Elliptical
Fourier Analysis - EFA Form & Worksheet
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GRAPHIC RESULT OF PARTIAL EFA ANALYSIS FOR A SINGLE Gomphonema OUTLINE. |
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The 4 Orientation Landmarks are also
located on the chart: |
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PREVIEW mode options (left and top right of form) | EFA ANALYSIS mode options (center and lower right of form) | ||
Valve Selection | ||||
Centering | ||||
Rotation (~apical axis) | ||||
Transapical Axis | ||||
Reflection (axis) | ||||
Trace Origin Normalization | ||||
Smoothing Coordinates | ||||
Sampling Coordinates | ||||
Size Invariance | ||||
PREVIEW | ||||
Number of Harmonics | ||||
Invariance (EFA) | ||||
Analysis | ||||
EXECUTE | ||||
EFA Results | ||||
Progress | ||||
Help | ||||
Chart axis scale control | ||||
Goodness-of-fit: reproduced vs. original outlines. (on worksheet) | ||||
Refresh Menu (on worksheet) | ||||
PREVIEW OPTIONS Options in the Preview section are effected when the Preview button is clicked. |
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Enter
valve number to select a valve for preview or analysis. Frame indicates
the valve number selected out of the entire set of valves, e.g., Valve 5
of 16, on the OrigOutline worksheet. The number must be an integer in the range of 1 to the maximum number of valves in the data set. The default is 1. |
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Outlines
are translated (centered) such that their "centers" coincide with
the coordinate system's origin, i.e., x= 0 , y = 0. THE DEFAULT is Option
Centroid. Option Centroid uses the centroid of the original outline to define the center. Option Landmarks uses the midpoint of the x-coordinates of Orientation Landmarks 1 and 2. Option Custom uses a customized procedure and requires a code be entered for the procedure in the text box adjacent to the option. Option None results in no centering; however, this option is not recommended as the EFA chart is constrained to having its scale maximally +/- 100 on both the x-axis and the y-axis, and thus the outlines will not neccesarily fall within the domain of the chart. |
The
position of the transapical axis is based on either Landmark 4 (its perpendicular
projection onto the apical axis) or a midpoint on the apical axis. The
transapical axis is used for reflection of the outline, should it be desired,
such as reversing the display of the head and tail poles in a Gomphonema.
The DEFAULT is Midpoint. |
The
outline is reflected about either the transapical axis or apical
axis or both. DEFAULT is reflection about both axes, such that Orientation
Landmark 1 is to the left (on the negative x-axis) and Orientation Landmark
3 is above the x-axis (on the positive y-axis). A checked box effects the reflection; an unchecked box disables it. The axes used will be those defined in the Rotation (~apical axis) Frame and the Transapical Axis Frame. Selecting Option None is those frames disables the reflection |
All traces are forced to proceed clockwise, regardless of how they were generated and how the outline may have been reflected in processing. Trace origins may preserve that of the original digitization (Option None), or be reset to coincide the intersection of the outline with the positive x-axis (Option x-axis) or be reset to originate at a specific landmark. Landmark-defined origins require a customized procedure and code; Option Custom). |
The entire set of original
outline coordinates is smoothed by either an average or median filter.
The window size is set in the textbox for both options. The DEFAULT option
is NONE. |
Sampling evenly spaces all the points in the outline and may reduces their number to a smaller set of evenly spaced points. The DEFAULT is to evenly space all the points if their number is <= 1000, and if there are > 1000 points to reduce them to 200 equally spaced points. Option
original: The choice of "original coordinates" is allowed
only if the number of outline coordinates is <= 200. In this case the
200 original coordinates are not resampled and their spacing is not changed;
the analysis proceeds using the original raw coordinates. |
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The DEFAULT is None.
