Line Detection Using Hough Transform

1. Introduction

Line detection is extremely crucial for numerous real-world applications, including driving systems, navigation robots, and image processing operations [1,2]. The Hough Transform. is a highly efficient computer vision algorithm that can be used especially here. It offers the ability to detect straight lines in images. The Hough Transform. works by converting points of the original image into a new parameter space. In this new area, straight lines correspond to plain recognizable patterns that make it simpler to identify line parameters even in noisy images.

We discuss two specific tasks of line detection based on the Hough Transform. in this report [4,5]. The first problem utilizes MATLAB's inbuilt functions [3] to carry out edge detection and line detection from an image. After extracting the lines, we study their nature, that is, verifying whether the lines are parallel or not. The second problem involves performing the same line detection but carries out a thorough analysis using special methods and comparing the outcome. Our objective is to explicitly demonstrate how the Hough Transform. can be efficiently used, both through regular MATLAB operations and custom procedures, and thus understand its actual-world performance and computational accuracy.

2. Task 1: Line Detection Using MATLAB Built-in Functions

2.1 Task Description

The line detection in this study is performed using MATLAB in-built functions [3], including edge, hough, and houghpeaks. These functions provide an easy way to detect edges in images using the Hough Transform. [4], as well as important line extraction. This section merely provides a general idea of how MATLAB's basic tools are utilized to efficiently detect and extract lines.

2.2 Image Preprocessing and Edge Detection

The first step in line detection is generally the edge detection in the original image. We apply MATLAB's edge function to this, which identifies areas of abrupt intensity changes in an image, which typically correspond to edges. The edge detection must be accurate since it directly affects the precision of the subsequent line detection. Thus, the preprocessing process involves selecting the appropriate parameters for the edge detector to get clear, visible edges.  

2.3 Hough Transform

The second step is the application of the Hough Transform. using MATLAB's inbuilt function hough after edge extraction [5]. It transforms points on detected edges from the original image into a collection of points in a second parameter space. Specifically, it parameters lines in two modes: by distance r and by angle θ. Each edge point in the image space corresponds to a set of points in this parameter space, and the intersections in the parameter space indicate the presence of straight lines in the original image. The locations of the salient lines can be accurately obtained by inspecting these intersections using the houghpeaks function..  

2.4 Line Extraction and Parallelism Analysis

The next step involves using the houghpeaks function to identify the peaks in the Hough accumulator array, which correspond to the most prominent lines in the image. We extract the parameters r and θ for these lines. To analyze whether the detected lines are parallel, we check if the values of θ are identical or very close. Parallel lines should have nearly the same θ values, differing only by small numerical inaccuracies. If the difference between the θ values of two lines is smaller than a defined threshold, we consider the lines to be parallel.

2.5 Results and Analysis

With the capabilities offered by MATLAB [3], we were able to select lines from the image and verify whether they are parallel or not. The experimental results show that in Figure 1a, all the lines detected from the image are of 0-degree angle, which indicates that these detected lines are parallel to each other. That is, a 0-degree line is horizontal, and hence in this case, all the image-detected lines are parallel to each other. Whereas in Figure 1b, the lines drawn are of angles 5°, 0°, -5°, and -10°. The lines are of different angles and thus the angle between them is not the same. Therefore, in conclusion, they are not parallel to each other. As can be seen from the above results, even slight differences in the angles can lead to significant differences in whether the lines are parallel or not.

Figure_1a

Figure_1b

3. Task 2: Line Detection Using Custom Functions

3.1 Task Description

The second task is to do edge detection, Hough Transform, and line extraction using our own customized functions. We wish to show a better grasp of the Hough Transform. process by implementing our own customized version of the algorithms. We develop our own customized versions of the edge, hough, and houghpeaks functions, namely myedge, myhough, and myhoughpeaks.

3.2 Custom Hough Transform. Implementation

Our program begins by defining our own function, myEdge.  To find edges, this function works like MATLAB's built-in edge detector but in a simpler way regarding how edges are found in the image.  It essentially finds edges by computing gradient data and thresholding functions.

