Convolution operators are applied in the linear domain.
Echoview's convolution operators fall into the following categories:
Dilation filter n x n
Median filter n x n
Erosion filter n x n
Beam dilation filter 3x3
Beam median filter 3x3
Beam erosion filter 3x3
Beam opening filter 3x3
Beam closing filter 3x3
n x n convolution
X x Y convolution
Blur
Sharpen
Beam convolution 3x3
The algorithms and sample boundary issues are described below.
Please see Convolution operators illustrated for a graphic comparison of these operators.
These operators apply a sliding window of either 3 x 3, 5 x 5 or 7 x 7 data points to the echogram. This window is centered in turn on each data point in the echogram and defines 9, 25 or 49 values including the data point at the center and its neighbors (data points near the edges are a special case).
The value at the center of this window is replaced by a value which is chosen as follows:
|
Type of operator |
Choice of value |
|
Dilation |
the maximum of the values within the window |
|
Median |
the median of the values within the window |
|
Erosion |
the minimum of the values within the window |
The beam opening filters apply an erosion to the entire echogram, followed by a dilation.
The beam closing filters apply a dilation to the entire echogram, followed by an erosion.
These operators also apply a sliding window of either 3 x 3, 5 x 5, 7 x 7 data or X x Y points to the echogram. This window is centered in turn on each data point in the echogram and defines 9, 25, 49 or XxY values including the data point at the center and its neighbors (data points near the edges are a special case).
Unlike the Convolution filter operators above they work with a kernel. This is an array of values, one value for each data point within the window - the kernel is the same size as the window.
For example a 3 x 3 convolution kernel may resemble:
|
0 |
0 |
0 |
|
0 |
1 |
0 |
|
0 |
0 |
0 |
The convolution is applied, by replacing the value of the data point at the center of the sliding window by a new value which is calculated by the convolution operator.
The new value is calculated differently depending on whether the sum of all the kernel cells is zero or non-zero. If the sum of all kernel cells:
is non-zero (the more usual case), then the kernel is normalized (each kernel cell is divided by the sum of all kernel cells) before it is applied.
is zero then the kernel is applied as-is; i.e. without normalization.
This operator applies a user specified sliding window of X x Y data points to the echogram, the convolution algorithm is also user specified.
The values in the X x Y convolution kernel are set to 1.
The Gaussian blur algorithm is an image-blurring filter that uses a normal distribution for calculating data values for the X x Y sliding window. The standard deviation s is used to classify the "strength" of the blurring. The values in the X x Y convolution kernel are calculated using the equation for a Gaussian distribution in two dimensions as follows:
Where:
u is the row index (relative to the center [0,0]) of the specified sliding window.
v
is the column index (relative to the center [0,0]) of the specified sliding window.
s
is the standard deviation of the Gaussian distribution.
0.6 = weak
1.0 = moderate
2.0 = strong
More information on the Gaussian blur convolution can be found in the Wikipedia topic Gaussian blur.
Where sum of all kernel cells is non-zero, the new value is calculated as follows:
Where:
|
R |
is the region of convolution, i.e. the 3x3, 5x5, or 7x7 sliding window. The following cells will be excluded from the region of convolution:
Note: If cells are excluded from the region of convolution, the kernel is re-normalized before the new value is calculated. If it is not possible to normalize the kernel (because the sum of the data values in the remaining cells is zero) the new value of the data point is set to "no data". |
|
i |
is the number of a kernel cell (being 1 to 9, 1 to 25 or 1 to 49 respectively) |
|
n |
is the number of kernel cells (being 9, 25 or 49 respectively) |
|
ki |
is the value of the kernel cell i |
|
vi |
is the value in the sliding window cell i (being the value of a data point in the source variable). |
Note: Although "no data" points are not included in the calculation of new values for surrounding data points, they will be processed as per other kernel cells by convolution operators and may receive a new value from the convolution process. This may "patch up" holes in the data, i.e. calculate values for data points that were previously assigned a "no data" value (e.g. the data points may have been assigned a "no data" value to remove a data spike)
Where sum of all kernel cells is zero, the new value is calculated as follows:
Note: If any cells are excluded from the region of convolution (R), the new value will be "no data". This will result in eroded boundaries and "no data" areas being enlarged on the resulting virtual echogram (in comparison with when the source echogram).
The Blur and Sharpen operators are two specific cases of the general 3x3 convolution operator.
The convolution kernel used in the Blur operator is:
|
1 |
2 |
1 |
|
2 |
1 |
2 |
|
1 |
2 |
1 |
The convolution kernel used in the Sharpen operator is:
|
-1 |
-1 |
-1 |
|
-1 |
9 |
-1 |
|
-1 |
-1 |
-1 |
In the diagram below the cells marked '-' do not contain a data point (i.e. the sliding window is at the edge of the echogram). These cells will be excluded from the convolution region.
If the ping range changes in an echogram, neighboring pings are unlikely to line up in a precise grid. When this occurs, the cells in the sliding window that are not on the same side of the range change as the center cell are excluded from the convolution region. In the diagram below, cells that will be excluded from the two convolution regions are marked with "-".
You can ensure this does not occur by:
Creating a virtual variable that resamples the original operand using settings that will ensure that all pings will line up.
Changing the operand for the convolution virtual variable from the original variable to the resampled virtual variable.
Note: To check for range changes in the echogram, on the Display page of the Variable Properties dialog box, increase the value in the Lower display limit (m) box, until you can see the full range of each ping.
Convolution operators illustrated
Operators
About virtual variables
References
