Digital Image Processing Examination 1. Fourier Transform problem. 1） For an image given by the function f(x,y)=(x+y)3 where x,y are continuous varibales; evaluate f(x,y)(x-1,y-2) and f(x,y)* (x-1,y-2),where is the Dorac Delta function.2） For the optical imaging system shoen below,consisting of an image scaling and two forward Fourier transforms show that the output image is a scale and inverted replica of the original image f(x,y).f(x,y)Scalingf(ax,by)FFg(x,y)_3) three binary images (with value 1 on black areas and value 0 elsewhere) are shown below. Sketch the continuous 2D FT of these images(dont do this mathematically, try to use instead the convolution theorem and knowledge of FTs of common functions)2. The rate distortion function of a zero memory Gaussian source of arbitary mean and variance 2 with respect to the mean-square error criterion isa) Plot this functionb) What is Dmaxc) If a distortion of no mor than 75% of the sources variance is allowed, what is the maximum compression that can be achieved?3. The PDF of an image is given by Pr(r) as shown below. Find the transform toconvert the images PDF to Pr(z). Assume continuity, and find the transform in terms of r and z. Explain the transformation.4. A certain inspection application gathers black & white images of parts as they travel along a conveyor belt. It is necessary to sort the parts into two categories: parts with holes and parts with-out holes. An example of an image that might be taken by the inspection camera is shown at the right.Propose a method to identify and locate the objects of each category in the image so that they can be picked up by a robotic system and placed in different bins. Assume that the imaging system knows where each image pixel is located on the conveyor belt at every point in time.Provide an annotated flow chart of the algorithm you propose.5. In a given application, an averaging mask is applied to input images to reduce noise and then a Laplacian mask is applied to enhance small details. Would mathematics predict that the result should be the same if the order of the operations were reversed? What practical issues would be encountered in computer implementation?Digital Image Processing Examination1. A preprocessing step in an application of microscopy is concerned with the issue of isolating individual round particles from similar particles that overlap in groups of two or more.Assuming that all particles are of the same size, propose a morphological algorithm that will produce an image that contains only the isolated (non-overlapping) particles that are not in contact with the boundary of the image.102. An image represented by a continuous function f(x, y) is w = 2 cm wide and h = 3 cm high. The image is to be converted to an array of pixels by a scanner whose response is zero above 80 lines/centimeter in both the horizontal and vertical directions. The discrete image is represented by an array f(n, m) where n and m take on integer values, 0 n N - 1, 0 m M-1.(a) Determine suitable values for N and M.(b) Assume that f(n, m) = f(na, mb). Determine the values of a and b.(c) Determine constants A, B, C, D, E such that the DFT of f can be expressed as(d) Find numbers (P1, P2) such that F(u + jP1, v + kP2) = F(u, v) for any integers j, k, u, v.3. The arithmetic decoding process is the reverse of the encoding procedure. Decode the message 0.23355 given the coding model. SymbolProbabilitya0.2e0.3i0.1o0.2u0.1!0.14. The gradient of a function f (x) is defined asComputationally, the first derivative is implemented by calculating the difference between adjacent pixels.(a) Is the following a linear operator?(b) State how would you implement the above operator using differences between pixels.(c) A Sobel operator uses two masks, Hx and Hy to process an image. Explain why are two masks needed and what do they measure?(d) Write down the masks Hx and Hy, and identify them in the following figures:5. The three images shown were blurred using square averaging masks of sizes n=23, 25 , and 45, respectively. The vertical bars on the left lower part of (a) and (c) are blurred, but a clear separation exists between them. However, the bars have merged in image (b), in spite of the fact that the mask that produced this image is significantly smaller than the mask that produced image (c). explain this.Digital Image Processing Examination1. An image array f(m, n) of size M1 N1 is to be convolved with a filter array h(m, n) of size M2 N2 to produce a new image array g(m, n).1) Write a pseudo code program that describes a method to compute g(m, n) through the use of Fourier transforms. The result should be the same size as would be achieved with direct convolution.2) Modify the