CS 691 Computational Photography

Instructor: Gianfranco DorettoCutting Images

This Lecture: Finding Seams and Boundaries

Segmentation

This Lecture: Finding Seams and Boundaries

Retargetting

http://swieskowski.net/carve/

This Lecture: Finding Seams and Boundaries

Stitching

This Lecture: Finding seams and boundaries

Fundamental Concept: The Image as a Graph

• Intelligent Scissors: Good boundary = short path

• Graph cuts: Good region has low cutting cost

Semi-automated segmentation

User provides imprecise and incomplete specification of region – your algorithm has to read his/her mind.

Key problems1. What groups of pixels form cohesive regions?2. What pixels are likely to be on the boundary of regions?3. Which region is the user trying to select?

What makes a good region?

• Contains small range of color/texture• Looks different than background• Compact

What makes a good boundary?

• High gradient along boundary• Gradient in right direction• Smooth

The Image as a Graph

Node: pixel

Edge: cost of path or cut between two pixels

Intelligent ScissorsMortenson and Barrett (SIGGRAPH 1995)

Intelligent Scissors

• Formulation: find good boundary between seed points

• Challenges– Minimize interaction time– Define what makes a good boundary– Efficiently find it

Intelligent Scissors

A good image boundary has a short path through the graph.

Mortenson and Barrett (SIGGRAPH 1995)

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Start

End

Intelligent Scissors: method

1. Define boundary cost between neighboring pixels

2. User specifies a starting point (seed)3. Compute lowest cost from seed to each

other pixel4. Get new seed, get path between seeds,

repeat

Intelligent Scissors: method

1. Define boundary cost between neighboring pixels

a) Lower if edge is present (e.g., with edge(im, ‘canny’))

b) Lower if gradient is strongc) Lower if gradient is in direction of boundary

Gradients, Edges, and Path Cost

Gradient Magnitude

Edge Image

Path Cost

Intelligent Scissors: method

1. Define boundary cost between neighboring pixels

2. User specifies a starting point (seed)– Snapping

Intelligent Scissors: method

1. Define boundary cost between neighboring pixels

2. User specifies a starting point (seed)3. Compute lowest cost from seed to each

other pixel– Djikstra’s shortest path algorithm

Djikstra’s shortest path algorithm

Initialize, given seed s:• Compute cost2(q, r) % cost for boundary from pixel q to

neighboring pixel r• cost(s) = 0 % total cost from seed to this point • A = {s} % set to be expanded• E = { } % set of expanded pixels• P(q) % pointer to pixel that leads to q

Loop while A is not empty1. q = pixel in A with lowest cost2. for each pixel r in neighborhood of q that is not in E

a) cost_tmp = cost(q) + cost2(q,r)

b) if (r is not in A) OR (cost_tmp < cost(r))i. cost(r) = cost_tmp ii. P(r) = qiii. Add r to A

Intelligent Scissors: method

1. Define boundary cost between neighboring pixels

2. User specifies a starting point (seed)3. Compute lowest cost from seed to each

other pixel4. Get new seed, get path between seeds,

repeat

Intelligent Scissors: improving interaction

1. Snap when placing first seed2. Automatically adjust as user drags3. Freeze stable boundary points to make

new seeds

Where will intelligent scissors work well, or have problems?

Grab cuts and graph cuts

User Input

Result

Magic Wand (198?)

Intelligent ScissorsMortensen and Barrett (1995)

GrabCut

Regions Boundary Regions & Boundary

Source: Rother

Segmentation with graph cuts

Energy(y;θ,data) = ψ 1(yi;θ,data)i∑ + ψ 2 (yi ,yj;θ,data)

i, j∈edges∑

Source (Label 0)

Sink (Label 1)

Cost to assign to 0

Cost to assign to 1

Cost to split nodes

Segmentation with graph cuts

Energy(y;θ,data) = ψ 1(yi;θ,data)i∑ + ψ 2 (yi ,yj;θ,data)

i, j∈edges∑

Source (Label 0)

Sink (Label 1)

Cost to assign to 0

Cost to assign to 1

Cost to split nodes

Interactive Graph Cuts [Boykov, Jolly ICCV’01]

Image Min Cut

Cut: separating source and sink; Energy: collection of edges

Min Cut: Global minimal enegry in polynomial time

Foreground (source)

Background(sink)

constraints

GrabCut Colour Model

Gaussian Mixture Model (typically 5-8 components)

Foreground &Background

Background

Foreground

BackgroundG

R

G

RIterated graph cut

Source: Rother

Graph cuts segmentation1. Define graph

– usually 4-connected or 8-connected

2. Set weights to foreground/background– Color histogram or mixture of Gaussians for

background and foreground

3. Set weights for edges between pixels

4. Apply min-cut/max-flow algorithm5. Return to 2, using current labels to

compute foreground, background models€

edge _ potential(i, j) ∝ [y i ≠ y j ]exp− c(i) − c( j)

2

2σ 2

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪€

unary _ potential(i) = −log(P(c(i);y i))

What is easy or hard about these cases for graphcut-based segmentation?

