

As such, there is not a direct topological relationship between points; however, for a variety of reasons, users consider these elevations to lie at the vertices of a regular grid, thus imposing an implicit representation of surface form.

For most GIS, a linear relationship between 'vertices' is assumed, while a bilinear representation is assumed within each DEM 'cell'. The consequences of imposing such assumptions can be critical for those applications that interpolate unsampled points from the DEM.

The first part of the paper demonstrates the problems of interpolation within a DEM and evaluates a variety of alternative approaches such as bi-quadratic, cubic, and quintic polynomials and splines that attempt to derive the shape of the surface at interpolated points.

Extrapolation is an extension of interpolation to locations outside the current spatial domain.

One can think of extrapolation as "standing in the terrain and given my field of view, what is my elevation at a location one step backwards? The demand for better data compression algorithms is a consequence of finer resolution data, e. In a similar manner to the interpolation algorithms, the basis of elevation prediction is to determine the local surface form by correlating values within the 'field of view'.

The extent of this field of view can be the nearest three DEM vertices that are used to bilinearly determine the next vertex. The second part of the paper evaluates this approach for more extensive fields of view, using both linear and non-linear techniques. Applications of digital terrain modelling abound in civil engineering, landscape planning, military planning, aircraft simulation, radiocommunications planning, visibility analysis, hydrological modelling, and more traditional cartographic applications, such as the production of contour, hill-shaded, slope and aspect maps.

In all such applications, the fundamental requirement of the digital elevation model DEM is to represent the terrain surface such that elevations can be retrieved for any given location.

As it is often unlikely that the sampled locations will coincide with the user's queried locations, elevations must be interpolated from the DEM. To date, most GIS applications only consider linear or bilinear interpolation as a means of estimating elevations from a regular grid DEM.

This is due mainly to their simplicity and efficiency in what may be a time-consuming operation, such as a viewshed calculation; however, some researchers have recently focused attention on the issue of error in digital terrain modelling applications, particularly for visibility analysis.

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To this extent, this paper examines the choice of interpolation technique within a regular grid DEM, in terms of producing the most suitable solution with respect to accuracy, computational efficiency, and smoothness.

The regular grid DEM data structure is comprised of a 2-D matrix of elevations, sampled at regular intervals in both the x and y planes. Theoretically, the resolution or density of measurements required to obtain a specified accuracy should be dependent on the variability of each terrain surface. The point density must be high enough to capture the smallest terrain features, yet not too fine so as to grossly over-sample the surface, in which case there will be unnecessary data redundancy in certain areas Petrie, The term regular grid DEM suggests that there is a formal topological structure to the elevation data.

In reality, the DEM is only representative of discrete points in the x, y planes Figure 1 and should not be thought of as a continuous surface Figure 2 ; however, it is often perceived as such when combined with a method of surface interpolation, such as a linear algorithm. We should not be constrained to thinking of interpolation within an imaginary grid cell, but rather of interpolation within a locally defined mathematical surface that fits our discrete samples - the extent and smoothness of which is related to the application and requirements of the user.

Leberl recognised that the performance of a DEM depends on the terrain itself, on the measuring pattern and point density in digitising the terrain surface, and on the method of interpolating a new point from the measurements. In the meantime, many researchers have focused on the relationship between sampling density and the geomorphology and characterisation of terrain surfaces; however, the subject of interpolation has largely been ignored.

It is our belief, that users should be aware of the limitations of linear interpolation or any other technique that restricts the surface function to just the vertices of the grid cell. In many instances, linear techniques can produce a diverse range of solutions for the same interpolated point.

A higher order interpolant that takes acount of the neighbouring vertices, either directly or indirectly as slope estimates, will always produce a better estimate than the worst of these linear algorithms. The paper will demonstrate that by extending the local neighbourhood of vertices used in the interpolation process, accuracy will improve.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Signal Processing Stack Exchange is a question and answer site for practitioners of the art and science of signal, image and video processing.

It only takes a minute to sign up. Related question: What are the practically relevant differences between various image resampling methods? Bilinear and bicubic interpolation for image resampling seem to be fairly common, but biquadratic is in my experience rarely heard of.

To be sure, it's available in some programs and libraries, but generally it doesn't seem to be popular. This is further evidenced by there being Wikipedia articles for bilinear and bicubic interpolation, but none for biquadratic. Why is this the case? That is, a function that is continuous but has a discontinuous first derivative.

That is, a function that is continuous and has a continuous first derivative the second derivative is discontinuous though. So you are not gaining anything in terms of "smoothness" continuity of higher derivatives by using biquadratic over bilinear, just more complexity. To get a smoother interpolation, you have to step up from bilinear to bicubic. You want to have an interpolation with a equal number of weights on both sides of the points you want to interpolate in between.

So you choose either one or two weights on each side, resulting in an interpolation of two linear or four cubic points. An quadratic interpolation would need three points, which would only make sense at the border of a grid, where you can choose only one point on one side and two on the other. More expensive computationally with marginal results. Most images don't require such interpolation technique. You can also look at this question: Higher order spline interpolation.

I looked through the internet, and in terms of Bicubic Interpolation, I can't find a simple equation for it. Wikipedia's page on the subject wasn't very helpful, so is there any easy method to learning how Bicubic Interpolation works and how to implement it? I'm using it to generate Perlin Noise, but using bilinear interpolation is way to choppy for my needs I already tried it. If anyone can point me in the right direction by either a good website or just an answer, I would greatly appreciate it.

