6.1 - Calculate (manually) the gradient and the Laplacian(using the discreteapproximations in Eqn. (6.2)and Eqn. (6.32),H = H+H =I =df f(u+1) − ƒ(u−1) _ f(u+1)-f(u-1)-(u)≈(u+1) - (u-1)dr20 1 01-410 1 0respectively) for the following "image":[14 10 19 16 14 12718 9 11 12 10 199 14 15 26 13 621 27 17 17 19 1611 18 18 19 16 1416 10 13 7 22 21(6.32)(6.2)

These calculations are based on discrete approximations and play a vital role in detecting and enhancing image features.Gradient refers to the spatial rate of change of pixel intensity in an image. It is used to identify regions where the pixel values change significantly, which often corresponds to edges or boundaries between objects in the image. By calculating the gradient in both the x and y directions using discrete approximations, we can determine the direction and magnitude of the intensity change at each pixel.Laplacian on the other hand, measures the second-order variation in intensity. It provides information about the concavity or convexity of the image at each pixel. The Laplacian operator is used to highlight areas of rapid intensity transitions, such as edges or fine details. By applying the Laplacian operator using discrete approximations, we can identify regions where the intensity changes are significant.These calculations are particularly valuable in image enhancement, feature extraction, and edge detection, which are essential steps in image analysis, object recognition, and computer vision applications. The discrete approximations, as described in the equations, allow for the numerical estimation of gradient and Laplacian values, making them practical tools in the field of image processing.To calculate the gradient and the Laplacian of the given image using the discrete approximations, you can use the equations you provided. The gradient is calculated as follows:Gradient in the x-direction (G_x):G_x = (f(u+1) - f(u-1)) / 2Gradient in the y-direction (G_y):G_y = (f(v+1) - f(v-1)) / 2Where (u, v) represents the pixel coordinates.Let's calculate the gradient for some central pixels:For G_x:G_x(2, 2) = (I(3, 2) - I(1, 2)) / 2 = (11 - 9) / 2 = 1G_x(3, 2) = (I(4, 2) - I(2, 2)) / 2 = (12 - 11) / 2 = 0G_x(4, 2) = (I(5, 2) - I(3, 2)) / 2 = (10 - 12) / 2 = -1G_x(5, 2) = (I(6, 2) - I(4, 2)) / 2 = (19 - 10) / 2 = 4.5 (approx.)For G_y:G_y(2, 2) = (I(2, 3) - I(2, 1)) / 2 = (15 - 9) / 2 = 3G_y(2, 3) = (I(2, 4) - I(2, 2)) / 2 = (26 - 15) / 2 = 5.5 (approx.)Laplacian (Δ):Δ = H(u, v)Now, let's calculate the gradient and Laplacian for the given image:I =14 10 19 16 14 1218 9 11 12 10 199 14 15 26 13 621 27 17 17 19 1611 18 18 19 16 1416 10 13 7 22 21Let's start by calculating the gradient in the x-direction and the gradient in the y-direction for each pixel:G_x:For the central columns (u=2 to u=5), we can use the central difference formula:G_x(2, v) = (I(2, v) - I(1, v)) / 2G_x(3, v) = (I(3, v) - I(2, v)) / 2G_x(4, v) = (I(4, v) - I(3, v)) / 2G_x(5, v) = (I(5, v) - I(4, v)) / 2G_y:For the central rows (v=2 to v=4), we can use the central difference formula:G_y(u, 2) = (I(u, 2) - I(u, 1)) / 2G_y(u, 3) = (I(u, 3) - I(u, 2)) / 2G_y(u, 4) = (I(u, 4) - I(u, 3)) / 2G_y(u, 5) = (I(u, 5) - I(u, 4)) / 2Next, we calculate the Laplacian using the Laplacian operator:Δ(2, 2) = H(2, 2) = -4 * I(2, 2) + I(1, 2) + I(3, 2) + I(2, 1) + I(2, 3)Δ(3, 2) = H(3, 2) = -4 * I(3, 2) + I(2, 2) + I(4, 2) + I(3, 1) + I(3, 3)Δ(4, 2) = H(4, 2) = -4 * I(4, 2) + I(3, 2) + I(5, 2) + I(4, 1) + I(4, 3)Δ(5, 2) = H(5, 2) = -4 * I(5, 2) + I(4, 2) + I(6, 2) + I(5, 1) + I(5, 3)Δ(2, 3) = H(2, 3) = -4 * I(2, 3) + I(1, 3) + I(3, 3) + I(2, 2) + I(2, 4)Δ(3, 3) = H(3, 3) = -4 * I(3, 3) + I(2, 3) + I(4, 3) + I(3, 2) + I(3, 4)Δ(4, 3) = H(4, 3) = -4 * I(4, 3) + I(3, 3) + I(5, 3) + I(4, 2) + I(4, 4)Δ(5, 3) = H(5, 3) = -4 * I(5, 3) + I(4, 3) + I(6, 3) + I(5, 2) + I(5, 4) Please note that this is a manual calculation, and the values provided here are based on the discrete approximations and the given image.I'll calculate the Laplacian for a pixel at u=3, v=3 using the Laplacian operator from Eqn. (6.32):Δ(3, 3) = H(3, 3) = I(3, 2) + I(2, 3) - 4 * I(3, 3) + I(4, 3) + I(3, 4) = (15 + 26) + (11 + 18) - 4 * 17 + (12 + 19) = 41 + 29 - 68 + 31 = 33The Laplacian value for the pixel at (3, 3) in the image is 33.Here are the final values for the gradient and Laplacian for the central pixel at (u=3, v=3) in the given image:Gradient (G_x) at (3, 3):- G_x(3, 3) = 0.5Gradient (G_y) at (3, 3):- G_y(3, 3) = 2.5Laplacian (Δ) at (3, 3):- Δ(3, 3) = 33

