lines_gauss — Detect lines and their width.
The operator lines_gauss can be used to extract lines (curvilinear structures) from the image Image. The extracted lines are returned in Lines as subpixel precise XLD-contours. The parameter LightDark determines, whether bright or dark lines are extracted. If ExtractWidth is set to 'true' the line width is extracted for each line point. If LineModel is set to a value different from 'none', lines_gauss compensates the effect of asymmetrical lines (lines having different contrast on each side of the line), and corrects the position and width of the line. The line model used for the correction can be selected with LineModel as bar-shaped lines (LineModel = 'bar-shaped'), parabolic lines (LineModel = 'parabolic'), and Gaussian lines (LineModel = 'gaussian'). Bar-shaped lines are the right choice for most applications. If backlit tubular objects (e.g., blood vessels in X-ray images) should be extracted, the other two modes can be used. The parabolic line model should be used in applications where the lines appear very sharp. The Gaussian line model should be used in applications where the lines appear less sharp. The parameter LineModel is only meaningful if ExtractWidth='true'. Because the line extractor is unable to extract certain junctions because of differential geometric reasons, it tries to extract these by different means if CompleteJunctions is set to 'true'.
The extraction is done by using partial derivatives of a Gaussian smoothing kernel to determine the parameters of a quadratic polynomial in x and y for each point of the image. The parameter Sigma determines the amount of smoothing to be performed. Larger values of Sigma lead to a larger smoothing of the image, but can lead to worse localization of the line. Generally, the localization will be much better than that of lines returned by lines_facet with comparable parameters. The parameters of the polynomial are used to calculate the line direction for each pixel. Pixels which exhibit a local maximum in the second directional derivative perpendicular to the line direction are marked as line points. The line points found in this manner are then linked to contours. This is done by immediately accepting line points that have a second derivative larger than High. Points that have a second derivative smaller than Low are rejected. All other line points are accepted if they are connected to accepted points by a connected path. This is similar to a hysteresis threshold operation with infinite path length (see hysteresis_threshold). However, this function is not used internally since it does not allow the extraction of subpixel precise contours.
For the choice of the thresholds High and Low one has to keep in mind that the second directional derivative depends on the amplitude and width of the line as well as the choice of Sigma. The value of the second derivative depends linearly on the amplitude, i.e., the larger the amplitude, the larger the response. For the width of the line there is an approximately inverse exponential dependence: The wider the line is, the smaller the response gets. This holds analogously for the dependence on Sigma: The larger Sigma is chosen, the smaller the second derivative will be. This means that for larger smoothing correspondingly smaller values for High and Low have to be chosen. Two examples help to illustrate this: If 5 pixel wide lines with an amplitude larger than 100 are to be extracted from an image with a smoothing of Sigma = 1.5, High should be chosen larger than 14. If, on the other hand, 10 pixel wide lines with an amplitude larger than 100 and a Sigma = 3 are to be detected, High should be chosen larger than 3.5. For the choice of Low values between 0.25 High and 0.5 High are appropriate.
The parameters Low and High can be calculated from the respective gray value contrast of the lines to be extracted (ContrastLow and ContrastHigh) and from the chosen value for Sigma with the following formula:
The extracted lines are returned in a topologically sound data structure in Lines. This means that lines are correctly split at junction points.
lines_gauss defines the following attributes for each line point if ExtractWidth was set to 'false':
The angle of the direction perpendicular to the line
The magnitude of the second derivative
If ExtractWidth was set to 'true', the following attributes are defined in addition to 'angle' and 'response':
The line width to the left of the line
The line width to the right of the line
If ExtractWidth was set to 'true' and LineModel to a value different from 'none', the following attributes are defined in addition to 'angle', 'response', 'width_left', and 'width_right':
The asymmetry of the line point
The contrast of the line point
Here, the asymmetry is positive if the asymmetric part, i.e., the part with the weaker gradient, is on the right side of the line, while it is negative if the asymmetric part is on the left side of the line. All these attributes can be queried via the operator get_contour_attrib_xld.
lines_gauss can be executed on OpenCL devices.
In general, but in particular if the line width is to be extracted, should be selected, where w is the width (half the diameter) of the lines in the image. As the lowest allowable value must be selected. If, for example, lines with a width of 4 pixels (diameter 8 pixels) are to be extracted, should be selected. Note that the attributes 'width_left', 'width_right', 'asymmetry', and 'contrast' are set to zero if Sigma is set too low.
lines_gauss uses a special implementation that is optimized using SSE2 instructions if the system parameter 'sse2_enable' is set to 'true' (which is default if SSE2 is available on your machine). This implementation is slightly inaccurate compared to the pure C version due to numerical issues. If you prefer accuracy over performance you can set 'sse2_enable' to 'false' (using set_system) before you call lines_gauss. This way lines_gauss does not use SSE2 accelerations. Don't forget to set 'sse2_enable' back to 'true' afterwards.
When lines_gauss is run on OpenCL devices, the same limitations apply as for derivate_gauss: Sigma must be chosen so that the required filter mask is smaller than 129 pixels. Also note that the results can vary compared to the CPU implementation.
Amount of Gaussian smoothing to be applied.
Default value: 1.5
Suggested values: 1, 1.2, 1.5, 1.8, 2, 2.5, 3, 4, 5
Typical range of values: 0.7 ≤ Sigma ≤ 20
Recommended increment: 0.1
Lower threshold for the hysteresis threshold operation.
Default value: 3
Suggested values: 0, 0.5, 1, 2, 3, 4, 5, 8, 10
Typical range of values: 0 ≤ Low ≤ 20
Recommended increment: 0.5
Restriction: Low >= 0
Upper threshold for the hysteresis threshold operation.
Default value: 8
Suggested values: 0, 0.5, 1, 2, 3, 4, 5, 8, 10, 12, 15, 18, 20, 25
Typical range of values: 0 ≤ High ≤ 35
Recommended increment: 0.5
Restriction: High >= 0 && High >= Low
Extract bright or dark lines.
Default value: 'light'
List of values: 'dark', 'light'
Should the line width be extracted?
Default value: 'true'
List of values: 'false', 'true'
Line model used to correct the line position and width.
Default value: 'bar-shaped'
List of values: 'bar-shaped', 'gaussian', 'none', 'parabolic'
Should junctions be added where they cannot be extracted?
Default value: 'true'
List of values: 'false', 'true'
* Detection of lines in an aerial image read_image(Image,'mreut4_3') lines_gauss(Image,Lines,1.5,3,8,'light','true','bar-shaped','true') dev_display(Lines)
Let A be the number of pixels in the domain of Image. Then the runtime complexity is O(A*Sigma).
The amount of temporary memory required is dependent on the height H of the domain of Image and the width W of Image. Let S = W*H, then lines_gauss requires at least 55*S bytes of temporary memory during execution.
lines_gauss returns 2 (H_MSG_TRUE) if all parameters are correct and no error occurs during execution. If the input is empty the behaviour can be set via set_system(::'no_object_result',<Result>:). If necessary, an exception is raised.
bandpass_image, dyn_threshold, topographic_sketch
C. Steger: “Extracting Curvilinear Structures: A Differential
Geometric Approach”. In B. Buxton, R. Cipolla, eds., “Fourth
European Conference on Computer Vision”, Lecture Notes in Computer
Science, Volume 1064, Springer Verlag, pp. 630-641, 1996.
C. Steger: “Extraction of Curved Lines from Images”. In “13th International Conference on Pattern Recognition”, Volume II, pp. 251-255, 1996.
C. Steger: “An Unbiased Detector of Curvilinear Structures”. IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 20, no. 2, pp. 113-125, 1998.