Obtaining Motion Blur Parameters Form The Frequency Spectrum
Fourier transform is applied on digital images to interprets their content in terms frequency information. To illustrate, Flat areas, where the intensity is slowly changing, result in low frequencies. Rough areas, on the other hand, result in high frequencies because of the dramatic change in the intensity value. this paper discusses the impact of manipulating the frequency information of digital images and how the frequency spectrum can be used to address a real world situation.
Filtering an image in the frequency domain is usually composed of three steps. First, the Fourier transform is calculated (DCT or DFT). Then, a certain operation is performed on the frequencies (detailed below). Finally, the inverse Fourier transform is applied on the frequency information, resulting in a modified image. The simplest category of filters (also known as the ideal filters) includes the low pass filter, the high pass filter, and the band pass filter. A low pass filter attenuates high frequencies resulting in a smoothing effect. On the contrary, a high pass filter eliminates low frequencies yielding an edge enhancement effect. Lastly, a band pass filter, which is a combination of a low pass and a high pass filters, retains a mid-range of frequencies and suppresses the low and high frequencies that fall out of the range. Band pass filtering can be used to enhance edges (suppressing low frequencies) while reducing the noise at the same time (attenuating high frequencies). Filtering is mathematically simpler to implement in the frequency domain compared to convolution in the spatial domain .
Also, the frequency data reflects the geometrical structure and orientation of an image. Different shapes are usually associated with different scattering patterns as shown Figure 1.
It can be observed that moving objects in still images exhibit a distorted look along their contours, Figure3.a. This distortion is defined as Motion Blur; it results from the relative motion of the object to the camera while its shutter is open. Huei and Kun  proposed that the speed of moving objects in a single image can be estimated using the blur parameters, the camera parameters and imaging geometry. To illustrate, the displacement of a moving object (d) can be determined using similar triangles according to the blur length (k) Figure2. And by knowing the shutter speed of the camera (T), the speed of the object is v =d/T(1). Since this paper is concerned with Fourier Transform, the details of equation (1) are omitted and calculating the blur parameter is detailed below.
The blur parameters, including the blurs’ direction and length, can be determined by examining their impact on the Fourier spectrum. As shown in figure, the Fourier spectrum of the motion blur contains strips of dark lines that are parallel and uniformly separated. Note that the Fourier spectrum...