ENERGY-MINIMIZATION BASED MOTION ESTIMATION

USING ADAPTIVE SMOOTHNESS PRIORS

Tarik Arici

and Vural Aksakalli

Department of Electrical Engineering, Istanbul Sehir University, Istanbul, Turkey

Department of Industrial Engineering, Istanbul Sehir University, Istanbul, Turkey

Keywords:

Energy Minimization, Approximation Algorithms, Primal-dual Method, Motion Estimation.

Abstract:

Energy minimization algorithms are used in low-level computer vision applications for labeling tasks such

as stereo-disparity estimation, image restoration, motion estimation, and optical ﬂow. The energy function

involves terms that evaluate the goodness of a solution in terms of a prior knowledge in addition to data terms.

The most widely used priors are smoothness-based priors, which enhance the quality signiﬁcantly. However,

the smoothness assumption is notvalid across discontinuities (e.g. motion boundaries). We present a method to

update the weights of smoothness terms using the dual problem when the approximation algorithm is iterative.

The dual of the primal energy minimization problem is used to infer about the validity of the smoothness prior

and impose it more correctly at each iteration. We demonstrate the effectiveness of this method against the

state-of-the-art in the optical ﬂow literature.

1 INTRODUCTION

Energy minimization refers to designing an energy

function that describes the desired properties of a

solution and its minimization to obtain a good so-

lution. Most problems in computer vision such as

motion/optical-ﬂow estimation, stereo-disparity esti-

mation, video synopsis, image formation modeling,

texture segmentation are ill-posed (Boykov et al.,

2001; Rav-Acha et al., 2008; Rav-Acha et al., 2006;

Hofmann et al., 1996) in the Hadamard sense: many

solutions exist and choosing one of the many solu-

tions based solely on data results in over-ﬁtting of the

parameters. A model with over-ﬁtted parameters is

not good in prediction and also not effective in gener-

alizing beyond data for truly explaining the observed

phenomenon.

An energy function consists of a data term and

a prior knowledge term to discriminate between the

large number of solutions. Energy functions are of-

ten difﬁcult to minimize because the solution space is

generally exponentially large depending on the size

of the problem. The data energy typically has a

large number of non-convexities, which exacerbates

the problem. One of the most commonly used prior

knowledge is the smoothness of the solution, which is

shown to be NP-hard (Boykov et al., 2001).

Energy minimization problems can be formulated

as labeling problems. A typical energy is of the form

E(v) =

∑

b∈B

) + λ

∑

b,a∈N

b,a

, l

), (1)

where D

is the data ﬁdelity term, V

b,a

represents the

prior knowledge as a penalty cost between two inter-

acting sites b and a. B is the set of sites, N is the set

of sites that interact with each other, and l

is the label

for site b. With n

sites and n

labels, the number of

possible labellings is n

NP-hard labeling problems will require exponen-

tial time to ﬁnd the global minimum. To achieve prac-

tical algorithms, local search methods can be used

to reﬁne an initial solution iteratively (Papadimitriou

and Steiglitz, 1998). When there are many local

minimums, approximation algorithms that do a lo-

cal search are likely to get stuck at a local minimum.

The local minimums are created mostly due to the

non-convexities in D

. Fortunately, V

b,a

is generally

convex because the prior knowledge is commonly a

norm of a distance measure on the solution space or

its derivatives. Hence, increasing the weight ofV

b,a

the energy will help with the convergence of the al-

gorithm. But labels have discontinuities, which make

b,a

invalid for speciﬁc interactions between sites and

increasing its weight will falsely impose an invalid

prior knowledge on the labels. To solve this problem,

we propose to use the dual problem of the primal en-

201

Arici T. and Aksakalli V..

ENERGY-MINIMIZATION BASED MOTION ESTIMATION USING ADAPTIVE SMOOTHNESS PRIORS.

DOI: 10.5220/0003805202010207

In Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP-2012), pages 201-207

ISBN: 978-989-8565-04-4

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

ergy problem. The primal problem for each iteration

of a label is expressed as a linear programming prob-

lem and the equivalence of its Lagrangian to the en-

ergy under speciﬁc choices of the Lagrangian multi-

pliers is shown. The dual solution is used to correctly

impose the prior knowledge by adapting the weights

in the smoothness prior via complementary slackness

conditions.