Easier examples

GrabCut – Interactive Foreground Extraction 10

More difficult Examples

Camouflage & Low Contrast Harder CaseFine structure

Initial Rectangle

InitialResult

GrabCut – Interactive Foreground Extraction 11

Lazy Snapping (Li et al. SG 2004)

Limitations of Graph Cuts

• Requires associative graphs– Connected nodes should prefer to have

the same label

• Is optimal only for binary problems

Other applications: Seam Carving

Demo: http://swieskowski.net/carve/

Seam Carving – Avidan and Shamir (2007)

Other applications: Seam Carving

• Find shortest path from top to bottom (or left to right), where cost = gradient magnitude

Demo: http://swieskowski.net/carve/

Seam Carving – Avidan and Shamir (2007)

http://www.youtube.com/watch?v=6NcIJXTlugc

Dynamic Programming

• Well known algorithm design techniques:.– Divide-and-conquer algorithms

• Another strategy for designing algorithms is dynamic programming.– Used when problem breaks down into recurring small

subproblems

• Dynamic programming is typically applied to optimization problems. In such problem there can be many solutions. Each solution has a value, and we wish to find a solution with the optimal value.

Dynamic Programming• Dynamic programming is a way of improving on

inefficient divide-and-conquer algorithms.

• By “inefficient”, we mean that the same recursive call is made over and over.

• If same subproblem is solved several times, we can use table to store result of a subproblem the first time it is computed and thus never have to recompute it again.

• Dynamic programming is applicable when the subproblems are dependent, that is, when subproblems share subsubproblems.

• “Programming” refers to a tabular method

Elements of Dynamic Programming (DP)

DP is used to solve problems with the following characteristics:

• Simple subproblems – We should be able to break the original problem to smaller

subproblems that have the same structure

• Optimal substructure of the problems – The optimal solution to the problem contains within

optimal solutions to its subproblems.

• Overlapping sub-problems – there exist some places where we solve the same

subproblem more than once.

Steps to Designing a Dynamic Programming Algorithm

1. Characterize optimal substructure

2. Recursively define the value of an optimal solution

3. Compute the value bottom up

4. (if needed) Construct an optimal solution

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Example: Matrix-chain Multiplication

• Suppose we have a sequence or chain A1, A2, …, An of n matrices to be multiplied

– That is, we want to compute the product A1A2…An

• There are many possible ways (parenthesizations) to compute the product

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Matrix-chain Multiplication …

contd

• Example: consider the chain A1, A2, A3, A4 of 4 matrices

– Let us compute the product A1A2A3A4

• There are 5 possible ways:1. (A1(A2(A3A4)))

2. (A1((A2A3)A4))

3. ((A1A2)(A3A4))

4. ((A1(A2A3))A4)

5. (((A1A2)A3)A4)

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Matrix-chain Multiplication …

contd

• To compute the number of scalar multiplications necessary, we must know:

– Algorithm to multiply two matrices– Matrix dimensions

• Can you write the algorithm to multiply two matrices?

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Algorithm to Multiply 2 Matrices

Input: Matrices Ap×q and Bq×r (with dimensions p×q and q×r)

Result: Matrix Cp×r resulting from the product A·B

MATRIX-MULTIPLY(Ap×q , Bq×r)1. for i ← 1 to p2. for j ← 1 to r3. C[i, j] ← 04. for k ← 1 to q5. C[i, j] ← C[i, j] + A[i, k] · B[k,

j] 6. return C

Scalar multiplication in line 5 dominates time to compute CNumber of scalar multiplications = pqr

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Matrix-chain Multiplication …

contd

• Example: Consider three matrices A10x100, B100x5, and C5x50

• There are 2 ways to parenthesize – ((AB)C) = D10x5 · C5x50

• AB 10·100·5=5,000 scalar multiplications• DC 10·5·50 =2,500 scalar multiplications

