graph-cuts is exploited and a novel object segmentation method is proposed based on the ... automatic object segmentation from still images. The proposed.
Automatic Object Segmentation by Quantum Cuts Çağlar Aytekin, Serkan Kiranyaz and Moncef Gabbouj, Fellow, IEEE Department of Signal Processing, Tampere University of Technology, Tampere, Finland. Abstract—In this study, the link between quantum mechanics and graph-cuts is exploited and a novel object segmentation method is proposed based on the ground state solution of a modified Hamiltonian. First, the graph representation of certain quantum mechanical operators is studied. This reveals strong connections with widely used graph-cut algorithms while quantum mechanical constraints exhibit crucial advantages over the existing graph-cut algorithms. Furhtermore, concepts such as potential field helps solving a particular singularity problem related to laplacian matrices. In the proposed approach, the ground state (wavefunction) corresponding to a subatomic particle of a modified Hamiltonian operator corresponds to a particular optimization problem, the solution of which yields the salient object segmentation in a digital image. This approach provides a parameter-free -hence dataset independent-, unsupervised and fully automatic segmentation, which outperforms many existing state-ofthe-art algorithms. The results of the proposed automatic object extraction method exhibit such a promising accuracy that pushes the frontier in this field to the borders of the input-driven processing only – without the use of “object knowledge” aided by long-term human memory and intelligence. Furthermore, with the novel technologies for measuring a quantum wavefunction, the proposed method has a unique potential: Automatic object segmentation in an actual physical setup in nano-scale. Such an unprecedendent property will not only produce segmentation results instantaneously, but may be a unique opportunity to achieve accurate object segmentation in real-time for the massive visual repositories of today’s “Big Data”. Keywords—Quantum Mechanics, Schrödinger’s equation, Salient Object Segmentation, Quantum Operators, Graph-Cut, Measuring the Quantum Wavefunction
I.
graph-cuts [7]. We demonstrate that the discovery of this link leads to a superior application for an ill-posed computer vision problem: automatic object segmentation from still images. The proposed method is physically realizable in quantum (nano) scales, fully automatic, unsupervised, and entirely parameter-free, and hence dataset independent, yet produces such results that are far superior to those of many sophisticated graph-cut methods including several salient object segmentation algorithms. A crucial property is that the speed of a possible physical realization will be instantenous in the sense that a wavefunction forms instantenously in a defined potential field. In this paper, we explore a quantum-mechanical operator in close relation with Laplacian matrix [6]. When a quantum mechanical concept such as potential field on a particle is integrated with this particular matrix, we obtain Hermitian operators, which satisfy the real eigenvalue constraint to be a quantum mechanical operator [8]. The quantum states, expressed by the wavefunctions that are obtained from performing eigen-vector analysis on these operators, lead to the definition of a probability density of occurrence of a particle in space with which we associate soft labels indicating the foreground (object) / background segmentation. The most significant phenomenon we observed is that especially the ground state of these operators corresponds to a solution of a unique energy function, which is more fitting to salient object extraction problem when compared to previous graph-based methods. Furthermore, this superiority is experimentally demonstrated by comparative evaluations against the complicated, parameterdependent and in some cases supervised state-of-the-art salient object segmentation algorithms.
INTRODUCTION
Quantum mechanics constitute a strong mathemathical background for physical events occurring in sub-atomic scale. However, concepts emerged from quantum mechanics have been excesively exploited in a variety of areas such as cryptography [1], computing [2], search algorithms [3], integer factorization [4], signal processing [5], etc. Some of these areas only exploit ideas from quantum mechanics and adapt them to the problem at hand, while some others directly utilize the quantum mechanical principals. Although a quantum computer utilizing quantum mechanical algorithms is still far from the practical use, the research on this field is immense. This is because concepts and approaches based on quantum mechanics provide significant advantages over classical problems, e.g., perhaps the most famous one is the search algorithm of Grover [3], which provides much faster search than any algorithm on a classical computer can achieve. Another example, equally well-known, is the integer factorization algorithm of Shor’s [4], which can solve integer factorization problem in polynomial time. Thus, research in this field is crucial in the sense of constituting the foundations of the future technologies. In this study, we draw the focus on a problem where quantum mechanics solely defines the environmental conditions. We explore the link between quantum mechanical operators and graphs, more specifically between ground state of quantum mechanical operators and
II.
