Schrödinger Transform of Image: A New Tool for Image Analysis

Image segmentation is the process of separating or grouping an image into different parts . These parts normally correspond to something that human beings can easily separate and view as individual objects. Computers have no means of intelligently for recognizing objects, and a large number of different methods have been developed in order to segment images, ranging from the simple thresholding method to advanced graph-cut methods. The segmentation process is based on various features found in the image. Those features might be histograms information, information about the pixels that indicate edges or boundaries or texture information and so on. Approaches of Image processing and analysis based on partial differential equation, such as deformable models or snakes (Terzopoulos et al., 1987; Kass, et al., 1987), balloon models (Cohen, L. D., 1991; Cohen, L. D. & Cohen, I., 1993), geometric models (Caselles et al., 1993), discrete dynamic contour models (Lobergt & Viegever, 1995), geodetic active contours (Caselles et al., 1995) and topology adaptive deformable model (McInerney & Terzopoulos, 1999), whose physical background is principle of minimum action or force equilibrium in classical mechanics, are being extensively applied to image segmentation, image smooth, image inpainting, extraction of boundary and so on. Xu and Prince analyzed the reason why snake methods have poor convergence to boundaries with large curvatures and replaced the gradient field with the gradient vector field (GVF), which has a larger capture region and slowly changes away from the boundaries (Xu & Prince, 1998). Consequently, the dependence on initial positions is decreased but the field can attract the moving contour to the right position. Parametric deformable models have high computational efficiency and can easily incorporate a priori knowledge. However, these models cannot naturally handle topological changes and are sensitive to initial conditions. Geometric deformable models are based on the level set method (Osher & Sethian, 1988), which was initially proposed to handle topological changes during the curve evolution. Geometric deformable models have the advantage of naturally handling the topological


Introduction
Image segmentation is the process of separating or grouping an image into different parts .These parts normally correspond to something that human beings can easily separate and view as individual objects.Computers have no means of intelligently for recognizing objects, and a large number of different methods have been developed in order to segment images, ranging from the simple thresholding method to advanced graph-cut methods.The segmentation process is based on various features found in the image.Those features might be histograms information, information about the pixels that indicate edges or boundaries or texture information and so on.Approaches of Image processing and analysis based on partial differential equation, such as deformable models or snakes (Terzopoulos et al., 1987;Kass, et al., 1987), balloon models (Cohen, L. D., 1991;Cohen, L. D. & Cohen, I., 1993), geometric models (Caselles et al., 1993), discrete dynamic contour models (Lobergt & Viegever, 1995), geodetic active contours (Caselles et al., 1995) and topology adaptive deformable model (McInerney & Terzopoulos, 1999), whose physical background is principle of minimum action or force equilibrium in classical mechanics, are being extensively applied to image segmentation, image smooth, image inpainting, extraction of boundary and so on.Xu and Prince analyzed the reason why snake methods have poor convergence to boundaries with large curvatures and replaced the gradient field with the gradient vector field (GVF), which has a larger capture region and slowly changes away from the boundaries (Xu & Prince, 1998).Consequently, the dependence on initial positions is decreased but the field can attract the moving contour to the right position.Parametric deformable models have high computational efficiency and can easily incorporate a priori knowledge.However, these models cannot naturally handle topological changes and are sensitive to initial conditions.Geometric deformable models are based on the level set method (Osher & Sethian, 1988), which was initially proposed to handle topological changes during the curve evolution.Geometric deformable models have the advantage of naturally handling the topological changes and are widely studied for medical segmentation (Malladi & Sethian, 1996).Another popular geometric model is proposed by Chan and Vese (Chan & Vese, 1999;2002).Chan-Vese's model is a simplified version of the Mumford-Shah energy model.The algorithm extracts the desired object through simultaneously minimizing the intensity variations inside and outside the contour.In 1997, Cohen and Kimmel described a method for integrating object boundaries by searching the path of a minimal active deformable model's energy between two points (Cohen, L. D. & Kimmel, 1997).But they are easy to fall into the local minimum, sensitive to noise, do not have topology adaptive, poor convergence to concave boundaries.Lou and Ding used point tracking by estimating the maximum probability of a particle in quantum mechanics moving from one point to another, and did not impose any smoothness constraints to ensure the extraction of the details of a concave contour (Lou & Ding, 2007a).To overcome the main drawbacks of global minimal for active contour models that the contour was only extracted partially for low SNR images, maximal probability method of boundary extraction based on particle motion was proposed (Lou et al, 2007b).Schrödinger transform of image was first given by Lou, Zhan, Fu and Ding, and the probability P(b,a) that a particle moved from a point a to another point was computed according to I-Type Schrödinger transform of image (Lou et al, 2008).In the chapter, a new tool for image, Schrödinger Transform of image, is investigated.The reminder of the chapter was organized as follows.First, we gave physical explanation of boundary extraction from the point view of classical mechanics and quantum mechanics in Section 2. Next, we defined Schrödinger transform of image, discussed its properties and computation in Section 3. Then we investigated scale parameter and potential function of Schrödinger transform in Section 4. And then, we constructed high and low pass filter and carried through automatic contour extraction for multiple objects using I-type Schrödinger transform of image, segmented image using deformation of II-type Schrödinger transform of image in Section 5. Finally, we gave our conclusion in Section 6.