Secondly, to use the Hough Transform, we call the main function, myEdge.  The function assigns each edge pixel detected in the image a location in a two-dimensional parameter space.  The parameter space keeps possible locations of lines in terms of parameters, typically by r and θ.  Each edge pixel votes for cells in an accumulator matrix based on its corresponding parameters.  Over time, the accumulator becomes saturated, with higher values representing greater potential lines.

Once the accumulator is filled, then comes detecting the dominant lines through a user-specified peak-detection function called myHoughPeaks.  The function finds the most voted cells within the accumulator matrix corresponding to the strongest and most probable lines in the input image.  Peaks are identified as local maxima and thus represent the strongest and most probable lines.  By reversing the parameter space correspondence to the original image coordinate, we have real values of the parameters r and θ for all the lines found.

3.3 Code Explanation

We provide a brief explanation of our own custom functions as given below:

 Edge Detection: The custom myEdge function uses gradient calculations to detect edges by comparing neighboring pixel intensities.  The output is a binary edge map.

 Hough Transform. The myHough function iterates through the edge points, calculates the corresponding r and θ, and stores these in an accumulator matrix.

 Peak Detection: The myHoughPeaks function scans the accumulator matrix for local maxima and extracts the parameters of the detected lines.

3.4 Computational Complexity Analysis

We study the complexity of the custom Hough Transform. algorithm on the basis of the following main steps:

 Edge Detection: The edge detection algorithm tests every pixel in the image.  This generates a time complexity of O(N×M), where M and N are the dimensions of the image.

 Hough Transform. Hough Transform. contributes a time complexity of O(N×M×T×R), where T and R are the discretization steps along the angle θ and distance r.

 Peak Detection: The peak detection process involves scanning the Hough accumulator matrix, which has a complexity of O(T×R).

From the analysis of the three major stages' complexity, the time complexity of the tailored implementation is O(N×M×T×R).  This makes the implementation relatively computationally expensive, especially for high discretization values of θ and r and large images.

3.5 Performance Comparison

When comparing the implementation of our own with MATLAB's native functions, we noticed that our own solution performs slower since it incorporates extra steps of computation which are less optimized compared to MATLAB's built-in processes.  However, the largest advantage of developing a custom procedure is having more control over parameterization and the way the Hough Transform. behaves.  On the other hand, MATLAB's built-in tools are carefully optimized for speed, efficiency, and accuracy and thus more appropriate for handling large data sets or operations that must be performed quickly.

We observed in our tests that nearby or parallel lines were well detected by the self-developed line detection routines at the expense of increased processing overhead.  For instance, in one of our test images (Figure 1a), eight lines were well detected to be perfectly horizontal (0°), and in our second test image (Figure 1b), the self-developed function was able to detect lines at 5°, 0°, -5°, and -10° successfully.  These results affirm that our own application can successfully detect lines accurately, closely simulating the real orientations present in the images.

4. Results and Discussion

Both our implementation and the built-in version of the Hough Transform. were able to detect the prominent lines in the test images. The results of both methods are comparable, showing that our own tailored algorithms function properly versus built-in functions. One of the advantages of using our self-created functions was that we were able to directly control and adjust some of the parameters in the algorithm such that we were able to test and better understand how selections of parameters would influence the detection outcome. In deciding whether the lines that were computed were parallel, we compared the angles θ obtained from both versions. The results show minimal variations in corresponding line angles, which means that both implementations were stable and consistent in determining line orientations, and it was easy to verify the parallelism of the detected lines. Another interesting fact is the real-world trade-off between convenience and control. While MATLAB's built-in functions give instantaneous and accurate results, the user-defined implementation is useful for providing flexibility, particularly in academic work and complex applications with special algorithmic modifications.

5. Conclusion and Future Work

Both the default and our own custom versions of the Hough Transform. performed adequately in line detection in the sample image. But implementing our own custom functions provided us with a deeper insight into the internal processes and mechanisms of the algorithm, and enabled finer tuning. Future development can make our custom implementation more efficient by optimizing for faster computation and lower processing time. Besides, future work might include making the algorithm more robust to handle unfavorable conditions, for instance, noisy images or images with complex backgrounds.

References

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