RELATED WORK
The foundations of salient region detection, which can be defined as the selection of visually interesting/important regions or objects are biologically inspired [9], [10], [11], [12]. However, other computationally oriented methods are proven to be fit to the problem. Most of the methods exploit local contrast concept [13], [14], [15], [16], [17], i.e., a salient region/object exhibits a significant contrast to its immediate surrounding. Hence center-surround distance or edge based algorithms are widely used. Another popular assumption is that the salient object is globally distinct; i.e possesses discriminative color distribution with respect to the rest of the image [16], [18], [19], [20]. A number of local contrast based algorithms only detects highfrequency edges and most of the time, the inner region of a salient object can be discarded [17]. In particular, the problem with the methods based on center-surround histogram difference [16] is the high computational complexity, i.e. a large number of analysis windows had to be processed since one does not have any information about the region size, aspect ratio and location of the salient region. Furthermore, the presence of the high-frequency noise degrades the performance of local contrast based algorithms.
Global contrast based approaches generally discard the location information and as a consequence the inter-similarities of color distributions between object and the background usually degrades their performances. Moreover, the high computational complexity is still a severe drawback of these approaches. Several efficient algorithms have recently been proposed such as saliency filters [35] contrast based filtering [21]. Despite the advances in the field, salient object segmentation still remains an ill-posed problem since the definition of the saliency is not clear. Most importantly, most of the salient object segmentation methods suffer from severe parameter dependencies and/or require a high degree of supervision and machine learning, which in turn makes them entirely dataset dependent. In other words there is no guarantee that they will show a similar performance on another dataset that they are not trained or tuned for. In order to address such drawbacks and deficiencies, in this study we present a novel, parameter free, graph-based approach, the quantum cuts, which takes into account background priors, local contrast –while preserving the inner area- whilst dealing with high frequency noises. Quantum cuts (QC) achieves this by optimizing a unique energy function which in turns minimizes the cut value while favoring large areas and restricting the boundary of the image as the background.. In order to clarify these properties, next, we shall explore the link between graph-cut [7] and quantum mechanics. III.
THE LINK BETWEEN QUANTUM MECHANICS AND GRAPH CUTS
Before going into details of the relationship between quantum mechanics and graph-cuts, we shall start with some descriptions about quantum mechanics for a basic understanding. The first postulate of quantum mechanics states that: “To any self-consistently and welldefined observable in physics (call it ) such as linear momentum, energy, mass, angular momentum, or number of particles, there corresponds an operator (call it ) such that measurement of yield values (call these measured values ) which are eigenvalues of ” [8]. Hence the following holds: (1) = where is the eigenfunction of and represents a wavefunction. Although on its own, has no physical meaning, it is sufficient to define any property of the particle with operators acting on it. Furthermore, ( )∗ ( ) defines the probability distribution function (PDF) of the particle’s location. ħ
When is the energy operator, = = − ∇ + ( ), the time independent Schrödinger’s equation can be expressed as follows: ħ (2) − ∇ = − ( ) + 2 where, is the energy of a particle, ( ) is a potential distribution, is particle mass and ħ is the reduced Planck constant. When the Laplacian operator is approximated with a finite-difference method, the solutions of this equation, i.e , are the eigenvectors of the Hamiltonian matrix with following entries: ħ ⎧ ( )+| | = ⎪ 2 ( , )= ħ (3) ⎨ − ⎪ 2 ⎩ 0 .
where | | is the cardinality of element . Note that the matrix
which is the set of neighbors for is similar to the Laplacian matrix
ħ
[6] of a graph while the edge weights are fixed to . However, the additive potential term on the diagonals, ( ), makes the difference as compared to the Laplacian matrix formulation. Therefore, we shall now discover the meaning of this potential by exploring a link to the Graph-Cut method [7]. First, we modify the Hamiltonian operator as follows: ħ ⎧ ( )+ , = 2 ⎪ ( , )= (4) ⎨ − ħ , ⎪ 2 ⎩ 0 .