Physical explanation of boundary extraction
Boundary extraction is belong to field of image progressing while classical mechanics and quntumn mechanics are pure physical concept.However, we can find their common ground from the point of view of particle motion.Boundary of object can be thought of as trajectory of moving particle while the law of motion of particle were investigated in classical mechanics and quntumn mechanics using two determinancy and nondeterminancy method.

Physical explanation of boundary extraction from the point of view of classical mechanics
Deformable models are the elastic curves defined within an image domain that can move under the influence of internal forces arising from curve smoothness and external forces computed from the image data.The internal and external forces are so defined that the deformable contour has a minimum energy at the true object boundary.The following mapping can represent the deformable contour model: is the parameterization variable of the object boundary and 0 and 1 correspond to the start and end points of the boundary.The deformable contour is a curve () (() , () ) sx s y s  x that moves in the spatial domain of the image to minimize the energy: where 12 , wware the weighting parameters controlling the contour's tension and rigidity respectively.pulls the contour towards the boundary.Thus, the object boundary is obtained either when the force equilibrium of Eq. ( 3) or a minimum of the energy in Eq. ( 2) is reached.

Physical explanation of boundary extraction from the point of view of quantum mechanics
The deformable contour () (() , () ) sx s y s  x can further be considered as the path of a moving particle in the image if the parameterization variable s is replaced by the time variable t .By referring the work of Feynman and Hibbs (Feynman & Hibbs, 1965), we explain boundary extraction from the point of view of quantum mechanics as follows: Suppose a particle moves from the position a at the time a t to the position b at the time b t , e.g., () , () . According to the theory of quantum mechanics, the amplitude, (,)  Kba , called kernel or propagator, contains the total contribution of all paths between a and b , which is different from Eq. ( 2) where only a specific path from a to b with the minimum energy is concerned.In order to distinguish these two types of contours, we refer to the contour determined by (,)  Kba as the quantum contour, denoted as () Qt , while the traditional deformable contour determined by minimizing the energy between a and b as the classical contour, denoted by () Xt .Obviously, the classical contour () Xt is considered as a specific case of the quantum contour ()  Qt when a single path is concerned.In physics, the energy functional of a path, E , is defined by: where L is the Lagrangian function of system.For a moving particle with the mass m , and potential (,)  Vt x , the Lagrangian function is determined by: (5) www.intechopen.com Comparing Eq. (4) to Eq. ( 2), we find that the Lagrangian functional of a path in Eq. ( 4) is similar to the energy functional of a contour curve in Eq. ( 2).Thus, we refer to deformable model as the motion of a particle described by classical mechanics.
In quantum mechanics, the total contribution of all paths between a to b is calculated by where (,) Rab is the set of all paths between a and b .(() )  xt is the contribution of a path with a phase proportional to its energy (() ) Et x , i.e., (2 / ) ( ( )) (() ) where h is the Planck's Constant and C is a constant.
According to the theory of quantum mechanics, the probability1 of a particle moving from the position a at time a t to the position b at b t , denoted by (,) Pba , is equal to the square amplitude of (,) For a system with a simple Lagrangian function, (,)  Kba can be calculated directly from the path integral ( (Feynman & Hibbs, 1965)) while for a system with a complex Lagrangian function, it is difficult and time-consuming to estimate the value of (,) Pba from (,) Kba .In order to avoid such difficulty, we estimate the probability of a particle moving from point a to point b directly from specific particle models, e.g., a free particle or a particle moving through a Gaussian slit, where their motions can be used to describe the boundary of an object of interest with a known probability density of appearing at a point.There is a stronger motivation of adapting the quantum mechanics than the similarity between equations ( 2) and (4).Snakes, Deformable models, level set methods, etc, are all based on classical mechanics in a form of partial differential equation.Although classical physics is adequate to explain virtually all phenomena one will ever directly experience in one's life, certain phenomena cannot be explained by classical physics.In many respects, quantum mechanics presents the physics that underlies physical reality at its most basic level.Quantum theory can be thought of as the generalization of classical mechanics and many non-classical phenomena that do not have a classical analog are known in the quantum physics world.The relationship between classical and quantum mechanics is of central importance to the philosophy of physics.Classical mechanics extends the elementary Newtonian concepts to the Lagrangian and Hamiltonian formulations, to the least action principle, to the angle-action variables, etc, in ways that are the essential framework of quantum mechanics.However, there are significant distinctions between the two theories that arise not because of quantization, but rather from the nonessential tendency to describe macroscopic systems by instantaneous values for position, speed and acceleration, and microscopic systems by time-averaged position probability densities.Probability is a bridge between classical mechanics and quantum mechanics.The detail discussion of this relationship is out of this chapter's scope, which can be founded by hundreds of literatures in quantum mechanics field.

