Summary of common interpolation methods in image processing

Common interpolation methods

1. Nearest Element Method

This is the simplest method of interpolation and does not need to be calculated. In the four adjacent pixels of the desired pixel, the gray scale of the adjacent pixel closest to the desired pixel is assigned to the waiting pixel. Set I + u, J + V (I, j is a positive integer, U, V is a decimal point greater than zero and less than 1, the same below) as the pixel coordinate to be obtained, shows the gray value f (I + u, J + V) of the pixel to be obtained:

 

 

If (I + u, J + V) falls into Area A, that is, u <0.5, v <0.5, the gray value of the pixel in the upper left corner is assigned to the pixel to be evaluated. Similarly, in Area B, the pixel gray value is assigned to the upper-right corner. in Area C, the gray value of the lower-left pixel is assigned to the lower-right pixel. In Area D, the gray value of the lower-right pixel is assigned to the lower-right pixel.

The nearest element method requires a small amount of computation, but may result in non-consecutive gray-scale images generated by interpolation. Obviously, jagged images may appear in the gray-scale areas.

 

2. bilinear interpolation

The bilinear interpolation method uses the gray scale of the four adjacent pixels of the desired pixel for linear interpolation in two directions, as shown in:

 

 

For gray-scale changes from (I, j + V), F (I, j) to f (I, j + 1) to linear relationships, there are:

F (I, j + V) = [f (I, j + 1)-f (I, j)] * V + f (I, j)

Similarly, for (I + 1, J + V), there are:

F (I + 1, J + V) = [f (I + 1, J + 1)-f (I + 1, J)] * V + f (I + 1, J)

Gray changes from f (I, j + V) to f (I + 1, J + V) are also linear, the formula for calculating the gray level of the pixel to be obtained is as follows:

F (I + u, J + V) = (1-u) * (1-V) * f (I, j) + (1-u) * V * f (I, j + 1) + u * (1-V) * f (I + 1, J) + u * V * f (I + 1, J + 1)

The bilinear interpolation method is more complex than the nearest vertex method and has a large amount of computation. However, it does not have the disadvantages of non-continuous gray scale, and the results are basically satisfactory. It has a low-pass filtering property, which damages the high-frequency components and may blur the image contour.

 

3. Three insertion Methods

This method uses cubic polynomial S (X) to obtain the best interpolation function sin (X)/X in approximation theory. The mathematical expression is as follows:

The gray value of the pixel (X, Y) is obtained by Weighted Interpolation of 16 gray values around it, for example:

 

 

The grayscale formula of the pixel to be calculated is as follows:

 

F (x, y) = f (I + u, J + V) = ABC

 

Where:

 

The cubic curve interpolation method requires a large amount of computing, but the interpolation results are the best.

Summary of interpolation methods:

"Inverse distance to a power (Inverse Distance Weighted Interpolation )",
[[Kerry gold interpolation method )",
"Minimum curvature (minimum curvature )",
"Modified Shepard's method (improved Xie bind's method )",
"Natural Neighbor (natural neighbor interpolation method )",
"Nearest neighbor (nearest neighbor interpolation method )",
"Polynomial regression (multivariate regression )",
"Radial Basis Function (Radial Basis Function )",
"Triangulation with linear interpolation (linear interpolation Triangle Method )",
"Moving average (moving average method )",
"Local polynomial (local polynomial method )"

1. distance reciprocal multiplication method
The distance reciprocal square gridded method is a weighted average interpolation method that can be used for exact or smooth interpolation. The square parameter controls how the weight coefficient decreases as the distance from a grid node increases. For a large cube, a relatively high weight share is given for a closer data point. For a smaller cube, the weights are evenly distributed to data points.
When a grid node is computed, the weight of a specific data point is proportional to the reciprocal of the distance from the specified node to the observation point. When a grid node is computed, the assigned weight is a score, and the sum of all weights equals 1.0. When an observation point and a grid node are duplicated, the observation point is given a weight of 1.0 actually, and all other observation points are given a weight of almost 0.0. In other words, the node is assigned a value consistent with the observation point. This is an accurate interpolation.
One of the features of the distance reciprocal method is to generate a "bull eye" around the observation point position in the grid area ". You can specify a smooth parameter when using the reciprocal distance grid. The smooth parameter guarantee is greater than zero. For a specific knot, no observation point is assigned all the weights, even if the observation point overlaps with the node. Smooth parameters reduce the effect of the "Ox eye" by smoothing the grid that has been interpolated.

2. Kerry kingfa
The kerkin method is a grid method of geological statistics that is useful in many fields. Kerry kingfa tries to show the trend that is hidden in your data. For example, a high point is connected along a ridge, rather than isolated by a ox-Eye contour.
The Kerry kingfa includes several factors: the change graph model, the drift type and the block effect.