– (A(BC)) = A10100 · E10050

• BC 100·5·50=25,000 scalar multiplications

• AE 10·100·50 =50,000 scalar multiplications

Total: 7,500

Total: 75,000

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Matrix-chain Multiplication …

contd

• Matrix-chain multiplication problem– Given a chain A1, A2, …, An of n

matrices, where for i=1, 2, …, n, matrix Ai has dimension pi-1xpi

– Parenthesize the product A1A2…An such that the total number of scalar multiplications is minimized

• Brute force method of exhaustive search takes time exponential in n

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Dynamic Programming Approach

• The structure of an optimal solution– Let us use the notation Ai..j for the

matrix that results from the product Ai Ai+1 … Aj

– An optimal parenthesization of the product A1A2…An splits the product between Ak and Ak+1 for some integer k where1 ≤ k < n

– First compute matrices A1..k and Ak+1..n ; then multiply them to get the final matrix A1..n

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Dynamic Programming Approach …contd

– Key observation: parenthesizations of the subchains A1A2…Ak and Ak+1Ak+2…An must also be optimal if the parenthesization of the chain A1A2…An is optimal (why?)

– That is, the optimal solution to the problem contains within it the optimal solution to subproblems

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Dynamic Programming Approach …contd

• Recursive definition of the value of an optimal solution

– Let m[i, j] be the minimum number of scalar multiplications necessary to compute Ai..j

– Minimum cost to compute A1..n is m[1, n]

– Suppose the optimal parenthesization of Ai..j splits the product between Ak and Ak+1 for some integer k where i ≤ k < j

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Dynamic Programming Approach …contd

– Ai..j = (Ai Ai+1…Ak)·(Ak+1Ak+2…Aj)= Ai..k · Ak+1..j

– Cost of computing Ai..j = cost of computing Ai..k + cost of computing Ak+1..j + cost of multiplying Ai..k and Ak+1..j

– Cost of multiplying Ai..k and Ak+1..j is pi-

1pk pj

– m[i, j ] = m[i, k] + m[k+1, j ] + pi-1pk pj for i ≤ k < j

– m[i, i ] = 0 for i=1,2,…,n

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Dynamic Programming Approach …contd

– But… optimal parenthesization occurs at one value of k among all possible i ≤ k < j

– Check all these and select the best one

m[i, j ] =0 if i=j

min {m[i, k] + m[k+1, j ] + pi-1pk pj } if i<ji ≤ k< j

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Dynamic Programming Approach …contd

• To keep track of how to construct an optimal solution, we use a table s

• s[i, j ] = value of k at which Ai Ai+1 … Aj is split for optimal parenthesization

• Algorithm: next slide– First computes costs for chains of

length l=1– Then for chains of length l=2,3, … and

so on– Computes the optimal cost bottom-up

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Algorithm to Compute Optimal Cost

Input: Array p[0…n] containing matrix dimensions and nResult: Minimum-cost table m and split table s

MATRIX-CHAIN-ORDER(p[ ], n)

for i ← 1 to nm[i, i] ← 0

for l ← 2 to nfor i ← 1 to n-l+1

j ← i+l-1m[i, j] ← ∞for k ← i to j-1

q ← m[i, k] + m[k+1, j] + p[i-1] p[k] p[j]if q < m[i, j]

m[i, j] ← qs[i, j] ← k

return m and s

Takes O(n3) time

Requires O(n2) space

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Constructing Optimal Solution

• Our algorithm computes the minimum-cost table m and the split table s

• The optimal solution can be constructed from the split table s

– Each entry s[i, j ]=k shows where to split the product Ai Ai+1 … Aj for the minimum cost

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Example

• Show how to multiply this matrix chain optimally

• Solution on the board– Minimum cost 15,125– Optimal

parenthesization ((A1(A2A3))((A4 A5)A6))

Matrix Dimension

A1 30×35

A2 35×15

A3 15×5

A4 5×10

A5 10×20

A6 20×25

min. error boundary

Minimal error boundaryoverlapping blocks vertical boundary

_ =2

overlap error

Other applications: stitching

Graphcut Textures – Kwatra et al. SIGGRAPH 2003

Other applications: stitching

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Graphcut Textures – Kwatra et al. SIGGRAPH 2003

Ideal boundary:1. Similar color in both images2. High gradient in both images

Summary of big ideas

• Treat image as a graph– Pixels are nodes– Between-pixel edge weights based on gradients– Sometimes per-pixel weights for affinity to

foreground/background

• Good boundaries are a short path through the graph (Intelligent Scissors, Seam Carving)

• Good regions are produced by a low-cost cut (GrabCuts, Graph Cut Stitching)

Slide Credits

• This set of sides also contains contributions kindly made available by the following authors– Alexei Efros– Carsten Rother– Derek Hoiem