In the next section we give a review of prior art in

energy minimization. In Section 3, we present our

primal-dual method for energy function design us-

ing a spatial smoothness prior. Finally, in Section 4,

we demonstrate the power of our proposed method

against the state-of-the-art in the optical ﬂow litera-

ture.

2 RELATED WORK

Prior works on energy-minimization determines the

energy function at the beginning by ﬁxing the pa-

rameters manually, and focus on efﬁcient minimiza-

tion techniques. Energy-minimization is commonly

used in computer vision applications (Rav-Acha et al.,

2008; Rav-Acha et al., 2006), and it is applied to

pixel level stereo-disparity estimation, which is simi-

lar to motion estimation in essence (Kolmogorov and

Boykov, 2002; Boykov and Veksler, 2006; Veksler,

1999).

The majority of prior art focuses on the mini-

mization problem rather than the design of the en-

ergy function. Graph-cut methods are powerful in

quickly converging to good local minimums due to

their ability to do large moves (Kolmogorov and

Boykov, 2002; Boykov and Veksler, 2006; Veksler,

1999). However, they are computationally complex

and are not hardware-friendly since they operate on

node lists to ﬁnd max ﬂows in a graph. Modifying

energy functions to avoid falsely imposing smooth-

ness priors across discontinuities is proposed for ob-

ject segmentation but the discontinuity is inferred us-

ing the contrast in the pixel intensities (Kohli and

Torr, 2007). For optical ﬂow changing the weights us-

ing the image gradients is also proposed (Werlberger

et al., 2009). In these approaches, although the en-

ergy is adapted to the image (or data in general), it is

still ﬁxed throughout the iterations. Also, the adapta-

tion may bias the solutions towards a bad local min-

imum since in some cases contrast in pixel intensity

may not produce discontinuities/boundaries in the la-

bels. Instead, we propose to modify the energy using

the labels from the previous iteration, which becomes

more reliable as the minimization algorithm improves

the solution.

Belief propagation techniques are also used for

minimizing the energy (Meltzer et al., 2005). The re-

sults achieved with belief propagation techniques are

similar to graph-cut methods in quality.

Application of energy minimization to various

computer vision problems is done by introducing

new terms in the energy function to utilize prior in-

formation for the speciﬁc problem. For example,

video synopsis tries to compress an image sequence

in time, hence, requires the use of energy terms that

imposes this via temporal energy terms (Rav-Acha

et al., 2006). Another application such as unwrap-

ping the surface of an object for video editing requires

some other energy terms related to tracking and mo-

saic stitching (Rav-Acha et al., 2008).

In this work, we study the energy design aspect of

energy minimization that uses ﬁrst-order smoothness

priors. We propose a primal-dual linear programming

method that utilizes labels from the previous iteration

to modify the smoothness priors in order to take the

discontinuities into account.

3 PRIMAL-DUAL METHOD FOR

ENERGY-MINIMIZATION

3.1 Formulation of a Move as a Linear

Program

We deﬁne a move as a change of labels to a particular

candidate label, ρ. A large move can change a group

of site’s labels to ρ, keeping all other labels the same

as the pre-move labels l

∗

, while a standard-move can

only change a single site’s label to ρ.

We formulate the move problem for candidate la-

bel ρ as a linear programming problem (move-LP) in

the canonical form

min

∑

b∈B

(ρ) + (1− x

)D)b(l

∗

) (2)

subject to

− x

≥ −p

, x

− x

≥ −p

, ∀{b, a} ∈ N ,

≥ 0, −x

≥ −1, ∀b ∈ B,

where l

∗

is the pre-move labels, ρ is the candidate

label of the move, and p

is the probability of b

and a to be on different objects

. x is the variable

of the move-LP, which is in [0, 1]. x

= 1 indicates

that l

= ρ, and x

= 0 indicates that l

= l

∗

. For

∈ (0, 1), a fractional move is indicated, which can

Generally, p

’s are neither known nor directly ob-

served from data. However as discussed in Section 3.3 our

technique does not require the knowledge of p.

VISAPP 2012 - International Conference on Computer Vision Theory and Applications

202

be interpreted as b takes on ρ with probability x

and

retains l

∗

with probability 1 − x

. The advantage of

expressing the move problem as a linear problem is

that it enables fractional moves during the iterations.

