Question

Problem 6 Define an image sharpening filter containing 30% low-pass filter frequency components and 70% high-pass filter frequency components. Problem 6 Define a Laplacian of Gaussian (LoG) function. Explain its key-advantages and limitations in the image processing

can someone please help me with these. im struggling to answer them. thank you for your help

Answer 1

(a) An image primarily consists of wide range of frequency components. Human eyes are highly sensitive to the edges and minute details of the image. An image will be in good quality if the edges and fine details of the image are very sharp. The edges and fine details are high frequency components which are essential component of the image. If somehow, this high frequency component gets attenuated or removed from the image, the images quality gets reduced. On the other hand, if this high frequency components gets enhanced with some scaling factor, the image quality gets better. The technique of enhancing the peak edges (high frequency components) and fine details of an image is called "Image Sharpening". Steps of Image sharpening: 1. The original image is first pass through High pass filter. The High pass filter extracts all the high frequency components of an image 2. The filtered image is then scaled with some positive quantity 3. The scaled high pass filtered image is then added with the original image to provide a sharper image. 4. The High pass filter is selected such that it gives better transition of edges of an image, simultaneously with suppression of noises in an image A tradeoff between noise attenuation and highlighted edges should be considered Consider the Block Diagram for Image sharpening. High Pass Filter Sharpened Image Original Image One can use more than one High pass filter with different cut-off frequencies and every different high frequency component can be scaled with different values. Hence, this provide image with more sharpening. Image sharpening filter containing 30% low-pass filter frequency components and 70% high-pass filter frequency components: 1. Use High pass filter that pass 70% high frequency component of the bandwidth of an image. 2. The remaining components that is low frequency components should be 30% of Bandwidth. 3. Suppose, an image contains frequency components ranging from 0 Hz to 100 MHz, then the cut off frequency of High pass filter is such that it passes the 70% upper frequency of 100 MHz. Here, the cutoff-frequency of high pass filter should be 30 MHZ. Therefore, the image filter with above requirements can be designed as,

High Pass Filter, f.= 30 MHz Sharpened Image Original Image (b) Laplacian of Gaussian (LoG) function is the Laplacian operator operated on gaussian function. These filters are useful to determine edges in an image. The LoG filter acts as Bandpass filter. The Gaussian function for two-dimensional space is expressed as, f(x,y)= 270 Le 202 When Laplacian operator operated on the 2-D gaussian function, it results in: x2 + y2 - 202 Vf(x,y)= - exp (208) The gaussian function provides smoothening, avoid false edges and minimise error in position of edges. The Laplacian filter provides or compute zero-crossings in an image. The Log operator computes the second derivative of an image and provide results based on analysis: 1. When the image is uniform, LoG is zero. 2. When there is change in image from lighter to darker or darker to lighter, LoG is positive towards darker side and negative towards lighter side. The result will be different for different values of o. 1. When LoG filter with a very narrow Gaussian is used, the resultant image is noisy and when added with original image, the filtered image is noisy. 2. When LoG filter with Gaussian with large o is used, the resultant image is noise-free, but the image is not sharpened and when added with original image, the filtered image is not sharpened. 3. One has to trade-off for values of o followed by Laplacian operator.