The illumination-reflection model for an image is compounded in the si.docx

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The illumination-reflection model for an image is compounded in the simple equation f(x, y) = i(x, y)r(x, y) where f is the image and i and r are the illumination and reflect ion components respectively. Suppose that a flat area with centre at (x_0, y_0) is illuminated by a light source with intensity distribution i(x, y) = Ke^-[(x - x_0)^2 + (y - y_0)^2]. If the reflectance characteristic of the area is r(x, y) = 10(x - x_0) + 10(y - y_0) + 20, what is the value of K that would yield an image intensity of 100 at (x_0, y_0)? Solution In an image processing context, the histogram of an image normally refers to a histogram of the pixel intensity values. This histogram is a graph showing the number of pixels in an image at each different intensity value found in that image. For an 8-bit grayscale image there are 256 different possible intensities, and so the histogram will graphically display 256 numbers showing the distribution of pixels amongst those grayscale values. Histograms can also be taken of color images --- either individual histograms of red, green and blue channels can be taken, or a 3-D histogram can be produced, with the three axes representing the red, blue and green channels, and brightness at each point representing the pixel count. The exact output from the operation depends upon the implementation --- it may simply be a picture of the required histogram in a suitable image format, or it may be a data file of some sort representing the histogram statistics. .

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Consider a l-D elastic bar problem defined on [0, 4]. The domain is devided into 4 linear 2-node elements as follows. Write down element shape functions N (x) for each element (e = 1, 2, 3), Give a sketch of the shape functions N_i(x) = (f = 1, 2, 3, 4, 5) on the entire domain [0, 4]. Given the nodal displacement matrix d = [4 8 6 2]^T, find the displacement solutions at x = 1.5 and x = 3.6. Given the nodal displacement matrix d = [4 8 6 2]^T, find the strain (defined as du/dx) solutions at x = 0.4 and x = 2.3. Solution Using the nodal-based smoothing operation, the strains to be used in Equation (23) is assumedto be the smoothed strain for node k dened by e k e ( x k ) = k e ( x ) W ( x x k ) d where W = W W W is a diagonal matrix of smoothing function W . For simplicity, the smoothingfunction is taken as W ( x x k ) = 1 / A k , x k 0 , x / k where A k = k d is the area of smoothing domain for node k .Substituting into Equation and integrating by parts, the smoothed strain canbe calculated using e k = 1 A k k e ( x ) d = 1 A k k L n u ( x ) d = e k ( u ) where k is the boundary of the smoothing domain for node k , and L n is the matrix of the outwardnormal vector on k . Equation states the fact that the assumed strain e k is a function of theassumed displacement u .Substituting Equation into Equation , the smoothed strain can be expressed in thefollowing matrix form of nodal displacements: e k = i N in B i ( x k ) d i where N in is the number of nodes in the inuence domain of node k (including node k ). Whenlinear shape functions are used, it is the number of nodes that is directly connected to node k inthe triangular mesh (see Figure 1). In Equation , the B i ( x k ) is termed as the smoothed strainmatrix that is calculated using B i ( x k ) = b ix ( x k ) 00 b iy ( x k ) b iy ( x k ) b ix ( x k ) Using the Gauss integration along each segment of boundary k , we have b il = 1 A k N s m = 1 N g n = 1 w n i ( x mn ) n l ( x m ) ( l = x , y ) (30)where N s is the number of segments of the boundary k , N g is the number of Gauss points usedin each segment, w n is the corresponding weight number of Gauss integration scheme, and n l isthe unit outward normal corresponding to each segment on the smoothing domain boundary. In theLC-PIM using linear shape functions, n g = 1 is used. The entries in sub-matrices of the stiffnessmatrix K in Equation are then expressed as K ij = N k = 1 K ij ( k where the summation means an assembly process as we practice in the FEM, and K ij ( k ) is thestiffness matrix associated with node k that is computed using K ij ( k ) = k B T i D B j d = B T i D B j A k The entries (in sub-vectors of nodal forces) of the force vector f in Equation can be simplyexpressed as f i = k N in f i ( k ) The above integration is also performed by a summation of integrals over smoothing domains;hence, f i is an assembly of nodal force vectors at the surrounding nodes of node k : f i ( k ) = t ( k ) U i ˆ t d + ( k ) U i b d Note again th.

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