To avoid confusion, it is important to note that the

above equation is not a relaxed version of the en-

ergy minimization problem given in (1), but its La-

grangian with speciﬁc set of multipliers is identical to

(1), which will be proved next.

The cost term in (2) is a weighted sum of data

terms with labels ρ and l

∗

. Two inequality con-

straints are imposed on the move to achieve smooth-

ness: x

− x

≥ −p

and x

− x

≥ −p

. They are

canonical form linear constraints for |x

− x

| ≤ p

With these constraints neighbor sites that are likely to

move together are forced to move together depending

on the probability, p

, of b and a to have different

labels. Hence, smoothness is enforced on actions but

not on the labels. However, a sequence of smooth

moves will create a smooth label set in the end.

We next write the Lagrangian for the move-LP in

(2),

L(x, β, γ, κ) =

∑

b∈B

(ρ) + (1−x

∗

) +

∑

{b,a}∈N

(−x

+ x

− p

) + β

(−x

+ x

− p

) +

∑

{b,a}∈(N)

−γ

+ κ

− 1), (3)

where β, γ, and κ are positive Lagrange multipliers

associated with their corresponding inequality con-

straints.

Lemma 1. A binary solution

x of the move-LP cor-

responds to a move from l

∗

to l

Proof. By the deﬁnition of x, l

= ρ if x

= 1, and

= l

∗

if x

= 0, which means a change of labels to

ρ while keeping all other labels same as in l

∗

. There-

fore, a binary x corresponds to a move from the pre-

vious label set l

∗

to a new label set l

, which is the

solution to our linear program.

Using Lemma 1, and the Lagrangian in (3) yields:

Theorem 2. The Lagrangian of the move-LP with a

binary solution x is equal to the energy, E(l

), if the

Lagrangian multipliers satisfy

= β

= −

λV

b,a

∗

, l

∗

) s.t. x

= 0, x

= 0

= 0, β

= 0 s.t. x

= 1, x

= 1

= 0.β

(1− p

) = λV

b,a

(ρ, l

∗

) s.t. x

= 1, x

= 0

(1− p

) = λV

b,a

∗

, ρ), β

= 0 s.t. x

= 0, x

= 1

= 0, κ

= 0,

(4)

A binary solution is deﬁned such that each component

of x is either zero or one.

where l

is a move away from l

∗

Proof. We skip the proof due to space limitations.

3.2 Sub-optimality of the Energy

Theorem 2 shows that for a speciﬁc choice of the La-

grangian multipliers, the Lagrangian of the move-LP

problem is equal to the energy under a binary move.

The conditions for the Lagrangian multipliers for this

equality may not be the best in terms of convergence

and energy function design, because the Lagrangian

multipliers are chosen without utilizing any informa-

tion learnt from the iterations. By intelligently choos-

ing the Lagrangian multipliers, contributions of the

constraints in the Lagrangian can be better adjusted.

This will especially improve the application of the

smoothness constraints for sites that are in the vicinity

of a discontinuity.

To achieve our goal, we utilize the primal-dual

schema (Papadimitriou and Steiglitz, 1998). This

technique constructs a feasible solution of the dual

problem and an integer solution of the primal prob-

lem iteratively. Since a solution of the dual problem

also provides lower bound on the primal problem by

the weak duality, one can compute a feasible solution

for the dual problem ﬁrst which then can be used to

ﬁnd the corresponding primal solution via the com-

plementary slackness conditions (Papadimitriou and

Steiglitz, 1998).

Our method differs from the primal-dual schema

by the way complementary slackness conditions

(CSCs) are utilized. In the conventional primal-dual

schema, CSCs are used to obtain the integer primal

solution from the dual solution. However, we want

to design a uniform algorithm, which has one type of

computation kernels, which iteratively update the la-

bels by substituting in the energy. Iterations of the

same forms of energy function are more suitable for

parallel implementations rather than an approach that

involves minimizing an energy followed by solving

the CSCs. In our method, CSCs are used to infer if

a constraint of the move-LP is tight or slack. A slack

constraint means the constraint is satisﬁed as a strict

inequality, and a tight constraint means the constraint

is satisﬁed with equality. By learning if a constraint

is slack or tight via the dual feasible solution and the

CSCs, the Lagrangian multipliers can be adjusted be-

cause a slack constraint means our prior information

agrees with our current solution, and a tight constraint

mean our prior information disagrees with our current

The proof can be found in the ﬁrst author’s Ph.D thesis.

