# A Novel GNSS Interference Detection Method Based on Smoothed Pseudo-Wigner-Hough Transform
## 4 Smoothed Pseudo-Wigner-Hough Transform  
The Hough transform is commonly used to isolate features of a particular shape within an image space, which can be particularly employed to change a difficult global detection of features in the image space into a relatively easy problem of local peak detection in the parameter space. In the parameter space, the noisy input points are usually randomly distributed, which cannot be effectively accumulated; therefore, the accumulated energy is fairly weak. On the other hand, in the image space, all the collinear points on the same straight line representing the frequency modulation law of the analyzed signal indicate their effective contributions to the feature extraction by accumulating the corresponding energy peak concentrated in the parameter space. The simple case of the Hough transform can be adopted to detect the straight lines; it is known that the straight line \(y=kx+b\) can be generally expressed as a point \((b,k)\) in the parameter domain; however, there are problems for the cases of the vertical lines which result in unbounded values of the slope parameter \(k\). Therefore, in Figure 5, the radius-and-angle parametrization of a straight line is employed due to computational complexity reasons, which can be defined algebraically by the following normal notion [25]:  
\[\rho=x\cos\theta+y\sin\theta \tag{13}\]  
where a point \(P(x,y)\) in the \(x-y\) image space can be mapped as \((\rho,\theta)\) in a \(\rho-\theta\) transform space, \(\rho\) is the length of a normal from the coordinate origin to the straight line, and \(\theta\) represents the orientation of \(\rho\) with respect to the \(x\) axis. In this way, it is reasonable to associate a pair \((\rho,\theta)\) in the polar Hough parameter space with each straight line in the Cartesian image space. If a single point \((x,y)\) is considered in the Cartesian image space, then the family of all the straight lines that pass through this point \((x,y)\) definitely maps to a particular curve with the sinusoidal form in the \(\rho-\theta\) polar Hough parameter space; furthermore, this mapping relationship is deterministic and unique. A set of two or more points that connect a straight line in the cartesian image space is bound to generate a series of sinusoidal curves that all intersect at the same point \((\rho,\theta)\) in the transform space corresponding to the determined straight line. Therefore, the problem of globally searching for collinear points on a straight line in the Cartesian image space is able to be effectively converted to the easily solved problem of detecting concurrent sinusoidal curves locally intersected in the same point \((\rho,\theta)\) within the polar Hough parameter space. Thus, an interesting and effective scheme for detecting straight lines can make use of this point-line transformation. The detection of the point \((\rho,\theta)\) in the polar transform space is relatively easier and more effective than detecting multiple collinear points in the Cartesian image space. When the position of the intersection point \((\rho,\theta)\)  
Figure 5: The normal parameterization of a straight line. of concurrent sinusoidal curves is determined by searching for the Hough-transformed energy peak in the \(\rho-\theta\) transform space, the parameters of the corresponding straight line in the Cartesian image space can then be estimated [16]. As an efficient solution to search for straight lines in the image space, the Hough transform can be used in combination with the Wigner-Ville distribution to develop the Wigner-Hough transform method [17; 18], which can be used to deal with the cross-term problem with the Wigner-Ville distribution; hence, the Wigner-Hough transform can be used in GNSS interference detection [23].