Schrödinger transform of image
For a complex system, however, it is difficult to calculate the value of (,) Pba .In 1995, Williams and Jacobs derived the probability that a particle with a random walk passes through a given position and orientation on a path joining two boundary fragments, which is obtained by the product of two vector-field convolutions (Williams & Jacobs, 1995).Although some specific particle motions have been considered, a general analytic expression of the probability for complex system is still open.In order to calculate the value of (,) Pba , a numerical approximation of (,)  Pba is needed.In this section, we will try to compute the probability (,)  Pba by using Schrödinger transform of image.

Definition of Schrödinger transform of image
The active contour model or Snake model had their profound physical background.If the parameter s in the deformable contour curve () (() , () ) sx s y s  x could be understood as time t , object contour curve () t x could be considered as the path of the particle in plane motion.
Suppose a particle moves from the position a at the time a t to the position at the time b t ,e.g., () According to the theory of quantum mechanics, the probability of a particle moving from the position a to b at b t , denoted by (,) Pba , is dependent on the kernel (,) Kba , which is the sum of all paths contribution between a x and b x , i.e., where (,) Rab is the set of all paths between a x and b is the contribution of a path We must solve the problem of computing the kernel (,) Kba to introduce law of particle motion in quantum mechanics into image processing and analysis.For a system with a simple Lagrangian function, (,) Kba can be calculated directly from the path integral (see 0) while for a system with a complex Lagrangian function, it is difficult and time-consuming to estimate the value of (,) Pba from (,) Kba .Replacing the kernel (,) Kba with the wave function (,)  ut x in the position x at the time t , then (,) ut x satisfied the following Schrödinger equation:  G x can be given in Fig. 1.