ENERGY-MINIMIZATION BASED MOTION ESTIMATION USING ADAPTIVE SMOOTHNESS PRIORS

203

solution. Compared to choosing the Lagrangian mul-

tipliers as speciﬁed by the condition of Theorem 2

in an ad hoc manner, our primal-dual method uses

the feasible dual solution to determine the Lagrangian

multipliers, which is then used in the Lagrangian that

is to be minimized. We present the dual problem of

energy minimization in the following section.

3.3 The Dual Problem

We start with rewriting the move-LP problem in (2) in

matrix form for a compact representation. The primal

problem is

min

x+ ¯c

(1− x) (5)

subject to

Ax ≥ −p

x ≥ 0, −x ≥ −1,

where c, ¯c, x ∈ R

|N |

, and A ∈ R

|N |×|B|

such that c

(ρ) and ¯c

= D

∗

), and A

= 1, A

= −1 for

any k

pair {i, j} ∈ N . It is straightforward to con-

struct the remaining labels so that the above problem

is equal to the original move-LP problem. The La-

grangian can be formed similar to (3):

L(x, β, γ, κ) = c

x+ ¯c

(1− x) +β

(−p− Ax)−

x+ κ

(x− 1)

= (c− ¯c− A

β− γ − κ)

x− β

p− κ

= m

x− β

p− κ

(6)

where m is substituted for c − ¯c − A

β − γ − κ for

compactness. One can see that because of the way

the constraints of the primal problem are utilized in

the Lagrangian, the Lagrangian is always smaller than

the primal problem’s objective

L(x, β, γ, κ) ≤ c

x+ ¯c

(1− x), (7)

given that x is a feasible point in (5).

The dual function is a minimization of (6) on x:

g(β, γ, κ) = inf

L(x, β, γ, κ)

(

− β

p −κ

1 if m = 0

− ∞ otherwise

(8)

For the dual function to exist m must be equal to 0.

It is straightforward to show that the dual function

provides lower bounds on the more complex primal

problem’s optimal value by observing (7) and (4).

The dual problem is the maximization of

the dual function, which becomes a minimization by

changing the sign of the objective function:

min

β,γ,κ

p+ κ

1 (9)

subject to

c− ¯c− A

β− γ− κ = 0, (10)

The dual problem turns out to be the minimum cost

ﬂow problem with costs given as the probabilities of

two sites belonging to different objects, which can be

solved using the maximum ﬂow algorithms. The re-

quired ﬂow is speciﬁed by c− ¯c− A

β− γ− κ, while

ﬂows across discontinuities (high p values) are dis-

couraged. We are looking for a feasible solution of

the dual problem, not necessarily the optimal solu-

tion. An optimal solution requires more computation

and knowledge of p, which speciﬁes the probability of

two sites having different blocks, which is not known.

p values can be inferred from data similar to (Kohli

and Torr, 2007) by using intensity gradient, but in this

work we will use a feasible solution of the dual prob-

lem that does not require the knowledge of p.

Dual feasibility condition is speciﬁed by the re-

quired ﬂow condition given by

c− ¯c− A

β− γ− κ = 0, (11)

And β can be estimated by

β = A

†

(c− ¯c− γ), (12)

where A

†

is the pseudo-inverse of A, κ is set to zero

in the second equation, because the dual function that

we want to minimize is an increasing function of κ

Since A

†

(c− ¯c) can be negative,the non-negativityas-

sumption of

β required for the derivation of the dual

function can be violated. Hence, we need to thresh-

old its components to zero from below. Since γ is

unknown, one can choose γ to minimize the need for

this clamping operation. At any rate, we can rewrite

(12) to get the dual feasibility condition (DFC)

β = max{0, A

†

(c− ¯c)}. (13)

Using a feasible solution for

β of the dual problem, we

can understand if the constraints of the primal prob-

lem are loose or tight, which can be used to re-design

the energy function for the next iteration. Next sec-

tion discusses this relation between

β and the smooth-

ness constraints, and also how it is used for energy

re-design.

