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Bicubic interpolation

Comparison of Bicubic interpolation with some 1- and 2-dimensional interpolations. Black and red/yellow/green/blue dots correspond to the interpolated point and neighbouring samples, respectively. Their heights above the ground correspond to their values.

In mathematics, bicubic interpolation is an extension of cubic interpolation for interpolating data points on a two-dimensional regular grid. The interpolated surface is smoother than corresponding surfaces obtained by bilinear interpolation or nearest-neighbor interpolation. Bicubic interpolation can be accomplished using either Lagrange polynomials, cubic splines, or cubic convolution algorithm.

In image processing, bicubic interpolation is often chosen over bilinear or nearest-neighbor interpolation in image resampling, when speed is not an issue. In contrast to bilinear interpolation, which only takes 4 pixels (2×2) into account, bicubic interpolation considers 16 pixels (4×4). Images resampled with bicubic interpolation are smoother and have fewer interpolation artifacts.

Contents

ComputationEdit

 
Bicubic interpolation on the square   consisting of 25 unit squares patched together. Bicubic interpolation as per Matplotlib's implementation. Colour indicates function value. The black dots are the locations of the prescribed data being interpolated. Note how the color samples are not radially symmetric.
 
Bilinear interpolation on the same dataset as above. Derivatives of the surface are not continuous over the square boundaries.
 
Nearest-neighbor interpolation on the same dataset as above.

Suppose the function values   and the derivatives  ,   and   are known at the four corners  ,  ,  , and   of the unit square. The interpolated surface can then be written as

 

The interpolation problem consists of determining the 16 coefficients  . Matching   with the function values yields four equations:

  1.  
  2.  
  3.  
  4.  

Likewise, eight equations for the derivatives in the   and the   directions:

  1.  
  2.  
  3.  
  4.  
  5.  
  6.  
  7.  
  8.  

And four equations for the   mixed partial derivative:

  1.  
  2.  
  3.  
  4.  

The expressions above have used the following identities:

 
 
 

This procedure yields a surface   on the unit square   that is continuous and has continuous derivatives. Bicubic interpolation on an arbitrarily sized regular grid can then be accomplished by patching together such bicubic surfaces, ensuring that the derivatives match on the boundaries.

Grouping the unknown parameters   in a vector

 

and letting

 

the above system of equations can be reformulated into a matrix for the linear equation  .

Inverting the matrix gives the more useful linear equation  , where

 

which allows   to be calculated quickly and easily.

There can be another concise matrix form for 16 coefficients:

 

or

 

where

 

Finding derivatives from function valuesEdit

If the derivatives are unknown, they are typically approximated from the function values at points neighbouring the corners of the unit square, e.g. using finite differences.

To find either of the single derivatives,   or  , using that method, find the slope between the two surrounding points in the appropriate axis. For example, to calculate   for one of the points, find   for the points to the left and right of the target point and calculate their slope, and similarly for  .

To find the cross derivative  , take the derivative in both axes, one at a time. For example, one can first use the   procedure to find the   derivatives of the points above and below the target point, then use the   procedure on those values (rather than, as usual, the values of   for those points) to obtain the value of   for the target point. (Or one can do it in the opposite direction, first calculating   and then   from those. The two give equivalent results.)

At the edges of the dataset, when one is missing some of the surrounding points, the missing points can be approximated by a number of methods. A simple and common method is to assume that the slope from the existing point to the target point continues without further change, and using this to calculate a hypothetical value for the missing point.

Bicubic convolution algorithmEdit

Bicubic spline interpolation requires the solution of the linear system described above for each grid cell. An interpolator with similar properties can be obtained by applying a convolution with the following kernel in both dimensions:

 

where   is usually set to −0.5 or −0.75. Note that   and   for all nonzero integers  .

This approach was proposed by Keys, who showed that   (which corresponds to cubic Hermite spline) produces third-order convergence with respect to the sampling interval of the original function.[1]

If we use the matrix notation for the common case  , we can express the equation in a more friendly manner:

 

for   between 0 and 1 for one dimension. Note that for 1-dimensional cubic convolution interpolation 4 sample points are required. For each inquiry two samples are located on its left and two samples on the right. These points are indexed from −1 to 2 in this text. The distance from the point indexed with 0 to the inquiry point is denoted by   here.

For two dimensions first applied once in   and again in  :

 
 
 
 
 

Use in computer graphicsEdit

 
The lower half of this figure is a magnification of the upper half, showing how the apparent sharpness of the left-hand line is created. Bicubic interpolation causes overshoot, which increases acutance.

The bicubic algorithm is frequently used for scaling images and video for display (see bitmap resampling). It preserves fine detail better than the common bilinear algorithm.

However, due to the negative lobes on the kernel, it causes overshoot (haloing). This can cause clipping, and is an artifact (see also ringing artifacts), but it increases acutance (apparent sharpness), and can be desirable.

See alsoEdit

ReferencesEdit

  1. ^ R. Keys (1981). "Cubic convolution interpolation for digital image processing". IEEE Transactions on Acoustics, Speech, and Signal Processing. 29 (6): 1153–1160. doi:10.1109/TASSP.1981.1163711.

External linksEdit