Fig. 1. the relation between the probability (,)
Pba and image gradient () We could rewrite Eq.( 11) as the initial-value problem: Continuous Schrödinger transform of image () v x is defined as the solution of Eq.( 12).And the transform is called I-type Schrödinger transform when () 0 v  x , otherwise the transform is called II-type Schrödinger transform.By applying Fourier transform to equation ( 12) and making use of the properties of Fourier transform, we have where the mark '  ' denotes convolution of two functions, ' ^' denotes Fourier transform of function.When () 0 v  x , both (,) ut x and ˆ(,) ut y have the following analytic solutions (see 0): ut x and ˆ(,) ut y also have analytic solutions (L.C. Evans, 1998), but they are too complex to be used to compute their numerical solutions.We give the following definition of discrete Schrödinger transform of image because of Eq.( 12) and Eq.( 13): Supposed both ()   x and () v x are mn  images, then two-dimensional discrete Schrödinger transform of image ()   x based on () v x is expressed with the following differential equation which its Fourier transform satisfies: www.intechopen.com where ˆt u  is mn -dimensional column vector formed by concatenating all the rows of mn  matrix ˆt u .mn mn  matrix y was diagonal matrix whose diagonal elements express distance.mn mn  matrix V is a block cyclic matrix, i.e., 01 1 where i V is a cyclic matrix, Discrete Schrödinger transform of image () Obviously, the solution of Eq.( 16) is  was diagonal matrix.Eq.( 19) degenerates into Eq.(14) when

Computation of Schrödinger transform
The transfer function of Schrödinger transform of image is It is difficult to directly compute II-type Schrödinger transform of image using Eq.( 19).We can compute II-type Schrödinger transform using a two step method since the matrix V is Block Circulant Matrix and the matrix 2 || y is diagonal matrix in Eq.( 19).The block circulant matrix V is similar to a diagonal matrix, that is, Here, , and V D is a mn mn  diagonal matrix.
We know that the exponential function satisfies x yy x ee e   for any real numbers (scalars) x and y .The same goes for commuting matrices: if the matrices X and Y commute (meaning that XY YX  ), then XY XY ee e   .It is usually necessary for A and B to commute for the law to still hold.However, in mathematics, the Lie product formula, named for Sophus Lie, holds for all matrices A and B , even ones which do not commute.That is, for arbitrary real or complex matrices A and B ,   The formula has applications, for example, in the path integral formulation of quantum mechanics.It allows one to separate the Schrödinger evolution operator into alternating increments of kinetic and potential operators.Hence, we rewrite (19) as Here, 1,2, , kN   , V D is the inverse Fourier transform of V while V is the Fourier transform of ()  v x , that is, V D is a mn mn  diagonal matrix with the diagonal elements () v x .So, we obtain the following two step method of computing II-type Schrödinger transform: Step 1: Using the formual  28), that is, Eq.( 29) means that discrete Schrödinger transform of image () and Obviously, for continous Schrödinger transform of image ()  x based on () v x , the conclusion mentioned above is not true proposition, that is, eq.( 12) can not decompose the following two initial-value problem: 0 0 () and 0 () Hence, there are essential differences betwwen continous Schrödinger transform continous and discrete Schrödinger transform.

Properties and meaning of Schrödinger transform
There are a variety of properties associated with the Schrödinger transform of image.The following are some of the most relevant for I-type Schrödinger transform of image.Energy conservation property also exists for the Schrödinger transform of image like the Fourier transform.Proposition 1. (Energy Conservation Theorem)Let (,)  ut x be Schrödinger transform of image () The proposition can be proved according to equation ( 14) and energy conservation properties of 2D Fourier Transform of image.The energy conservation properties of Schrödinger transform show that energy will diffuse from high energy to low energy while total energy is invariable(Fig.2)., that is, () H y is high pass filter.
High and low pass filter can be obtained using Schrödinger transform of image according to the above two propositions.And Schrödinger transform of image can be applied to image processing and analysis, such as, boundary extraction, edge enhancement, etc.The energy conservation properties of Schrödinger transform show that energy will diffuse from high energy to low energy while total energy is invariable(Fig.2).The following experiments (see Fig. 3 and Fig. 4) show the meaning and function of Schrödinger transformation of image, that is, Schrödinger transformation of image can be seen as the result of primitive image shrinking inside and spreading outward at the center of object, like as interference wave.The bigger at is, the more obvious the interference is.On the other hand, contour curves of object in the transformed image are similar to contour curves of object in the original image, and they are too similar to draw them manually.Hence, I-type Schrödinger transform of image is isotropic.

Scale parameter and potential function of Schrödinger transform
It is important to select the parameter at of Schrödinger transformation while the potential function is selecte.In the section, we'll discuss the effects of scale parameter and potential function for Schrödinger Transform of image.