We note that κ = 0 does not necessarily produce the

optimal solution of the dual problem.

VISAPP 2012 - International Conference on Computer Vision Theory and Applications

204

3.4 The Primal-dual Relation

First, we describe a method to solve for β. To have

a unique solution for the DFC equation, A

†

must be

well-deﬁned. However, by the way we deﬁned the

constraints in the move-LP each block pair {a, b} has

two constraints

− x

≥ −p

(14)

− x

≥ −p

. (15)

To ﬁnd a relation between β and move-LP con-

straints, we utilize the dual complementary slackness

conditions (Papadimitriou and Steiglitz, 1998). If the

complementary slackness conditions are satisﬁed, any

feasible solution x of the primal and β of the dual

problem are optimal. The dual complementary slack-

ness condition (DCSC) is given below.

Either β

= 0 or A

x = −p

, (16)

where {b, a} is any site pair and A

is the row for the

constraint associated with β

. The complementary

slackness conditions are important in the design of

efﬁcient approximation algorithms for complex prob-

lems. These algorithms are called primal-dual based

methods and they try to improve x and β by modi-

fying them in a way that more of their components

satisfy the complementary slackness conditions (Pa-

padimitriou and Steiglitz, 1998).

Our goal for applying the primal-dual method is

to re-design the energy function, which corresponds

to the Lagrangian of the primal problem. The La-

grangian is showed to be equivalent to the energy

function under the conditions on the Lagrangian mul-

tipliers given in Theorem 2. However, our choice of

the constraints were imposed for all sites pairing with

a site, independent of the discontinuity since the un-

known l is what we want to estimate. Fortunately,

β reveals some information on the connectedness of

sites with their neighbors by the use of DCSC. DCSC

dictates that if

β is non-zero, then the constraint is

binding and must be satisﬁed with equality. If the con-

straint had been relaxed, the interacting sites would

have chosen different labels. This implies that the two

sites are on different label segments. Hence, the con-

straint that is binding is conﬂicting with our smooth-

ness of l assumption in that locality, and should be

imposed less by decreasing its contribution in the en-

ergy. To this end, the Lagrangian multipliers must

be inversely related to

β, for example by multiplying

with a function f(

β) that is decreasing with

β and has

range [0, 1]. Hence, by DCSC we propose to update

the β values as

DCSC

= f(

)β

, (17)

where

β is a feasible solution of the dual problem

given in (13), and f is a decreasing function of

β. Re-

placing Lagrangian multiplier β with β

DCSC

and sub-

stituting κ = 0 from the solution of the dual problem,

Lagrangian in (3) after some simpliﬁcation becomes

E(v) =

∑

b∈B

) + λ

∑

{b,a}∈N

β)V

b,a

, v

). (18)

The above energy formulation does not isotropi-

cally enforce a smoothness constraint, but adapts the

weights of constraints in the energy with information

derived from the data via the dual problem. This will

enable us to obtain an energy function that is more

powerful to explain labeling across discontinuities.

4 EXPERIMENT RESULTS

We present experiment results using the proposed

method and compare our results with the state of the

art in the optical ﬂow literature. Our implementation

of the proposed method is a hierarchical motion es-

timation algorithm that uses a full resolution and a

half resolution image produced by down-sampling the

full resolution image by two. For each half resolution

block, a motion search is performed to pick the two

best motion vectors to minimize a cost. The cost is

a sum of absolute deviation (SAD) based cost and all

motion vectors in a 2-D search window are evaluated

to ﬁnd the minimum cost vectors. This way an ini-

tial motion-vector ﬁeld v

is created and reﬁned using

N standard-move iterations. Candidate vectors in re-

duced search window S is used to pick the best-two

standard move for each b. S consists of 18 vectors ob-

tained from a block’s and its eight-connectivityneigh-

borhoods’ best-two vectors. Smoothness cost is de-

rived from blocks in four-connectivity neighborhood

and weights of the neighboring blocks is adapted us-

ing information from the dual problem. Each block is

partitioned to quarter blocks so that a half-resolution

quarter-block matches with a full resolution block in

size. This will increase the reliability of the centered

motion search in full resolution. Also, partitioning

to quarter blocks increases the quality of half reso-

lution vectors in general, since with a smaller block

size block-based translational-motion model is less

problematic for rotation, zooming, and motion bound-

aries. After the partitioning, standard-move iterations

are again applied to reﬁne further, before passing to

full resolution. Execution of our algorithm on a sam-

ple instance of the underlying problem is illustrated

in Figures 1 and 2. FR-FULL, FR-QUARTER, and

FR-QUARTER

images in Figures 1 and 2 have res-

olutions of 16x16, 8x8, 4x4 pixel blocks respectively.

