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Gaussian

The Gaussian filter applies either a Lowpass or a Highpass digital filter to an image along either the Horizontal or the Vertical axis.

 Lowpass Gaussian Filter

The Lowpass Gaussian Filter eliminates high frequency (sharp) features oriented along either the X or Y axis of the scan. The practical effect upon the image is a loss of detail or "blurring" effect.

The Gaussian filter can average features running parallel to an image’s Y scan axis while leaving features relatively unchanged along the X axis, or vice versa. This is a similar capability to the Spectrum 2D function, although applied to only the X or Y axis.

Consider the scan of the grating shown in Figure 1, with prominent features oriented along the X axis.

Figure 1: Grating with Prominent X-Axis Features

Applying a Lowpass Gaussian Filter along the Vertical (Y) axis results in elimination of noise in the image.

Figure 2: Noise Removed by Applying Lowpass Gaussian Filter to Y Axis

NOTE: Notice that rulings running parallel to the Y axis are smoothed along their length by the filter, while features oriented orthogonal to the Y axis remain relatively unchanged. This results in an idealized (averaged) profile of the X axis.

Applying the Lowpass Gaussian Filter to the Horiztonal (X) axis destroys the ruling features in the image by averaging across their profile.

Figure 3: Lowpass Gaussian Filter Applied to Horizontal X Axis

NOTE: Notice how the filter has eliminated the prominent high frequency features along the axis which the filter has been applied, resulting in an almost flat surface.

 Highpass Gaussian Filter

The Highpass Gaussian Filter eliminates low frequency (dull) features oriented along either the X or Y axis of the scan. The "DC" (average) value is also eliminated, resulting in an image containing only the transitions from one region to the next. This filter is typically used for edge detection of different regions or grain bounders.

One example of applying a Gaussian Highpass filter is the magnetic domains in a permalloy specimen, as shown below.

Figure 4: Permalloy Specimen

NOTE: The figure shows the magnetic force microscopy (MFM) image in its original form. This is an early MFM image of a permalloy specimen, and contains artifacts which are significantly reduced in phase analyzed images. Magnetic force is represented in the image as height data. Suppose the microscopist wanted to highlight the magnetic boundaries without regard to magnetic force (height data). A Gaussian Highpass filter would be appropriate.

When the image is filtered using the Horizontal setting, the high frequency features along the X-axis are highlighted.

Figure 5: Highpass Gaussian Filter Using Horizontal Setting

NOTE: Notice that features running parallel to the X-axis (e.g., the tips of each oval area) are washed out, while features running perpendicular to the X-axis are enhanced.

When the image is filtered using the Vertical setting, the high frequency features along the Y-axis are highlighted.

Figure 6: Highpass Gaussian Filter Using Vertical Setting

NOTE: Notice that features running parallel to the Y-axis (e.g., the sides of each oval area) are washed out, while features running perpendicular to the Y-axis are enhanced.

A composite of the two images shows the domain boundaries clearly. All of the low frequency features have been removed, leaving only the transitions between the grain boundaries.

Figure 7: Highpass Gaussian Filter Using Horizontal & Vertical Settings

NOTE: To construct a composite image of the two Gaussian-filtered images, add them together. (This can be accomplished by using the subtract feature with an inverted image). See Subtract Image for details.

Gaussian Kernel Algorithm

Gaussian filters utilize a 1 x N matrix, where N is determined by the filter size parameter. In this instance, image data is analyzed in two-dimensional matrices which are shaped to a Gaussian curve where the sigma value (σ) is determined by the filter size parameter.

Figure 8: Gaussian Filter Depiction

The general equation used to generate a 1-by-(N+1) Gaussian kernel is:

Where i is in units of pixels, and σ is set by the Filter size value. Using this kernel, the filter output is:

Gaussian Filter Size

The Filter Size value corresponds to the sigma (σ) value of the Gaussian curve, encompassing approximately 68 percent of the data with the symmetric Gaussian curve centered over the operated-upon pixel.

Larger Filter size values distribute the curve broadly.

During Lowpass filtering, this lends greater weight to values farther away from the pixel and increases the Gaussian filter’s averaging effects upon the image.

During Highpass filtering, this subtracts a decreased average from each pixel, lessening the filter’s impact.

Figure 9: Larger Filter Size

Smaller Filter size values concentrate curve data around the center value.

Figure 10: Smaller Filter Size

During Lowpass filtering, this lends less weight to pixels distant from the center, decreasing the Gaussian filter’s ability to average local pixels with distant ones—the filter’s impact is lessened.

During Highpass filtering, the larger and more localized pixel average being subtracted from the operated-upon pixel value yields an enhanced impact upon the image.

Filter size is specified in units of Distance, Spatial Frequency, Time, Temporal Frequency, and #pixels.

Gaussian Filtering Procedure

 
  1. Select an image file from the file browsing window at the right of the main window Double click the thumbnail image to select and open the image.
  2. You can open the Gaussian view, shown in , using one of the following methods:
    • Right-click on the image name in the Workspace and select Add View > Gaussian from the popup menu. See Figure 11.

Figure 11: Select Gaussian from the Workspace.

 

Or

  • Right-click on a thumbnail in the Multiple Channel window and select Gaussian.

Or

  • Select Modify > Gaussian from the menu bar.

Or
Click the Gaussianicon in the toolbar.
 
  1. A separate window, shown in Figure 12, opens, also displaying the image.

Figure 12: The Gaussian filter panel

 
  1. Configure the Input parameters.
  2. Complete the Output File Name field to save the result.
  3. Click Execute to perform the Gaussian filtering operation.
  4. To restore the unprocessed image, click the Reload button.

Gaussian Filter Interface

Parameter Description

Filter Size

Size of the scan line to be operated upon by the Gaussian filter kernel. This value is expressed in the Cutoff Units specified below.

Range and Settings:

  • Minimum = 3 pixels
  • Maximum = one-half scan size
Filter Axis

Settings:

  • Horizontal—Applies the one dimensional Gaussian filter along the X axis.
  • Vertical—Applies the one dimensional Gaussian filter along the Y axis.
Cutoff Units

Selected units are applied simultaneously to the Filter size. (The #pixels field displays the pixel equivalent of the current Filter Size value.)

Range and Settings:

  • Distance
  • Spatial frequency
  • Time
  • Temporal Frequency
Filter Type

Range and Settings:

  • Lowpass filtering allows longer wavelength features through while filtering out shorter wavelength features. The net effect is to remove noise in the form of spikes and fuzz on the image.
  • Highpass filtering allows shorter wavelength features through while filtering out longer wavelength features.
#pixels

The current Filter size in pixel units. This value may be used to both enter and monitor the Filter size.

Range and Settings:

  • Minimum = 3 pixels
  • Maximum = one-half scan size
Output File Name Select the path of the extracted image file. Leave blank for immediate view/use without saving the altered image file
Write File Upon Execute Writes the output file(s) when the Execute button is clicked.

Table 1: Parameters in the Gaussian Filter Panel

Button Action

Execute

Applies the Gaussian Filter to the currently loaded image.

Reload

Restores the image to its original form by reloading the original file.

Table 2: Buttons on the Gaussian Panel

 

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