I'm using C by the way. For those also looking for the answer, here is what I used:. Took Eske Rahn answer and made a single call note, the code below uses matrix dimensions convention of j, i rather than image of x, y but that shouldn't matter for interpolation sake :. Yes it gives the correct values in 0 and 1, but the derivates of neighbouring cells does not fit, as far as I can calculate.

If the grid-data is linear, it does not even return a line The polynomial that fits in 0 and 1 AND also have the same derivates for neighbouring cells, and thus is smooth is almost as easy to calculate. How are we doing? Please help us improve Stack Overflow.

Take our short survey. Did you checked github about Perlin Noise? I hope that may be useful: github. Active Oldest Votes. Hello, Nicholas! Could you join me on this question? VenushkaT Hm, I apparently made the correct edits in Sept '16, but didn't ping you about it. This is a very late ping, but my final comment should be more readable now.

I'm a bit confused on the third degree polynomial used. Eske Rahn Eske Rahn 7 7 silver badges 8 8 bronze badges. Interpolation is an ill posed problem, so I think it is on the problem that what works better. If this was antialiasing we might be able say this works better or that. The idea is to fit the function to points but you don't know the original signal so there is no "best fit" so to say. A modified FFT and biquadratic interpolation based algorithm for carrier frequency offset estimation in MC-CDMA based ad hoc networks Abstract: The future wireless communication systems demand very high data rates, anti-jamming ability and multiuser support.

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People want large amount of data to be continuously accessible in their personal devices. One of the major problems with MC-CDMA is the high sensitivity towards carrier frequency offsets caused due to the inherent inaccuracy of crystal oscillators. This carrier frequency offset destroys the orthogonality of the subcarriers resulting in Intercarrier Interference ICI.

In this paper, we propose a computationally efficient algorithm based on Fast Fourier Transform FFT and biquadratic Lagrange interpolation.

The FFT is based on the use of overlapping windows for each frame of the data instead of non-overlapping windows.

This gives a coarse estimate of the frequency offset which is refined by the successive application of Lagrange quadratic interpolation to the samples in the vicinity of FFT peak. The proposed algorithm has been applied to the multiuser ad hoc network and simulated in Stanford University Interim SUI channels. Published in: International Conference on Emerging Technologies. Article :.

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DOI: Need Help?This thread on StackExchange asks a great question: why is quadratic interpolation so rarely used, especially in computer graphics? Linear interpolation provides fast, low quality results using 2 samples, while cubic interpolation provides slower, higher quality results with 4 samples. It seems that quadratic interpolation, with 3 samples, would offer more control on cost vs quality, especially in the 2D case at 4 vs 9 vs 16 samples.

And while bicubic interpolation can be obtained with only 4 bilinear filtered texture samples, this cannot be used when one needs to dynamically modulate the individual weights of each sample, such as is the case for bilateral or depth-aware upsampling filters. I happened to be precisely in that case when working on volumetric cloud depth-aware upsampling.

I wondered why and started trying it out myself when I quickly realized the reason : the symmetry of the problem didn't seem to allow it. When interpolating at a position between two values on a 1D grid, it's straightforward to either take the 2 or 4 neighbouring control points around it and interpolate:.

Linear and cubic interpolation. But if you want to take 3, which do you choose? You get an additional value on either the left or the right, and you end up not centered around the interpolation interval anymore:. Alain Brobecker details a simple fix to this issue on his post about quadratic interpolation. Basically shift the problem half a unit to the left and use two new virtual control points, created by linearly interpolating the real ones.

Note this means the curve no longer passes through the control points of your data set, but through the virtual control points:. Quadratic interpolation. Where x is the local interpolation coordinate varying from 0 to 1 across the [ i Apply the same process to the next intervals and you get a C1 continuous quadratic interpolation of the data set, with a continuous derivative.

Now we can extend this to biquadratic interpolation, following the regular 1D to 2D scheme used to go from linear to bilinear and cubic to bicubic. We get the biquadratic algorithm by interpolating quadratically three times along one coordinate, and one last time along the other:.

Using the following layout of control points and local x, y coordinates. This is a handy setup: it requires using the half-texel offset to correctly align the pixel values with their pixel centers, which should be done anyway.

There's a much better visualization of this thanks to paniq on shadertoy. Next is a comparison between bilinear, biquadratic and bicubic using upsampled volumetric clouds as an example.

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In this setup, biquadratic interpolation achieves largely similar results to bicubic with only the 3x3 neighbouring values I already had available, saving me 7 additional samples. Bilinear vs biquadratic vs bicubic upsampling.

However, what I needed was a depth-aware upsampling filter. This adds another constraint for the interpolation scheme: separate weights for each control points.

We need the biquadratic upsampling to be expressible in terms of a single equation: a weighted sum of the control points. In this form, each control point is only weighted by polynomials of x and y. If you then normalize this weight sum, this allows dynamically modulating each sample's weight.

To achieve this, we can first simplify the 1D quadratic equation into the form we want:. Plugging this into the 2D sampling pattern gives :.

Which can thank you Wolfram Alpha be developed into our desired form. Note that since this is a separable filter, many computations are reusable:.

This allows using smooth biquadratic upsampling in a depth-aware fashion, with discardable invalid samples:.