Schrödinger transform of rectangle image
Fig. 2 shows the parameter at is scale parameter of Schrödinger transformation.To make clear the relation between the parameter at and Schrödinger transformation, without loss of generality, take Schrödinger transform of rectangle image for example.And, the parameter at is denoted by the parameter a .Let a rectangle image be  .We can obtain 88580 samples using Equation (37).The Remark: Schrödinger Transform of image can be directly computed for a little number a by using Equation ( 14) while it needs to use Equation ( 14) repeatedly for a big unmber a , and use a little scale parameter every times, so that interference effect of Schrödinger Transform can be avoided for using a big scale parameter a .We should pay attention to two issues using the above method to detect image edge, enhance image and smooth image:

Potential function of Schrödinger transform
1.The parameter at should be appropriate.If the parameter is too small ,the filtering effect would not be obvious.Contrariwise, Schrödinger transforme will cause interference which effects the filtering.The parameter at should not exceed y .For a given mn  image and constant at , I-type Schrödinger transform of image (,) xy  c a n b e computed as the following steps: 1. Suppose the low frequency component be in the center of image, we construct a mn  distance matrix (the Fourier transform of (,) xy  , ˆ(,) uv  www.intechopen.com4. Compute the Fourier transform of () I-type Schrödinger transform of image (,) xy  is the modulus of the inverse Fourier transform of ˆ()   .
domain of image (see step (1) to step (4) of computing I-type Schrödinger transform of image ); Step 2: Compute II-type Schrödinger transform of image (,) xy  in the spatial domain of image according to Eq.( original image, (b), (c), (d) and (e) are Schrödinger transforms with parameters 0.00001, 0.00005, 0.0005 and 0.001, respectively.

Fig. 2 .
Fig. 2. Schrödinger transforms of a circular disc image with different parameters at Fig. 3. I-type Schrödinger transform of image.(a) The original image, (b), (c) are Schrödinger transforms.The constant at is 0.0005, 0.001, respectively.

Fig. 4 .
Fig. 4. Schrödinger transforms of a irregular closed image with different parameters at

Fig. 5 .Fig. 6 .
Fig. 5.The function 640,120,280 |( ) | gm .a = 0.00001 and 0.0001, respectively It doesn't need to use potential function for I-type Schrödinger Transform of image, which is isotropic.II-type Schrödinger Transform is anisotropic since nonzero potential function is applied.It is necessary that a right potential function is chosed so that we can obtain a perfect deformation processing according to anisotropic property of II-type Schrödinger Transform.However, if the potential function V D is real, from Equation (29)I-type Schrödinger Transform N times for image  .Obviously, that is surely not the result we want.So, the potential function V D of II-type Schrödinger Transform must be imaginary so that / 1 V itD N e   .Meanwhile, if we hope that deformation www.intechopen.com , Schrödinger transform of image is completed by using Schrödinger transform with smaller at repeatly So that the interference can be avoided.2. The origin of coordinates of frequency domain is the center rather than the top left corner of the image when the results of image edge detection by using Schrödinger Transform in Fig.7.The experements show the comparison results of a fan image by several edge detection operators in Fig.8.According to the comparison results of traditional edge detection operators, the high-pass filter designed by Schrödinger transform can better detect image edge and it would not increase noise simultaneously.In fact, filter designed by Schrödinger transform consider both the local and global feature of image, so filtering effect is better.

Fig. 7 .
Fig. 7. Edges detected using Schrödinger Transforms of image The original simulated image, (b) The interior points of objects (white pixels), (c) The exterior points of objects (white pixels), (d) the extracted contours.

Fig. 9 .
Fig. 9.The extracted contours of multiple objects using Schrödinger transform of image.

Fig. 10 .
Fig. 10.The extracted contours of beads using Schrödinger transform of image.

Table 1 .
Table 1 lists some descriptive statistics of sample datas.Pearson correlation coefficents between propagation distance d and a , M ,And standard error of the estimate, which is less than one pixel, is 0.88556.From Fig.6and Equation (38), we know the biggest affect on propagation distance d is the scale parameter a , size of image M takes second place while the propagation distance d and scale parameter a have a strong positive linear correlation, the propagation distance d and size of image M have a weak positive linear correlation,.Descriptive Statistics