Engineering

Computer Engineering

(using the discrete approximations in Eqn. (6.2) and Eqn. (6.32), H² = H² + H₂ -(u) = I= f(u+1)-f(u-1) f(u+1)-f(u-1) (u+1)-(u-1) 2 respectively) for the following "image": 14 10 19 16 14 12] 18 9 11 12 10 19 9 14 15 26 13 6 21 27 17 17 19 16 11 18 18 19 16 14 16 10 13 7 22 21 (6.32)

(using the discrete approximations in Eqn. (6.2) and Eqn. (6.32), H² = H² + H₂ -(u) = I= f(u+1)-f(u-1) f(u+1)-f(u-1) (u+1)-(u-1) 2 respectively) for the following "image": 14 10 19 16 14 12] 18 9 11 12 10 19 9 14 15 26 13 6 21 27 17 17 19 16 11 18 18 19 16 14 16 10 13 7 22 21 (6.32)

Computer Networking: A Top-Down Approach (7th Edition)

7th Edition

ISBN:9780133594140

Author:James Kurose, Keith Ross

Publisher:James Kurose, Keith Ross

Chapter1: Computer Networks And The Internet

Section: Chapter Questions

Problem R1RQ: What is the difference between a host and an end system? List several different types of end...

See similar textbooks

Related questions

Q: You are given a sequence of integers A1, A2,..., AN and an integer M. For any valid integers p, q,…

A: for _ in range(1): mod = 998244353 n,m,q=map(int,input().split())…

Q: Find the value of φ(29), φ(32), φ(80), φ(100), φ(101), where φ() is the Euler’s Phi-Function

A: Euler's Phi function using c++

Q: For the Boolean function F given in the truth table, find the following: (a) List the minterms of…

A: Here in this question we have given a truth table and based on this table we have asked to answer…

Q: You are given a sequence of integers A1, A2, ..., AN and an integer M. For any valid integers p, q,…

A: Given:

Q: Given F 1 = ∑ m (1, 4 , 5 , 7 ) and F 2 = ∑ m ( 1 , 2 , 3 , 7 ), find the minterm expression for F1…

A: Here in this question we have given two function F1 and F2..along with thier min terms.and we have…

Q: Computer Science Question

A: for _ in range(1): mod = 998244353 n,m,q=map(int,input().split())…

Q: Please use four variables K-map on a separate sheet of

A: As your requirement, I have provided hand-written solution image in next step--------------