ENERGY-MINIMIZATION BASED MOTION ESTIMATION USING ADAPTIVE SMOOTHNESS PRIORS

205

Table 1: Comparison of our DCSC-based method to Weickert et al.’s method on the Middlebury dataset. Mean endpoint error

of our method is comparable, if not superior, to that of Weickert et al.’s method even though our method uses only half the

pixel accuracy.

Mean Endpoint Error Schefﬂera Wooden Grove Urban Teddy

all disc untext all disc untext all disc untext all disc untext all disc untext

Method of Weickert et al. (2005) 1.12 1.80 0.99 1.07 2.06 1.12 1.23 1.52 1.62 1.54 2.15 0.96 1.38 2.26 1.83

DCSC-based method 1.21 1.77 1.18 0.94 2.03 0.97 1.20 1.57 1.08 1.73 1.90 1.12 1.37 2.16 1.81

The super-imposed image in Figure 4 shows the video

frame with color plane modulated using the estimated

motion vectors.

(a) (b)

Figure 1: (a) FR-FULL and (b) FR-QUARTER.

(a) (b)

Figure 2: (a) FR-QUARTER

and (b) Super-imposed im-

age.

We now compare our method’s performance

against that of Weickert et al.’s method (We-

ickert et al., 2005), which is a state-of-the-art

technique in optical ﬂow literature. We use

the Middlebury dataset available on the Web at

http://vision.middlebury.edu/ﬂow for benchmarking

purposes. This dataset has four types of data with

different characteristics and error measures distin-

guished for motion discontinuities and textureless re-

gions. Comparison of our DCSC-based method to

Weickert et al.’s method is given in Table 1. Although

performance of our method is comparable to that of

Weickert et al. (2005) in mean endpoint error, our

method performs better in 4 out of 5 discontinuity

cases; as expected. Our motion vectors operate with

only half pixel accuracy because we are searching in

a reference image upscaled by two. It is therefore

impressive that even though our current implemen-

tation uses only of half available pixels, our method

performs better than that of Weickert et al. (2005) in

10 out of the total 15 cases. This shows that adapt-

ing the smoothness prior using the dual problem is

a promising technique. It is, however, important to

observe that we are not ﬁnding the optical ﬂow, but

rather performing motion estimation via motion vec-

tor searching. As future work, we plan to perform

further benchmarking on different datasets and com-

pare our method to other optical ﬂow techniques in

the literature.

5 CONCLUSIONS

Energy minimization enables incorporation of prior

knowledge for improving parameter estimation. Un-

fortunately, the weights of the prior knowledge in the

energy function is not known. We proposed to use a

feasible solution of the dual of the primal minimiza-

tion problem to adjust the weights, which improves

the energy when the prior knowledge is not valid.

In the case of a smoothness prior knowledge, which

is commonly used in computer vision problems, the

dual feasible solution is used to infer about the discon-

tinuities via the dual complementary slackness condi-

tions.

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video editing. ACM Transactions on Graphics (SIG-

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Rav-Acha, A., Pritch, Y., and Peleg, S. (2006). Making

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Conference on Computer Vision and Pattern Recogni-

tion.

Veksler, O. (1999). Efﬁcient Graph-Based Energy Mini-

mizatation Methods in Computer Vision. Cornell Uni-

versity.

Weickert, J., Bruhn, A., and Schnorr, C. (2005). Lu-

cas/kanade meets horn/schunck: Combining local and

global optic ﬂow methods. International Journal of

Computer Vision, 61(3):211–231.

Werlberger, M., Trobin, W., Pock, T., Wedel, A., Cremers,

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tical ﬂow. In British Machine Vision Conference.

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