3. Least Curvature Method
The least curvature method is widely used in Earth science. The interpolation surface generated by the least curvature method is similar to a long thin elastic slice with the minimum bending volume through each data value. The least curvature method tries to generate as smooth a surface as possible while respecting data as strictly as possible.
When using the least curvature method, two parameters are involved: the maximum residual parameter and the maximum number of cycles parameter to control the convergence standard of the minimum curvature.

4. Multiple Regression
Multiple regression is used to determine the large-scale trend and pattern of your data. You can use several options to determine the type of the trend surface you need. In fact, multivariate regression is not an interpolation tool because it does not attempt to predict unknown Z values. It is actually a trend surface analysis and plotting program.
When using multivariate regression, you must define the surface and specify the highest level of xy. The surface definition is the Polynomial Type of the selected data, these types are simple plane, bilinear saddle, quadratic surface, cubic surface, and user-defined polynomials. The parameter setting is the highest level of the X and Y elements in the polynomial equation.

5. Radial Basic Function Method
The radial basic function method is a combination of multiple data interpolation methods. According to the ability to adapt to your data and generate a smooth surface, the complex quadratic function is considered by many people as the best method. All basic radial functions are accurate interpolation tools, and they must work to respect your data. To try to generate a smoother surface, you can introduce a smoother coefficient to all of these methods. The function that you can specify is similar to the Change chart in Kerry gold. When interpolation is performed on a grid node, these functions specify a set of Optimal Weights for data points.

6. Xie binfa
Xie bind's method uses the least squares weighted by reciprocal distance. Therefore, it is similar to the reciprocal square interpolation, but it uses the local Least Square to remove or reduce the appearance of the generated contour. Xie beide can be an accurate or smooth interpolation tool.
When using the Xie bind method as the grid method, the smooth parameter settings must be involved. Smooth parameters enable Xie bind to work like a smooth interpolation device. When you increase the value of a smooth parameter, the better the smooth effect is.

7. Tin/linear interpolation
The Triangle network interpolation tool is a strict interpolation tool, and its working route is similar to that drawn by hand. This method works by establishing several triangles by connecting data points. The connection method of the original data points is as follows: the edges of all triangles cannot interwork with other triangles. The result is a mesh that covers the grid and is spliced by triangles.
Each triangle defines an area that covers the grid nodes in the triangle. The Skew and elevation of a triangle are determined by the three original data points that define the triangle. All nodes in a given triangle are restricted by the surface of the triangle. Because the original data points are used to define various triangles, your data is very respected.


8. natural neighbor interpolation
The natural neighboring point interpolation method (naturalneighbor) is a new method of mesh developed by surfer7.0. The natural neighbor interpolation method is widely used in some research fields. The basic principle is that for a group of Tyson (Thiessen) polygon, when a new data point (target) is added to the data set, these Tucson polygon will be modified, the weighted average value of neighboring points determines the weight of the point to be inserted. The weight of the point to be inserted is proportional to the weight of the target taisen polygon. In fact, in these polygon, some polygon will be reduced, and no polygon will increase in size. At the same time, the natural neighbor interpolation method does not push out the contour lines (such as the contour lines of the taisen polygon) when the data point is raised ).

9. Nearest Neighbor Interpolation
Nearestneighbor (nearestneighbor) is also known as the Tucson Polygon method. The thiesen (also called Dirichlet or KNN polygon) analysis method is an analytical method proposed by Dutch Meteorological scientist A. H. Thiessen. It was originally used to calculate the average rainfall from the rainfall data of the discrete distribution weather station. in GIS and geographic analysis, it is often used to quickly assign values using the tysen polygon. In fact, an implicit assumption of nearest neighbor interpolation is that the attribute values of P (x, y) of any mesh point use the attribute values of the point closest to it, use the neighborhood value of each grid node as the node value to be added. When the data is evenly distributed at intervals, you must first convert the data to a surfer GRID file. You can use the nearest neighbor interpolation method.
Data is closely complete, and only a few points have no value. You can use the nearest neighbor interpolation method to fill in the data points without value. Sometimes it is necessary to exclude the areas with no value data in the grid file, set a value in the search elliptic (searchellipse), and assign the blank value to the area without data in the grid file. The size of the search radius is smaller than the distance between the data values of the grid file. All countless Data Grid nodes are assigned a blank value. When using the nearest neighbor interpolation gridded method to convert XYZ data at a regular interval to a grid file, you can set an equal distance between the grid interval and the data points of XYZ data. The nearest neighbor interpolation gridded method has no options. It is homogeneous and non-changing, and is useful for interpolation of data with even intervals. At the same time, it is very effective for filling areas with no value data.

Reference

Http://blog.csdn.net/coy_wang/article/details/5027872

Http://blog.sina.com.cn/s/blog_6e51df7f0100vb4b.html