Exercise 1
Particular features of the Fourier Image
- The Fourier analysis of the images in pre-set preview form the run program showcase the transformation action performed on the image. From the exercise one lab program the Fourier image shows how one can gain access to the geometric features of the real domain imagery from the original image.
- The particulars of the Fourier imagery from the image previews are lifted in such a way that image mean, otherwise denoted to as the DC-value is shown in the middle part or focus of the imagery.
- The Fourier image transform showcased contains the scale premeditated from the compound result. One can appropriately note and can recognize that the given DC-value is greatly abundant in the module of the image.
- Though, the dynamic series for the Fourier constants, which is the concentration standards in the Fourier imagery, can be noted to be far off big to be shown on the monitor screen as seen from the Fourier image, thus, all other consequential values give the impress as black on the screen as we can see from the Fourier image.
Fourier transforms of an image with little periodicity
- As showcased from the given images from the lab activities the effect highlight on the impact that edges on an image show. The greatness of the spectrum is significant in this case, and thus we can note that images that are pure cosines have mainly simple FTs.
How the Fourier transformations relate to the original Images
- You may start to notice there is a lot of proportion. For all REAL, as disparate to complex images, the FT is balanced about the starting point so the 1st and 3rd quadrants are similar and the 2nd and 4th quadrants are also identical. If the image is proportioned about the x-axis, as the cosine imageries are then 4-fold proportion outcomes.
- The consistency in white lines and dots are as the resulting image after a transform has a lowpass filtered version of the original real space image.
- The vertical image displayed from the transformation in the first exercise is due to the threshold scale of the Fourier image. One can note that the highest values lie within the vertical line, which signifies that the text lines in the original image are horizontal.
- The image of the ship outputs a Fourier transformation that has points on the center and the corner show that it is a logarithmic transform of the original image. This explains that the output image contains many minor frequencies hence the scattered lines at the center and corners.
Reason for logarithmic transformation
- The outcome demonstrates that the image comprises modules of all frequency occurrences; nonetheless the scale gets lesser for advanced occurrences. Therefore, low frequency occurrences have more imagery information evidence than the advanced and higher ones in the setup.
- The altered or transform imagery moreover conveys to us that there are two dictating commands that can be noted from the Fourier image, that is, the one crossing horizontally through the center and the one crossing vertically.
- The logarithmic transformation is aimed at compressing the dynamic range of an image through substituting each pixel rate with its logarithm.
- This resultant imagery initiates from the consistent arrays in the given background of the initial image.
- This resultant image has the outcome that the low power pixel value standards are improved.
- When relating the pixel logarithm to an imagery preview it can be convenient in the uses where the active range may too outsized thus cannot be shown on a screen monitor.
Exercise 2
PART A
The relationship between images of the original image, the transform, and the inverse transform of the transform.
- First, the basic functions for the Fourier Transform (FT) gotten from the original try to exemplify all images as an abstract of cosine-like images. Therefore images that are pure cosines have particularly simple FTs.
- The image transformation is resultant to the original shape as illustrated in the showcased images recognizable after the inverse transform of the transform.
- Subsequently, the resulting image is a lowpass filtered version of the original spatial domain image.
- Thus, applying the inverse Fourier Transform to the complex image yields the same image relatively same to the original image.
- Conferring to the mentioned distributive law, the resultant imagery is similar as the straight quantity of the double unique real space imagery.
PART B
- The noise circle and respective transform image represent the noise that consists largely of high frequencies, which are attenuated by a lowpass filter.
- The construction of a filter would in a procedural approach entail the attenuation of low frequencies to enhance the edges of the image.
- This is because all frequency occurrence filters can also be applied in the spatial domain and, if there is the existence of a mask for the anticipated filter outcome effect, it is computationally less expensive to perform the filtering in the real domain.
- The frequency incidence filtering is more suitable if no direct mask can be obtained in the spatial initiated domain.
- The frequency occurrence filtering is established on the Fourier Transform; the operative agent typically takes an imagery snapshot and filters the given function in the particular Fourier domain.
- This input image is then increased with the filter particular equation function in a pixel by pixel technique.
- Then the provided inverse filtering of the imagery is accomplished and executed through utilizing the initial image so as to de-blur thus it produces the low comparison contrast outcome.
PART C
- The application of the ideal low-pass is aimed at constructing an efficient spectrum of the circle with incoherent noise. The implications of this to the final image cause a distortion on spectrum.
- Although one can succeed to find an inception which splits the foremost peaks from the contextual background, we have a sensible or rational amount of noise in the Fourier image resultant from the uneven design of the letters.
- One might reduce these contextual standard values and therefore escalate the variance to the central top most values. If we remained able to make solid blocks out of the portrayed text-lines. This could, for instance, be completed by exhausting the utility of a morphological operative agent.
- We can decrease the noise by a lowpass filter, since noise entails mostly high occurring frequencies, which are mitigated by a lowpass filter.
PART D
- The Butterworth filter has proven to provide better results can be achieved with an instantaneous cut-off frequency of maybe 1/3 for example.
- With an intention to preserve more detailed specifics, one should escalate the cut-off occurrence or frequency to for example 0.5, as can be recognized from the portrayed images.
- This image copy is less blurry; nonetheless it also covers a rational quantity of noise.
- In overall, when utilizing a lowpass filter to decrease the high frequency noise, one has to reduce certain necessary high frequency statistics if one needs to regulate a significant volume of noise.
- This image does not show any visible ringing and only little noise.
- However, it also lost some image information, that is, the edges are blurred and the image contains fewer details than the original.
- The Butterworth filter is a better implementation for wide lowpass filter.
References
Marks, R. J. (2009). Handbook of Fourier analysis & its applications. Oxford: Oxford University Press.
Bracewell, R. N. (2003). Fourier analysis and imaging. New York: Kluwer Academic/Plenum Publishers.