In other words, we defined another discrete representation of the Laplacian using a weighted graph where , is the edge weight between nodes and . Note that is also Hermitian, hence a valid quantum mechanical operator. Now the graph structure of the image nodes is similar to the one in [7]; however, it suggests two additional terminal nodes the so-called source - S - and sink - T - nodes, respectively, which represent the foreground (i.e. the object) and the background nodes. Each image node is a pixel that has connection to both terminals and connected with weights which are defined as boundary penalties, , = , . Similarly, regional penalties to the object and the ("obj") and ("bkg"), are the edge weights from an background, image node to S and T terminals, respectively. IGC deals with the minimization of the following: argmin ( ) + ( ) (5) where regulates the importance of regional and boundary terms which are defined as, ( )= ( )=
{, }
( )
(6)
,
where = , … , , … | | is a binary indicator vector where each component , can be either corresponding to object (“obj”) or to corresponds to edges where labels of background (“bkg”) and nodes are different in a neighborhood, N. Accordingly, we can now define the vector L, similar to , but instead of abbreviations, it directly enumarates the binary “label” values as 1 or 0, corresponding to the object (L=1) or the background (L=0), respectively. Then, the goal is to minimize: argmin
("obj")(1 − +
= argmin
("obj") − +
)+
("bkg")
−
.
,
(7) ("obj") − −
("bkg") .
,
("obj") term since it has no effect on minWe can drop the ∑ imization over the vector L, nor does multiplying each side with a
constant. For the purpose of compatibility, we multiply each side with 1 . At this point we turn back to Schrödinger’s equation where the modified energy operator (Hamiltonian), is defined as in Eq. 10. Let the right part of modified Hamiltonian be, = . Then from Eq. 10 and considering that , = , we can explicitly write: ()= ()
ħ ( )+ 2
ħ = ( ) ( )+ 2
= ( ) ( )+
ħ 2
,
()
ħ − 2
,
ħ () 2
−
( )− ( )
()
,
,
Now let us define the potential as the penalty term between the foreground (object) and background, ( )= ("bkg") − ("obj"). Then, ( ) = ( )[ ("bkg") − ħ + 2
("obj")]
( )− ( )
(9)
,
Now, multiplying both sides of the Schrödinger’s equation with leads to the following: ∗ = ∗ where ∗ = , =
[ ( )∗ ( )]. [ ("bkg") − ħ + 2
∗
("obj")]
∗
(10)
[ ( ) − ( )].
()
,
Let us then define = ( )∗ ( ). Note that at this point we assume that can only have values {-1, 0, 1} but this is not the case in quantum mechanics; however, this assumption will lead to the true analysis shortly. Also, note that = ∗ = , since ∗ = 1 simply following from the properties of a PDF. On the other hand, ( )∗ ( ) corresponds to the PDF of the quantum particle’s occurrence and thus it can be thought of a soft labeling function. Now considering the second term in Eq. 10: ħ ∗( ) [ ( ) − ( )]. , 2 =
=
=
ħ 2
[
ħ 2
(
ħ 2
ħ = 2
(
∗(
−
−
−
∗(
) ( )− +
ħ + 2
−
+
ħ 2
).
− 1).
∗(
−
).
∗(
) ( )].
,
) ( ).
(11)
) ( ) .
(
,
,
,
(
∗
Hence we can write Eq. 10 as follows:
) () ()
∗(
)
( ) ( )( ( )
∗(
)
+
ħ 2
ħ 2
( (
−
∗
("obj")]
).
(12)
,
( ) ( )). (( ( )
∗(
) − 1).
,
ħ
[7] if is selected as, . However, there is an additional third term, which we define as the phase penalty. Note further that alone is not a labeling vector, but a wavefunction. Hence, it can have negative values too. Whereas this term penalizes an oscillation throughout the wave function, it has no effect on labeling vector , but on only. Indeed, this term only penalizes a [1, −1] or [−1, 1] transition in a neighbourhood while all other combinations yield no penalty. Since we care about the labeling only for the object segmentation purpose we can therefore simply omit this term. Or in other words, for any labeling , there can be found a yielding a zero phase penalty. In terms of labeling the ground state of the operator minimizes the same energy function as in [7] with an additional constraint: ∗ = 1. This constraint corresponds to minimization of Rayleigh ∗ quotient ( , ) = ∗ which will impose an area normalization over the cut. Hence, while [7] only minimizes ( , )̅ , the ground state of the operator minimizes the following: ( , ̅) ( )
(13)
Also, note that = ∗ = , since ∗ = 1 simply following from the properties of a PDF. This analysis demonstrated that the interpreation of the potential field in quantum mechanics as, ( )= ("bkg") − ("obj") leads to a segmentation similar to the graph cuts approach where the cut is obtained by minimizing the expression in Eq. 13.
