Video 9.1 Jianbo Shi. Property of Penn Engineering, Jianbo Shi
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1 Video 9.1 Jianbo Shi 1
2 Exmples of resizing 2
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4 (a) (b) (c) Guess We use crop, scaling and carving for resizing the given image Guess which one is for carving? 4
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7 Guess which one is for carving? 7
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9 Guess which one is for carving? 9
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14 Expanded 14
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19 Video 9.2 Jianbo Shi 19
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23 k Energy matrix e M i-1 i N Seam: 1 y S : s.t. (i,y(i)) i = 1,...,M y(i) y(i ) k 23
24 k Energy matrix e M i-1 i N 1 y S : s.t. (i,y(i)) i = 1,...,M y(i) y(i ) k Seam Cost: y S M y S i i 1 E ( ) = e ( ( )) 24
25 > k Energy matrix e M i-1 i N Seam S y : i, y i i = 1,, M s. t. y i y(i 1) k Seam Cost: y S M y S i i 1 E ( ) = e ( ( )) Goal: S * = E y min ( S )
26 Where does the energy matrix come from? 26
27 abs Energy matrix e + = abs Energy matrix For example: L1 norm of the edge gradients for the energy function 27
28 How to find the best seam? 28
29 M i-1 i k Energy matrix e N Idea 1: Brute force search i-1 1 st row: N i i+1 2 nd row: 3 rd row: M th row: ( 2k 1)N 2 ( 2k 1) N 1 ( 2k 1) ( M ) N How many possibilities for Seam S y? 29
30 M i-1 i k Energy matrix e N Idea 1: Brute force search i-1 M th row: 1 ( 2k 1) ( M ) N i Given M = 768, N = 1024, k = 1 i+1 How many possibilities for Seam S y?
31 Too many possibilities!
32 M N Idea 2: Find the shortest path from first row to the the last row in the energy domain! 32
33 i row i + 1 row Construct the directed graph where each pixel (node) is connected to the (2k+1) neighbors in the next row 33
34 1 st row V (u) Create an `interior set S, initialize with the first row Growing S with `least-resistant step Construct a Value matrix with value V (u) encoding the shortest path cost to each node in S 34
35 1 st row S Initialize to include the first row, and set the Value function V (u) = e(u),u S For the first row, the shortest path contains only itself 35
36 1 st row S u ~ S v Iterate: find the step expansion of least resistant from v = argmin[ V(u) e (v)] where (u v) is a graph edge ~ u S,v S S ~ S 36
37 1 st row S u ~ S v Iterate: find the step expansion of least resistant from u S,v S Remember the path back from (v u) S ~ v = argmin[ V(u) e (v)] where (u v) is a graph edge ~ P(v) u S 37
38 1 st row S ~ S Updating the interior set: ~ ~ S= S v, S S- v. Updating the Value function: V (v) = min[ V(u) e (v)] V(v) : Is the contingent cost E of the shortest path connecting the pixel v to the first row v S ~ 38
39 1 st row S ~ S Updating the interior set: ~ ~ S= S v, S S- v. Updating the Value function: V (v) = min[ V(u) e (v)] ~ v S Until reaching one of the pixels on the last row. 39
40 Energy Matrix e Value functionv(u) Path Matrix P 40
41 Video 9.3 Jianbo Shi 41
42 Dynamic Programming Frontier growing row by row 42
43 M N Idea 3: Dynamic programming! 43
44 i row i + 1 row Use the same graph structure Propagate the frontier row by row to construct the Value Matrix 44
45 1 st row Value Matrix V Path Matrix P Still start from first row, and initialize the Value function V ( 1, j) = e( 1, j) Set the Path function P ( 1, j) = 0 45
46 1 st row Updated Propagate to the second row, and update the value matrix V ( 2, j) = e( 2, j ) min( V(1,j-k),, V(1,j),, V(1,j+k)) V( 2j, ): Is the contingent cost of the shortest path connecting the pixel (2,j) to the first row 46
47 1 st row Value Matrix V Updated Path Matrix P Propagate to the second row, and update the value matrix V ( 2, j) = e( 2, j ) min( V(1,j-k),, V(1,j),, V(1,j+k)) Set the Path function P ( 2, j ) = argmin( V(1,j-k),, V(1,j),, V(1,j+k)) 47
48 Updated Iteratively propagate to the ith row, and update the value matrix V ( i, j) = e( i, j ) min( V(i-1,j-k),, V(i-1,j),, V(i-1,j+k)) Set the Path function P ( i, j ) = argmin( V(i-1,j-k),, V(i-1,j),, V(i-1,j+k)) 48
49 Value Matrix V Energy Matrix e Path Matrix P 49
50 Value Matrix V Path Matrix P Localize the minimum of the last row of argmin V ( M:, ) 50
51 Value Matrix V Path Matrix P
52 Value Matrix V Path Matrix P
53 Value Matrix V Path Matrix P
54 Value Matrix V Path Matrix P
55 Value Matrix V Path Matrix P Since V is the contingency table, we can start from any pixel and find a shortest path to the first row! 55
56 Video 9.4 Jianbo Shi 56
57 Illustration for synthetic case 57
58 Energy matrix 58
59 Value matrix Energy matrix Goal: Construct value and path matrix from the energy matrix Value matrix records the energy of the shortest path from the starting row to the current pixel 59
60 Value matrix Energy matrix Goal 1: Construct Value matrix from the energy matrix Value matrix records the energy of the shortest path from the starting row to the current pixel Property: every entry encodes the minimal shortest path to that node from starting row 60