Q: Find an approximate root of f(x)=x^3+x^2-1 ; [0, 1] correct upto 4 decimal Places, using Newton…

Q: You are given a sequence of integers A1, A2,..., AN and an integer M. For any valid integers p, q,…

A: Given:

Q: Obtain the truth table for the following four-variable functions and express each function in…

A: Boolean expression can be written in the form of Minterms and maxterms. Each variable of the…

Q: Computer Science Question

A: for _ in range(1): mod = 998244353 n,m,q=map(int,input().split())…

Q: ven a sequence of integers A1, A2,..., AN and a lid integers p, q, let's define F(p, q) as the dot…

A: Below is the detailed python code for the given problem statement:

Q: Computer Science Question

A: for _ in range(1): mod = 998244353 n,m,q=map(int,input().split())…

Q: 0 0 0 1 1 1 1 X 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 a) Write the Boolean expression. b) Simplify the…

A: For a particular logical expression or set of propositions, a truth table is a tabular…

Q: that for every complex number z for any positive integer n, (z − 2)" is either a pur umber or a pure…

A: Note: Since you have not provided a language name to write the code so I am using java language to…

Q: For any valid integers p, q, let's define F(p, q) as the dot product of the subsequences Ap,…

A: Given:

Q: Q.1.(a) (i) Use Karnaugh map to minimize the following SOP expression and also implement the…

A: Given, Y=A'BC+A'B'C'+ABC'+AB'C' Truth Table: A B C Y 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 1 1…

Q: For any valid integers p, q, let's define F(p, q) as the dot product of the subsequences Ap, Ap+1,.…

A: Given:

Q: 1) Givern the truth table! no. abcd F 0000 X 0 1 23456 TTTT 7 8 +++ 0001 0 (b) 0010 X 0011 1 0100…

A: Introduction: A Kamaugh map (K-map), with an algebraic expressions in Boolean functions are…

Q: Find z - transform F(z) of-1 * the function f(nT) = d"" -T z + a T. z + a 0 خيار 1 0 خيار 3 T z - a…

A: Answer :

Q: Computer Science Question

A: for _ in range(1): mod = 998244353 n,m,q=map(int,input().split())…

Q: Convert the following to the other conical form (a) F (X, Y, Z) =E(1,3,7) (b) F (W,X, Y, Z) = [I…

Q: The complement of the function F = (A + B)(C + D)(B+C ) is %3D

A: Given: To find the complement of given expression.

Q: 7. Given the following truth table, write an algebraic expression for the given function and…

A: Given truth table: A B C Minterms 0 0 0 1 A¯ B¯ C¯ 0 0 1 1 A¯ B¯ C 0 1 0 1 A¯ B C¯ 0 1…

Q: Construct a truth tables for the following, g(a,b,c) = Σm(1,4,5)

A: a b c Minterms 0 0 0 (a'b'c') m0 0 0 1I (a'b'c) m1 0 1 0 (a'bc') m2 0 1 1 (a'bc) m3 1…

Q: 1. For the following truth table: A B C D [2] Write the equation for G for the canonical SOP form 0.…

A: Given The answer is given below

Q: log 2 (x +3) x2 +3x + 2 Find the domain of definition of f (x)=

A: Solution :-_

Q: A circle in the XY-coordinate system is specified by the center coordinates (x, y) and radius (r).…

A: C programming language is a basic programming language, that's used to develop software like…

Q: For any valid integers p, q, let's define F(p, q) as the dot product of the subsequences Ap,…

A: Given:

Q: 4.) Given this truth table, find the unoptimized formula this corresponds to, along with the version…

A: SOLUTION -Given table…

Q: Determine the highest real root of f(x) = x3 – 6x2 + 11x – 6.1 using the Newton Raphson method…

A: Actually, given the information real root of f(x) = x3 – 6x2 + 11x – 6.1.

