Hof transformation Principle

Development of research and application of Hough transformation at Home and Abroad
He was proposed by Paul Hough in 1962 and published as a patent in the United States. It implements a ing from image space to parameter space. Because of its obvious advantages and valuable properties, it has attracted the attention of many domestic and foreign scholars and engineering technicians. For example, because it calculates a full description parameter based on a local metric, some interruptions occur to the boundary caused by noise interference on the region boundary or being covered by other targets, it has good fault tolerance and robustness. Over the years, experts have conducted in-depth and extensive research on the theoretical nature and application methods of the Hough transformation, and have made many valuable achievements.

The essence of the Hough transformation is to cluster pixels with certain relationships in the image space, and find the parameter space to accumulate corresponding points that can associate these pixels in a certain resolution form. When the parameter space does not exceed two dimensions, this transformation has an ideal effect. However, as the parameter space increases, the computing workload increases sharply, and the storage space consumes a huge amount of time. In this regard, over the years, many scholars at home and abroad have explored many aspects of the conventional hough transformation based on specific situations, and put forward many valuable improvement methods.

The basic principle of Hough transformation is to convert the given curve of the original image space into a point in the parameter space through curve expression by using the parity between the point and the line. In this way, the detection problem of the given curve in the original image is transformed into the peak problem in the search parameter space. That is, the overall detection feature is transformed into a local detection feature. For example, straight lines, ovans, circles, and arcs.

II. The basic idea of Hough Transformation

If a black and white image is known, a straight line is drawn, and the position of the straight line is required. We know that the linear equation can use y = K * x + B.
K and B are parameters, which are slope and intercept. All linear parameters of a certain point (x0, y0) will satisfy the equation Y0 = kx0 + B. Point (x0, y0) determines a family of straight lines. Equation Y0 = kx0 + B is a straight line on the K -- B plane of the parameter (you can also be a straight line corresponding to Equation B =-x0 * k + y0 ). In this way, a foreground pixel on the x-y plane corresponds to a straight line on the parameter plane. Let's take an example to illustrate how to solve the previous problem. Set the line on the image to Y = X,
Let's first take the above three points: A (), B (), C (22 ). We can find that the parameter of the straight line passing through the point must satisfy the equation B = 0,
The parameters of a straight line passing through B must satisfy equation 1 = K + B,
The parameter of the straight line passing through the C point must satisfy the equation 2 = 2 K + B,
These three equations correspond to three straight lines on the parameter plane, and these three straight lines will intersect at one point (k = 1, B = 0 ). Similarly, other points (such as (3, 3) and (4, 4) on the X line in the original image) correspond to the straight lines on the parameter plane through the points (k = 1, B = 0 ). This feature provides a solution for us to solve the problem, that is, to map the points on the image plane to the line on the parameter plane, and finally solve the problem through statistical features. There are two straight lines on the dummy plane, so two peak points will be seen on the parameter plane, and so on.

In short, the idea of the Hough transformation is: a point in the original image coordinate system corresponds to a straight line in the parameter coordinate system, and a straight line in the same parameter coordinate system corresponds to a point in the original coordinate system. Then, all vertices of a straight line are displayed in the original coordinate system. Their slope and intercept are the same, so they correspond to the same point in the parameter coordinate system. In this way, after each point in the original coordinate system is projected under the parameter coordinate system, check whether there are any clustering points under the parameter coordinate system. Such a clustering point corresponds to a straight line in the original coordinate system.

In practical application, the linear equation in the form of Y = K * x + B cannot represent a straight line in the form of X = C (at this time, the slope of the straight line is infinite ). In practice, the parameter equation P = x * Cos (theta) + y * sin (theta) is used ). In this way, a point on the image plane corresponds to a curve on the p-Theta plane. The others are the same.

Iii. Promotion of Hough Transformation

1. Circle with known radius

In fact, the best way to detect any curves with known expressions is to look at the choice of its parameter space. The choice of the parameter space can be determined by its expression form. For example, the circle is expressed in the form, so when detecting a circle with a certain radius, you can select the same space as the original image space as the parameter space. Then, a circle in the circular image space corresponds to a point in the parameter space. A point in the parameter space corresponds to a circle in the image space, and a point in the circular image space on the same circle, they have the same parameters, that isA, BSame, then their corresponding circles in the parameter space will pass through the same point (A, B), So after all vertices in the original image space are transformed to the parameter space, based on the aggregation degree of the midpoint of the parameter space, you can determine whether the image space is similar to the circle. If yes, this parameter is the circle parameter.

