## Interactive Resolution Enhancement (Peak Sharpening)

This is a Matlab implementation of a mathematically simple, quickly-computable resolution enhancement technique for time-series signals consisting of partially overlapping peaks. It is based on subtracting the scaled, smoothed 2nd derivative and adding the scaled, smoothed 4th derivative to the original signal. The algorithm can be customized for signals of different types by adjusting the scaling factors for the derivatives and the width of the smoothing function. The interactive routines here have controls that allow you to adjust the resolution enhancement parameters continuously while observing the effect on your signal dynamically.

Video Demonstration
This 15-second, 1.7 MByte video in WMV format (ResEnhance3.wmv ) demonstrates this resolution enhancement technique. The signal consists of four overlapping, poorly-resolved Lorentzian bands. First, the 2nd derivative factor (Factor 1) is adjusted, then the 4th derivative factor (Factor 2) is adjusted, then the smooth width (Smooth) is adjusted, and finally the Factor 2 is tweeked again.

The "enhance" command-line function
function Enhancedsignal=enhance(signal,factor1,factor2,SmoothWidth)
Basic function for resolution enhancement by the even-derivative method. The arguments factor1 and factor2 are the weighting factors for the 2nd and 4th derivatives. SmoothWidth is the width of the smoothing function applied to the derivatives. Optimum values for factor1 and factor2 depend on the width and the shape of the peaks in the signal, and also on the desired trade-off between resolution enhancement (peak width reduction) and baseline artifacts that are a by-product of the method. As a starting point, a reasonable value for factor1 is PeakWidth2/25 and for factor 2 is PeakWidth4/833 for peaks of Gaussian shape (or PeakWidth2/6 and PeakWidth4/700 for Lorentzian peaks), where PeakWidth is the full-width at half maximum of the peaks expressed in number of data points. The easiest way to determine the optimum values for your data is to use iSignal or InteractiveResEnhance, described below.

Interactive Resolution Enhancement using the iSignal function
iSignal is a Matlab function that performs resolution enhancement for time-series signals, using the above enhance function, with keystrokes that allow you to adjust the 2nd and 4th derivative weighting factors and the smoothing continuously while observing the effect on your signal dynamically.
View the code here or dowload the ZIP file with sample data for testing. Just place isignal.m in the Matlab path and type

`             >> isignal(DataMatrix); or             >> isignal(x,y);  or             >> isignal(y);  `

where DataMatrix is a matrix with x values in the first row or column and y values in the second.  Press K to see all the keyboard commands.  Use the cursor arrow keys to pan and zoom.

The E key turns the resolution enhancement function on and off.  iSignal is particularly convenient to use because it calculates the sharpening and smoothing settings for Gaussian and for Lorentzian peak shapes using the Y and U keys, respectively. Just isolate a single typical peak in the upper window using the pan and zoom keys, then press Y for Gaussian or U for Lorentzian peaks.  (The optimum settings depends on the width of the peak, so if your signal has peaks of widely different widths, one setting will not be optimum for all the peaks). You can fine-tune the sharpening with the F/V and G/B keys and the smoothing with the A/Z keys. Press K to see all the keyboard commands.

The graphic example on the right is the result of the following commands:

>> x=[0:.005:2];y=humps(x);Data=[x;y];

>> isignal(Data,0.3,0.5,1,3,1,0,1,220,5400);

Press the 'E' key to toggle sharpening ON/OFF to compare before and after peak sharpening.

There is also an older version (which works only in Matlab version 6.5, but not in more recent versions) that uses sliders rather than keyboard controls. This version, however, does not have the automatic calculation of the sharpening and smoothing settings for Gaussian and for Lorentzian peaks.  Click here to download the ZIP file "InteractiveResEnhance.zip" that also includes supporting functions, self-contained demos to show how it works. You can also download it from the Matlab File Exchange.

### InteractiveResEnhance

 Click to view enlarged figure Interactive optimization of derivative resolution enhancement for your own data. Requires Matlab 6.5. To use this, place the data to be enhanced in the global vector "signal", then execute this file. It plots the data and displays sliders for separate real-time control of 2nd and 4th derivative weighting factors (factor and factor2) and smooth width. (Larger values of factor1 and factor2 will reduce the peak widths but will cause artifacts in the baseline near the peak. Adjust the factors for the best trade-off). Use the minimum smooth width needed to reduce excess noise. The resolution-enhanced signal is placed in the global vector "Enhancedsignal". (If the range of the sliders is inappropriate for your signal, you can adjust the slider ranges in lines 27-29).

### DemoResEnhance

 Click to view enlarged figure Self-contained demo of resolution enhancement for a simulated signal of four overlapping peaks. Requires Matlab 6.5. Displays sliders for separate real-time control of 2nd and 4th derivative weighting factors (factor and factor2) and smooth width. Larger values of factor1 and factor2 will reduce the peak width but will also cause artifacts in the baseline near the peak. Adjust these factors for the the best compromise. Use the minimum smooth width needed to reduce excess noise (too much smoothing will reduce the resolution enhancement).

### DemoResEnhance2G

 Click to view enlarged figure Similar to DemoResEnhance, but for a single Gaussian peak. This allows you to experiment with the adjustable parameters that work best for a peak of Gaussian shape. You can change the width of the peak in line 26. The estimated width of the resolution-enhanced peak is computed and displayed above the graph, to make it easier to determine the extent of resolution enhancement quantitatively. Try to adjust the parameters until the estimated peak width is as small as possible, while still giving acceptable baseline flatness.

### DemoResEnhance2L

 Click to view enlarged figure Similar to DemoResEnhance, but for a single Lorentzian peak. Requires Matlab 6.5. This allows you to experiment with the adjustable parameters that work best for a peak of Lorentzian shape. You can change the width of the peak in line 26. The estimated width of the resolution-enhanced peak is computed and displayed above the graph, to make it easier to determine the extent of resolution enhancement quantitatively. Try to adjust the parameters until the estimated peak width is as small as possible, while still giving acceptable baseline flatness.

### ZIP file containing all of the above Interactive Resolution Enhancement functions and demos.

Tom O'Haver
Professor Emeritus
Department of Chemistry and Biochemistry
The University of Maryland at College Park
toh@umd.edu
http://www.wam.umd.edu/~toh

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