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Clicking
here performs the reorientation, smoothing, sampling, refection and size
normalization procedures chosen and then plots the resulting subset of outline
coordinates (blue markers) on top of the original coordinates (yellow markers)
in the chart on the EFA worksheet. This subset of coordinates will be the
subject of the EFA, when it is executed. The coordinates of this subset
are in columns D & E on the worksheet under "Final Outline". Reorientation, selecting, smoothing, sampling and varying the size sormalization methods may be previewed repeatedly to examine the effects of different combinations of these processes on different valves. |
EFA ANALYSIS OPTIONS | ||
Enter
the number of harmonics to be used to the describe the outline. The number of harmonics must be an integer in the range 1 to 30. The default is 20. On each side of the box indicating the number of harmonics is a button, one with a "+" sign and the other "-" sign. These buttons EXECUTE the EFA after increasing (+) or decreasing (-) the number of harmonics-to-be-extracted by 1, respectively. |
Centering is based on the centroid, rotation and size on "the [slope and] magnitude of the semimajor axis of the best-fitting ellipse, that is, the first order Fourier coefficients corrected to the trace starting point" (Rohlf & Archie 1984). The trace starting point is normalized as the intersection of the positive x-axis and the outline. The scale of specimens for which size has been normalized by the above will generally be different from that of the original and smoothed-and sampled outlines, and so when plotted together these outlines will not "superimpose". One of the major reasons for drafting this program was to provide a visual means of interpreting the effects of different harmonics on the characterization of the outlines. To do this, leave the size box in this section unchecked and examine the results of the EFA while you explore what different harmonics do. When you are ready to export results of the EFA to the worksheets, that is, actually store the data, check the size box is you wish size invariance for the specimens. This will effect the scale of the results, not the effects of the harmonics. |
Option Selected valve specifices an EFA be conducted on a single selected valve (see Select Valve Frame) which displays the Fourier coefficients and the harmonic amplitudes on the EFA worksheet and then plots the reproduced outlines using k [worksheet columns F & G] and k-1 [worksheet columns H & I] harmonics [k is the number of harmonics used in the analysis ("Number of harmonics" frame above)] along with the original and smoothed-and-sampled outlines. The colors of the chart points correspond to colored columns on the EFA worksheet holding the respective outline coordinates. Option All valves specifies EFAs on all valves in the data set on the ImportOutline worksheet. No graphic output occurs during this processing, except for updating of the Progress Bar, unless the Export box is checked. No tabulated output occurs to the EFA worksheet, except for coordinate columns B to I. Selecting Export causes the EFA results of either a single valve or all valves to be written to two worksheets. Data exported to the EFAResults worksheet include basic valve identification and descriptive information, processing modes, Fourier coefficients and harmonic amplitudes. Data exported to the EllipDescr worksheet are basic valve identification and descriptive information, processing modes and the elliptical descriptors of Schmittbuhl et al. (2003). The EXECUTE Button executes the EFA using the smoothing, sampling, normalization and number of harmonics chosen for either a "Selected valve" or "All valves." Reproduced outlines are not stored but are generated independently. |
The CLEAR LAST button deletes information of the last valve processed from both EFAResults and EllipDescr worksheets. CLEAR LAST produces a complete, single-row deletion of all data from the last row on these worksheets. The button can be used repetitively to delete a series of "last" valves. The CLEAR ALL button deletes all outline information on the EFAResults and EllipDescr worksheets. If you choose this option, a message will inquire if you are sure that you want to do this. You might want to copy these data to a new worksheet before deletion. |
When calculations are expected to take more than a few seconds, this progress indicator is activated. This will usually occur when Median Filters are selected for smoothing or when option All Valves is selected and outlines are processed as a batch in the EFA. |
Clicking displays abbreviated versions of this Help page. The green Help button accesses the Preview sections, the red Help button the EFA execution sections. |
The scroll bar at the bottom right of the EFA form controls the upper and lower limits of the axes of the chart of the valve outlines. Both abscissa and ordinate are constrained to have the same upper and lower limits. The axes in this Chart should ONLY be modified using this scroll bar. |
For ever analysis the absolute linear distance ("error") between members of each pair of coordinates in the reproduced (k harmonics) and the smoothed-and sampled outline is recorded in column J on the EFA worksheet. Cells J3:J9 provide a summary of the statistical descriptors of the distribution of these "errors" . These provide a quantitative measure of the differences between the outlines seen on the chart. | ![]() |
On occasion the CDiatom item in the Main Menu diappears. Usually this is a result of the cursor focus being on an embedded chart. Selecting anything other than the chart will usually restore CDiatom. But not always, for reasons I am unable to fathom, so I have provided a Refresh Menu button which will do so. |
Updated 1 July 2009 RKE
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