CS6643 Homework4-Discriminant functions for a two-class classification Solved

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  1. ย The discriminant functions for a two-class classification problem are given below:

Class 1: ย ย ย ๐ท๐ท1(๐‘ฅ๐‘ฅ) = ๐‘ฅ๐‘ฅ12 + ๐‘ฅ๐‘ฅ1 + ๐‘ฅ๐‘ฅ2 + 4ย  ย ย ย ย ย ย  Class 2:ย ย ย  ๐ท๐ท2(๐‘ฅ๐‘ฅ) = ๐‘ฅ๐‘ฅ1 + 2๐‘ฅ๐‘ฅ2 + 3

  • Find the equation of the decision boundary between the two classes.
  • Plot the equation you have found on a graph and label the regions on either side of the plot with ๐œ”๐œ”1 if samples within the region belong to class 1, and ๐œ”๐œ”2 if samples within the region belong to class 2.
  1. ย We would like to use a minimum-distance classifier formulated using linear discriminant functions ๐ท๐ท๐‘–๐‘–(๐‘‹๐‘‹) to classify input X into one of three classes. Input X and the prototype vectors for the three classes are given below:

X = ๏ฃฎ๏ฃฏxx12๏ฃบ๏ฃน๏ฃป ,ย ย  R1 =๏ฃฐ๏ฃฎ๏ฃฏ32..50๏ฃน๏ฃป๏ฃบ , R2 =๏ฃฏ๏ฃฐ๏ฃฎโˆ’02.5.5๏ฃบ๏ฃน๏ฃป ,ย ย ย  R3 =๏ฃฎ๏ฃฐ๏ฃฏโˆ’23.5๏ฃน๏ฃป๏ฃบ

๏ฃฐ

  • Write the mathematical formulas for the discriminant functions ๐ท๐ท๐‘–๐‘–(๐‘‹๐‘‹) for the three classes.
  • Classify the input X = ๏ฃฎ๏ฃฏ๏ฃฐโˆ’57.5.0๏ฃน๏ฃบ๏ฃป into one of the three classes using your discriminant

functions in (a).

  1. In face recognition using eigenfaces, we use a set of training face images to derive the eigenface matrix ๐‘ˆ๐‘ˆ that forms the face space. (a) Given the eigenface matrix U, write the formula for computing the PCA coefficients of an input face image (b) Suppose 20 training images, each of size 360 ร— 480 ย (height ร— width), were used to derive U, what is the dimension of U and what is the dimension of the computed PCA coefficient?
  2. ย We would like to use the unsigned representation of the Histogram of Oriented Gradients (HOG) descriptor to detect human in images.
  • What is the dimension of the descriptor if we assume the following parameter settings: detection window size = 136 x 80 pixels (rows x columns), cell size = 8 x 8 pixels, block size = 2 x 2 cells, block overlap = 8 pixels, and number of histogram bins per cell = 9.
  • The bin centers Center(i) for the 9 histogram bins are given in the table below. Given the gradient magnitude M and gradient angle ๐œƒ๐œƒ of an edge that lies between two bin centers

Center(i) and Center(j), with ๐‘—๐‘— = ๐‘–๐‘– + 1 for ๐‘–๐‘– = 1 ๐‘ก๐‘ก๐‘ก๐‘ก 8, ๐‘—๐‘— = 1 for ๐‘–๐‘– = 9. ย derive the formulas that allow you to compute the increments to histogram bins ๐ป๐ป(๐‘–๐‘–) and ๐ป๐ป(๐‘—๐‘—).ย  The input gradient ๐œƒ๐œƒ can range from 0 to 360 degrees. If ๐œƒ๐œƒ is greater than or equal to 180, subtract by 180 first. Your formulas should be expressed in terms of

๐‘€๐‘€, ๐œƒ๐œƒ, ๐‘–๐‘–, ๐‘—๐‘—, ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ๐‘ก๐‘ก๐ถ๐ถ๐ถ๐ถ(๐‘–๐‘–), ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ๐‘ก๐‘ก๐ถ๐ถ๐ถ๐ถ(๐‘—๐‘—), ๐ป๐ป(๐‘–๐‘–) and ๐ป๐ป(๐‘—๐‘—), and should be able to handle angles that lie between bin centers 9 and 1. You can give more than one formulas.

  • Given the gradient magnitudes and gradient angles of an 8 x 8 cell as shown in the figures below, compute the histogram of the cell (before block normalization.)

 

Bin # 1 2 3 4 5 6 7 8 9
Center(i) (in degrees) 0 20 40 60 80 100 120 140 160

 

0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 100 0 0 0 0 0 0
0 0 200 0 0 0 0 0
0 0 0 0 160 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0

ย ย ย ย ย ย ย ย ย ย ย  Gradient Magnitudes

ย 

200 45 23 98 130 260 255 250
125 295 85 90 130 265 249 240
123 35 85 95 125 260 250 240
100 90 45 90 120 265 240 230
95 99 105 106 355 120 100 110
90 100 110 120 120 130 125 120
85 90 100 110 110 120 120 110
80 80 100 110 100 100 100 110

ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย  Gradient Angles

 

 

ย 

 

 

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