61 Goal 2: Construct Path matrix The Path matrix records the immediate predecessor in the shortest path, for every node 61
62 e: energy function Energy function Energy function records the cost of a pixel. Typically it is the image gradient magnitude
63 How to generate the two matrices? 63
64 Energy matrix Step1: Initializing two matrices Set value and path matrix the same size as the energy matrix Initialize its first row of Value matrix with that of the energy matrix Initialize its first row of path matrix to zero 64
65 Value matrix V Step2: Propagation Start with 2 nd row, and propagate row by row 65
66 Value matrix V ? Step2: Propagation Start with 2 nd row 66
67 Neighborhoods of the pixel in previous row Value matrix Step2: Propagation Start with 2 nd row Find the neighbors of the pixel in the previous row 67
68 Value matrix V V 2,1 = min V (1,1) V(1,2) + e 2, Step2: Propagation Start with 2 nd row Find the neighbors of the pixel in the previous row Find the minimum among neighbors 68
69 Step2: Propagation Start with 2 nd row Find the neighbors of the pixel in the previous row Find the minimum among neighbors, record it in Path matrix 69
70 Energy matrix Value matrix V V 2,1 = min V (1,1) V(1,2) + e 2, = Step2: Propagation Start with 2 nd row, Find the neighbors of the pixel in the previous row Find the minimum among neighbors Add the energy value of the pixel with the minimum 70
71 Value matrix V ? Step2: Propagation for every row 71
72 Neighborhoods of the pixel in previous row Value matrix V ? Step2: Propagation for every row Find the neighbors of the pixel in the previous row 72
73 Value matrix V min V(i 1, j 1) V (i 1, j) V(i 1, j + 1) + e i, j ? Find the neighbors of the pixel in the previous row Find the minimum among neighbors 73
74 Find the neighbors of the pixel in the previous row Find the minimum among neighbors, record it in Path matrix 74
75 Value matrix V min V(i 1, j 1) V (i 1, j) V(i 1, j + 1) + e i, j ? Find the neighbors of the pixel in the previous row Find the minimum among neighbors Add the energy value of the pixel with the minimum 75
76 Value matrix V ? 6 = Find the neighbors of the pixel in the previous row Get the minimum among neighbors Add the energy value of the pixel with the minimum 76
77 Value matrix Step3: path resolving Path matrix
78 Value matrix Find the minimum of the last row of the Value matrix, Find the predecessor of that pixel 78
79 Value matrix Path matrix Find the minimum of the last row of the Value matrix, Find the predecessor of that pixel, using Path matrix 79
80 Value matrix Path matrix Move to its predecessor
81 Value matrix Path matrix Follow predecessor of current pixel,
82 Value matrix Path matrix Move to its predecessor
83 Value matrix Path matrix Find the predecessor of current pixel
84 Value matrix Path matrix Move to its predecessor
85 Value matrix Path matrix Find the predecessor of current pixel
86 Value matrix Path matrix Stop when reaching the first row
87 Value matrix Path matrix Seam carving is to delete the path with minimum cost 87
88 Value matrix Path matrix Seam carving is to delete the path with minimum cost 88
89 Energy matrix Seam carving is to delete the path with minimum cost 89
90 Carved energy matrix Seam carving is to delete the path with minimum cost 90
91 Value matrix Path matrix What if we want to remove a seam that ends on particular pixel? 91
92 Video 9.5 Jianbo Shi 92
93 Illustration on a real image 93
94 Step 0: prepare the energy function Transform a color image into gray, Im = rgb2gray(im) Calculate image gradients Use the L1 norm of the gradients for the energy function 94
95 abs Energy matrix e + = abs Step 0: prepare the energy function Transform a color image into gray, Im = rgb2gray(im) Calculate the gradients Use the L1 norm of the gradients for the energy function 95
96 Value Matrix V Energy matrix e Path Matrix P V :, 1 = e(:, 1) P :, 1 = 0 Step 1: Initializing two matrices Set value and path matrix the same size as the energy matrix Initialize the first column of value matrix with that of the energy matrix Initialize the first column of path matrix to zero 96
97 min V(i 1, j 1) V (i, j 1) V(i + 1, j 1) + e i, j Step 2: Propagation Start with 2 nd column Find the neighbors of the pixel in the previous column Find the minimum among neighbors Add the energy value of the pixel with the minimum 97
98 Step 2: Propagation Start with 2 nd column Find the neighbors of the pixel in the previous column Find the minimum among neighbors Add the energy value of the pixel with the minimum Assign the direction of the minimum to path matrix 98
99 Step 3: Path Resolving Find the minimum of the last column of the Value matrix, Find the predecessor of that pixel, using Path matrix Trace back to complete the path using the Path matrix 99
100 Step3: Path Resolving Find the minimum of the last column of the Value matrix, Find the predecessor of that pixel, using Path matrix Trace back to complete the path using the Path matrix 100
101 Step 4: Eliminate the Seam 101
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