2. Circle with unknown radius

When the radius of a circle is unknown, it can be regarded as the detection, center, and radius of a circle with three parameters. At this time, the principle is still the same, but the dimension of the parameter space increases and the computing workload increases. Any point in the image space corresponds to a cluster of circular curves in the parameter space ., It is actually a cone type. Any point in the parameter space corresponds to a circle in the image space.

3. elliptic

An elliptic has five free parameters, so its parameter space is five dimensions. Therefore, the calculation workload is very large, so many improved algorithms are proposed.

Iv. Summary

In an image space, points in the same circle, straight line, and elliptic point correspond to a graph in the parameter space. In the image space, these points meet the conditions of their equations, therefore, each projected image goes through the points in this parameter space. That is, in the parameter space, they will intersect at one point. Therefore, the larger the intersection point in the parameter space, the greater the number of images that meet this parameter in the image space. More like what we want to detect.

If you specify the equation, you can find any curve by using the Hough transform. In terms of determining the target of a known shape, the Hough transformation has the advantage of being affected by curve interruptions and not by image rotation, and can be correctly identified even if the target has a slight defect or pollution.

 

L extend the application scope and propose multiple parameterization methods
In earlier studies, the Hough transformation only detects the straight lines in the image and expands to the shapes of detected arcs, quadratic curves, and arbitrary curves; line parameterization methods have also evolved from initial intercept parameters to slope inclination and intercept parameters, dual-hough space parameters, as well as circle center coordinates, radius parameters, and complex shape detection. template-based multi-dimensional key point parameters.
L improve real-time performance and propose a variety of methods to reduce computational workload
Aiming at the deficiencies in the large amount of computing in the Hough transformation, we have successively proposed the four-tree structure of the like, the gradient information-guided, the like, the layered like, the adaptive like, the fast adaptive like, and the random like the like; for high-dimensional Hough transformations, dimensionality reduction is adopted, and data structures are mostly quantified dynamically.
L enhance anti-interference capability and improve detection accuracy
Abstract: The accuracy of the extraction by using the key-value-transform algorithm has always been widely concerned, such as the discretization error, overlapping interference, and anti-noise interference performance of the key-value-transform algorithm. For example, kiryati and buckstein proposed to use the best kaider window function to smoothly filter the parameter fields to reduce the mixing Error; hunt, Nolte, and others used the Signal Detection Theory to compare the anti-interference performance of the Hough transform and the best algorithm based on the maximum posterior probability, and pointed out the reasons that affect the anti-interference performance of the Hough transform.
L various Peak Detection Methods

The peak value detection of parameter space is a clustering detection problem in the Hough transformation, and the selection of threshold is the key to success or failure. One method is to weight the image space to change the peak distribution of the parameter space, and the other is to directly search for the maximum value of the parameter space.

Although conventional hough transformation has significant advantages, its shortcomings cannot be ignored. Without improvement, it may be at a loss. For example, the detection speed is too slow to achieve real-time control; the accuracy is not high enough. If the expected information cannot be detected, the system makes incorrect judgments, resulting in a large amount of redundant data. In terms of Circle Detection, there are mainly the following shortcomings in conventional Hough Transformation:
1. The parameters are raised from the two parameters of a straight line, namely the intercept and slope, to three, that is, the Center Coordinate and radius. Each point is mapped to a surface in the parameter space, is one to multiple ing, so the amount of computing increases dramatically;
2. A large amount of memory space is required, which takes a long time and has poor real-time performance;
3. In reality, images are generally subject to external noise interference, and the signal-to-noise ratio is low. In this case, the performance of conventional hough transformation will be greatly reduced. It is difficult to determine the maximum parameter space for search due to the appropriate threshold value, there are often "virtual peaks" and "missed checks.
This topic aims at the above issues. Due to the considerable efforts made by our predecessors, there are many types of improvement algorithms in the current process. Here, we only choose a mainstream improvement direction, that is, Random Hough Transformation (RHT, in comparison with conventional hough transformation, this paper analyzes its performance characteristics. Although rht also has obvious defects, there are also many improvement algorithms for it. However, due to limited time and energy, the most perfect effect cannot be achieved for the moment, however, this algorithm is very effective when the number of detected circles is small and the edge extraction effect is good.