IV. ,
, ∗(
+
[ ("bkg") −
When the ground state for the modified Shcrödinger’s equation is considered, i.e the wavefunction correspoinding to the minimum energy, Eq. 12 turns into an optimization problem where the goal is to minimize the modified energy, which is the eigenvalue of the modified Hamiltonian, , . Note that the first two terms of the above minimization is exactly the same as the one in the Graph-Cut method
(8)
,
=
THE QUANTUM CUTS
In order to make use of Eq. (13) in an automatic object segmentation, one has to define the potential field manually for every image. Recall that this was acquired by the human defined foreground and background seeds in [7] which is not applicable in our case. We assume to have no clue regarding the location or the color distribution of the object; however, we shall assume that the image boundary belongs to the background. Inspired by this assumption we define the potential as follows: , ∈ ()= (14) 0 . where is a 1-pixel wide boundary of the image and = ∞ but in practice, it is a sufficiently high value to prevent wavefunction (and hence the object) from occurring on the boundary. This clearly indicates a certainity that the boundaries are background but we have no assumption about the rest of the image. The overall QC algorithm for saliency map generation can now be summarized, as follows:
I. II. III.
Construct matrix, i.e the graph structure. Assign the potential ( ) as in Eq. 14 Compute the eigenvector with the smallest eigenvalue, i.e., the ground state wavefunction, . IV. Compute the soft labeling vector, = ( )∗ ( ). The final step of the QC, the salient object segmentation on the other hand, can then be obtained by a naïve or adaptive thresholding over the soft labels, L. As a preprocessing step we have observed that an edge preserving smoothing algorithm such as the one proposed in [22] can further improve the performance. It should also be noted that the only parameter in QC, , which is the mass of the particle can also be omitted, since Eq. (14) combined with Eq. (4) clearly shows that the change in within the potential ( ) can be reflected as a proportional change in the value. Hence, as long as this value is kept sufficiently high, the particle mass will have no effect and can, therefore, be selected as any value. However, we will select this particle mass equal to the mass of an electron, since we will further propose and analyze an actual physical setup for the proposed QC method next.
V.
where is the RGB vector of pixel which is normalized to (0,1) interval, (. , . ) is the Euclidian distance where a positive prevents division by zero. First, we perform visual analysis on the object extraction performance of the QC algorithm. For this, the saliency maps are normalized to [0, 1] interval and then thresholded with a fixed value, 0.027 in order to obtain the salient object segments. We obtained this threshold experimentally yielding 0.8614 precision (P) and 0.8642 recall (R) values for MSRA1000 dataset. In Fig.1 some typical results over the cluttered images (with the presence of strong edges) are shown. Note that the area constraint indeed prevents the algorithm to cut small regions with strong boundaries, hence resulting in accurate object segmentation. Note further that strong edges may be present in the object or outside, or even both. In either case, the most salient region generally corresponds to the object.
THE PHYSICAL SETUP FOR QUANTUM CUTS IN NANO-SCALE
There exist several detailed proposals to measure both the magnitude and the phase of a wave-function of quantum particles [23][26]. However, the actual physical realization of the QC can be even a simpler setup since it only requires the measurement of the square of the magnitude of the wave-function, i.e the PDF of the quantum particle. Aside from the classical measurements, any measurement in the quantum scale disturbs the actual state of the particle. Hence, it is a common practice to prepare a large number of physical setups that are identical to each other and measure each one of them, then combine the results. In each system, we shall measure the position of a particle in this constructed system e.g., by taking a microscopic photo of an electron, and as a final step we shall accumulate the results. Note that each outcome is a sample from a PDF, and simply cumulating the outcomes in a histogram will approximate the square of the wavefunction [27]. Although a complete description is necessary for the realization since the graph structure in QC is not regular, we have to omit this description in this study due to space limitation.