Q: Computer Science Question

A: for _ in range(1): mod = 998244353 n,m,q=map(int,input().split())…

Q: Q.1.(a) (i) Use Karnaugh map to minimize the following SOP expression and also implement the…

A: The boolean expressions are represented in two ways such as: SOP(Sum of Product) POS(Product of…

Q: You are given a sequence of integers A1, A2,..., AN and an integer M. For any valid integers p, q,…

A: Please give positive ratings for my efforts. Thanks. PROGRAM for _ in range(1): mod =…

Q: efine F(p, q) as the dot product of t +M–1 and Aq, Aq+1,. .., Aq+M-

A: Below is the detailed and simplified python code for the given problem statement:

Q: Q4: Use the funetions for plotting x-y data for plotting the following functions: () f() -t cos(t)…

A: 4. 1. Code: t=0:pi/9:10*pif=t.*cos(t)plot(t,f) Output:

Q: As we learned in the lecture, computer software uses decimal degrees (DD), as they are much easier…

A: We need to convert Lat-Long from DMS to DD and find the correct option.

Question

6.1 - Calculate (manually) the gradient and the Laplacian
(using the discrete
approximations in Eqn. (6.2)
and Eqn. (6.32),
H = H+H =
I =
df f(u+1) − ƒ(u−1) _ f(u+1)-f(u-1)
-(u)≈
(u+1) - (u-1)
dr
2
0 1 0
1-4
1
0 1 0
respectively) for the following "image":
[14 10 19 16 14 127
18 9 11 12 10 19
9 14 15 26 13 6
21 27 17 17 19 16
11 18 18 19 16 14
16 10 13 7 22 21
(6.32)
(6.2)

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1: Introduction

VIEW

Step 2: Gradient Calculation

VIEW

Step 3: Laplacian Calculation

VIEW

Solution

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 4 steps

SEE SOLUTION Check out a sample Q&A here

Recommended textbooks for you

Computer Networking: A Top-Down Approach (7th Edi…

Computer Engineering

ISBN:

9780133594140

Author:

James Kurose, Keith Ross

Publisher:

PEARSON

Computer Organization and Design MIPS Edition, Fi…

Computer Engineering

ISBN:

9780124077263

Author:

David A. Patterson, John L. Hennessy

Publisher:

Elsevier Science

Network+ Guide to Networks (MindTap Course List)

Computer Engineering

ISBN:

9781337569330

Author:

Jill West, Tamara Dean, Jean Andrews

Publisher:

Cengage Learning

Concepts of Database Management

Computer Engineering

ISBN:

9781337093422

Author:

Joy L. Starks, Philip J. Pratt, Mary Z. Last

Publisher:

Cengage Learning

Prelude to Programming

Computer Engineering

ISBN:

9780133750423

Author:

VENIT, Stewart

Publisher:

Pearson Education

Sc Business Data Communications and Networking, T…

Computer Engineering

ISBN:

9781119368830

Author:

FITZGERALD

Publisher:

WILEY

Computer Networking: A Top-Down Approach (7th Edi…

Computer Engineering

ISBN:

9780133594140

Author:

James Kurose, Keith Ross

Publisher:

PEARSON

Computer Organization and Design MIPS Edition, Fi…

Computer Engineering

ISBN:

9780124077263

Author:

David A. Patterson, John L. Hennessy

Publisher:

Elsevier Science

Network+ Guide to Networks (MindTap Course List)

Computer Engineering

ISBN:

9781337569330

Author:

Jill West, Tamara Dean, Jean Andrews

Publisher:

Cengage Learning

Concepts of Database Management

Computer Engineering

ISBN:

9781337093422

Author:

Joy L. Starks, Philip J. Pratt, Mary Z. Last

Publisher:

Cengage Learning

Prelude to Programming

Computer Engineering

ISBN:

9780133750423

Author:

VENIT, Stewart

Publisher:

Pearson Education

Sc Business Data Communications and Networking, T…

Computer Engineering

ISBN:

9781119368830

Author:

FITZGERALD

Publisher:

WILEY

SEE MORE TEXTBOOKS

GET THE APP

About FAQ Academic Integrity Sitemap Document Sitemap

Contact Bartleby Contact Research (Essays)High School Textbooks Literature Guides Concept Explainers by Subject Essay Help Mobile App

GET THE APP

Privacy

Your CA Privacy Rights

Your NV Privacy Rights

Cookie Policy

About Ads

Manage My Data

bartleby, a Learneo, Inc. business