Arduino library for fast non-linear mapping or interpolation of values.
In Arduino applications often the 'raw' value of a sensor is mapped upon a more usable value. E.g. the value of analogRead() 0 .. 1023 is mapped onto 0 .. 5.0 Volt. This is often done by the map() function which does a linear interpolation with integer truncating.
This means in code:
output = C1 + input * C2As C1 and C2 are to be determined, Arduino has the map() function that calculates the two variables runtime from two given mapping points (I1, O1) and (I2, O2).
output = map(input, I1, I2, O1, O2):In many cases when there is no linear mapping possible as the 'points' are not on a single straight line. Think a logarithmic or some sine wave form. To solve this one needs non-linear math to calculate the output from a given input.
The multiMap() function simulates this math by approximating the non-linear function with multiple linear line segments. Of course this approximation introduces an error. By increasing the number of points and choose their position strategically the average error can and will be reduced. An important feature of the multiMap() is that the points do not need to have the same distance (non-equidistant). This allows to have more pointe where needed (curvy line) and less point where possible (straight lines).
Note: some functions are hard to approximate even with multiMap() as they go to infinity or have a singularity. Think of tan(x) around x = PI/2 (90°) or sin(1/x) around zero.
The function to approximate must be (mathematically) an injection. This means that for every input value (in the given range) there is an output value. However there might be more than one input value mapping onto the same output value. See - https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection
Other mapping libraries and interesting links.
- https://github.com/RobTillaart/FastMap
- https://github.com/RobTillaart/Gamma
- https://github.com/RobTillaart/map2bits
- https://github.com/RobTillaart/map2colour
- https://github.com/RobTillaart/moduloMap
- https://github.com/RobTillaart/MultiMap
- https://mycurvefit.com/
- https://www.desmos.com/calculator
#include "MultiMap.h"The basic call for multiMap() is:
output = Multimap<datatype>(input, inputArray, outputArray, size);multiMap() needs two equally sized arrays representing the reference 'points' named inputArray[] and outputArray[] both of the datatype.
multiMap() will do a lookup of the input value in the inputArray[]. If it cannot find the index of an exact point it will determine a weighted position between two points. This optional weighted point is used to interpolate a value from the data in the outputArray[].
- The inputArray[] must have increasing values, there is no such restriction for the output[] array.
- The values of the inputArray[] do not need to have the same distance (non-equidistant). E.g a sort of logarithmic input array like { 1, 10, 100, 1000 } is valid.
- multiMap() automatically constrains the output to the first and last value in the outputArray[]. This is a explicit difference with the map() function. Therefore it is important to extend the range of the arrays to cover all possible input and output values.
The multiMap() function does a linear search for the inputValue in the inputArray[]. This implies that usage of larger and more precise arrays will take more time. Furthermore "low" input values will be found faster than "high" values, so the function has no constant time execution.
To optimize performance a binary search version exists, see multiMapBS() below.
As every usage of multiMap() is unique one should always do a performance check to see if there is a substantial gain in the case at hand. In my experience there often is.
Experimental 0.1.7 => use with care.
multiMapBS() stands for MultiMapBinarySearch or MMBS for short. It is a very similar function as multiMap() with the same interface. The main difference is that MMBS uses binary search instead of linear search.
Performance tests indicate that for array sizes of about 10 elements, the multiMapBS() is on par with multiMap(). This is an expected value as both need on average about 5 steps to find the right interval to interpolate.
Be sure to do your own tests to see if MMBS improves your performance.
Experimental 0.1.7 => use with care.
multiMapCache() or MMC for short, is a very similar function as multiMap(). The main difference is that MMC caches the last input and output value. The goal is to improve the performance by preventing searching a repeating input value over and over again.
If the input sequence has a lot of repeating values e.g. 2 2 2 2 2 2 5 5 5 5 5 4 4 4 4 2 2 2 2 2 2 MMC will be able to return the value from cache often. Otherwise keeping cache is overhead.
Be sure to do your own tests to see if MMC improves your performance.
A possible variation is to cache the last interval - lower and upper index. It would allow a to test that value and improve the linear search. (not implemented, to be investigated).
Experimental 0.2.0 => use with care.
multiMap<T1, T2>() or MMTT for short, is a very similar function as multiMap(). The main difference is that MMTT uses two different types, typical the input is an integer type and the output is a float or double type. It is expected that there will be a gain if two different sized integer types are used. So performance is platform specific.
See the example multimap_distance_two_types.ino
// for a sharp distance range finder (based upon graph datasheet).
float sharp2cm2(int value)
{
// out[] holds the distances in cm
float out[] = {150, 140, 130, 120, 110, 100, 90, 80, 70, 60, 50, 40, 30, 20};
// in[] holds the measured analogRead() values for that distance
int in[] = { 90, 97, 105, 113, 124, 134, 147, 164, 185, 218, 255, 317, 408, 506};
float distance = multiMap<int, float>(value, in, out, 14);
return distance;
}First tests indicate that using the int type for the input in the example is substantial (~37%) faster per call. Tested on UNO R3, time in micros per call.
| types | time us | call |
|---|---|---|
| 1 | 194.93 | float dist = multiMap<float>(value, in, out, 14); |
| 2 | 121.97 | float dist = multiMap<int, float>(value, in, out, 14); |
Furthermore it is obvious that there is less need for RAM if the integer type is smaller in size than the float type.
Be sure to do your own tests to see if MMTT improves your performance.
See examples
Please note the fail example as this shows that in the intern math overflow can happen.
- improve documentation
- investigate multiMapCache behaviour
- determine overhead.
- extend unit tests
- multi type versions
- Investigate class implementation
- basic call
out = mm.map(value); - runtime adjusting input and output array begin(in[], out[])
- performance / footprint
- less parameter passing
- isInRange(value)?
- caching last value / position / index (does that help?)
- flag if input value was "IN_MIN" < input < "IN_MAX", now it is constrained without user being informed.
- basic call
- Investigate a 2D multiMap, 3D multiMap
- is it possible / feasible? (YES | ??)
- float ==> complex (Y)
- e.g. time ==> XY position (Y)
- complex ==> complex (?)
- X ==> { Y, Z } in fact 2 output arrays.
- needs other API call.
- can be implemented with multiple single MM calls, one per dimension.
- can the single type version be implemented with the two type version?
- should the lookup tables be merged into one array of pairs?
- you cannot reuse e.g. the input array or the output array then. this would not improve the memory footprint.
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