VI.
EXPERIMENTAL RESULTS
In this section, the proposed salient object segmentation algorithm, QC, is first tested visually for a large set of images and the results are analyzed in detail to point out the advantages and flaws of the proposed algorithm. Second, QC is compared against 19 state-of the art object segmentation algorithms over the benchmark MSRA-1000 dataset. In all experiments, we halved both width and height of the images to reduce the computational cost. For a 200x150 image, an non-optimized simulation of the algorithm in MATLAB R2013a, in a 8GB RAM, Intel i7 2,20 GHz computer with 64-bit OS, takes about 1.5 seconds per image. The boundary penalties are selected as follows: 1 , = (15) + ( , )
Fig. 1: (Left) Original images (middle) ground truth segmentation (right) object segmentation with QC.
Next, we perform comparative evaluations of the QC against the other state-of-the-art salient object segmentation methods over the benchmark MSRA 1000 database. In Fig. 2, the precision-recall (P-R) curves of the 15 state-of-the-art methods are presented. It is evident from this figure that QC clearly outperforms them. In Fig. 3, we have visually compared the saliency maps of the state-of-the-art methods which are comparable with QC. It is evident from this comparison that even a simple minimization with background priors can result in accurate segments in some cluttered scenes over which several state-ofthe-art methods fail. Also it should be noted that, besides the optional pre-processing step, QC is entirely realizable within a quantum mechanical system. Hence, further improvements that may boost the performance such as segment selection, iterative running of algorithm for multiple segment extraction and superpixel implementations are intentionally omitted in this study.
1
AC AIM CA CB FT GB IM IT MSS SEG SeR SF SR SUN SW D QC
0.9
0.8
precision
0.7
0.6
0.5
0.4
0.3
0.2
0
0.1
0.2
0.3
0.4
0.5 recall
0.6
0.7
0.8
0.9
1
Fig. 2 Precision-recall curve of performances of state-of-the art algorithms AC[34], AIM[28], CA[19], CB[21], FT[17], GB[12], IM[33], IT[9], MSS[29], SEG[30], SeR[31], SF[35], SR[14], SUN[32], SWD[18] and QC method.
Fig. 3. Visual comparison of saliency maps with state-of-the-art algorithms that are comparible with the proposed method: Original Image, Ground Truth and saliency maps of the proposed method, CB [21] and SF [35] (From left to right).
VII.
CONCLUSIONS AND FUTURE WORK
In this study, we have demonstrated that the (salient) object segmentation problem can be efficiently and accurately modelled by the quantum mechanics thanks to its rich mathematical strucuture and unique properties. In particular, we have explored the link between graph-cut algorithms and ground-state wave functions obtained from the Schrödinger’s equation where a modified energy operator (Hamiltonian) is used. After an extensive mathematical analysis, we have proposed a quantum-mechanical approach for salient object segmentation, namely QC that is fully automatic, parameter free and has a solid link between graph-cut algorithms and quantum mechanical operators. Experimental results demonstrated that QC significantly surpassed all of the stateof-the-art object segmentation methods over the MSRA1000 dataset despite of the fact that most of them are strictly parameter dependent and optimized for the benchmark dataset. The most crucial and unique property of QC is that it is not only inspired from quantum-mechanics, but also it can be realized at such small scales that would actually work under the physical constraints that would approximate the solutions proposed in this paper. This is why we have not performed any computational complexity analysis and in particular note that the actual speed of the algorithm is only limited with the measurement time since forming the wavefunctions in this scale is instantenous. Moreover, not to lose this propery, we have not employed any heuristics that would result in further performance gain such as iterative execution of the method to segment multiple objects and the selection from - or combination with - several wave-functions according to some learned shape priors. As future studies, we plan to exploit different quantum mechanical operators and form corresponding experimental setups. We already have the initial findings indicating that the superposition of some low-energy wavefunctions may result in segments with an even higher accuracy. Furthermore, we also consider applying this framework to more challenging tasks such as object segmentation (not necessarily a salient one) in cluttered scenes, via selection from wave-functions or combining them in a superposition framework which may also contain multiple objects. How to select and combine different wave-functions is also a topic for future work.
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[22] [23]
[24] [25]
[26]
[27] [28]
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