JP4669993B2 - Extreme value detection method and extreme value detection program using optimal smoothing spline - Google Patents

Description

  The present invention relates to an extremum detection method and an extremum detection program for discrete data points and functions, and more particularly to an extremum detection method for detecting all extremums by designing an optimal smoothing spline for discrete data points and functions. The present invention relates to an extreme value detection program, an extreme value detection method thereof, and an edge detection method and an edge detection program of a digital image using the extreme value detection program.

  The problem of extreme value detection concerning discrete data points and functions appears in various fields including engineering, and is known as an important problem in the field of so-called “mathematical programming” or “optimization method”. As a method for detecting extreme values related to discrete data points and functions, a numerical extreme value search method represented by a hill-climbing method (hill-descent method) is generally used, and is therefore a local method.

  However, the extreme value search by such a local method is easily influenced by local noise of discrete data points, and it is difficult to detect the extreme value accurately and stably. Therefore, there is a need for an extreme value detection method that can be globally evaluated for discrete data points.

As an extreme value detection method capable of globally evaluating discrete data points, for example, there is a method described in Patent Document 1. According to this method, a method of globally sampling a set of data points, generating a single polyline based on the set of data points, and obtaining an extreme value of the polyline is disclosed.
JP 2005-222554 A

  In the method described in Patent Document 1, curvature (second derivative) is used to select “specific points” and “dividing points” used when creating a curve. In particular, the “specific point” is selected as a point where the curvature takes an extreme value (minimum value, maximum value) or becomes zero. On the other hand, the “division point” is selected based on the integral value of the curvature.

  However, the calculation method for the extreme value of the curvature and the point at which the curvature becomes zero is not specifically shown. It is estimated that the curvature value is calculated at several points, and the maximum, minimum, and zero points are selected from the obtained values, but this is not certain.

  In addition, the method described in Patent Document 1 is intended only for a user who does not have a high level of skill in curve design and assists in drawing and correcting a curve desired by the user. It is not possible to evaluate globally.

  The present invention solves such a conventional problem relating to extreme values, and by designing an optimal smoothing spline for discrete data points and functions, the extreme value for the optimal smoothing spline can be detected accurately and stably. Is to provide a method. In addition, since all the extreme values are obtained, a method for obtaining the maximum value and the minimum value is to be provided.

The present invention is executed by a processing apparatus comprising input means for inputting data, storage means for storing a program, arithmetic means for executing a program and processing data, and output means for outputting a processing result of the arithmetic means. An extreme value detection method, wherein the calculation means generates an optimal smoothing spline curve having a plurality of nodes from a discrete data group or function in a plane input from the input means, and the calculation means includes the optimum by each determining an extreme value for each of the nodes section is smoothed spline curve or on a section which is divided by the nodes adjacent on its derivatives, the optimum smoothing spline curve or extremes of its derivatives And an extreme value detecting method using an optimal smoothing spline curve, characterized in that the method includes a step of obtaining all and outputting the extreme value to an output means.

Further, the present invention provides a processing apparatus comprising input means for inputting data, storage means for storing a program, arithmetic means for executing a program and processing data, and output means for outputting a processing result of the arithmetic means. In the extreme value detection method executed in step (b), the calculation unit generates an optimal smoothing spline surface having a plurality of nodes from a discrete data group or function in a space input from the input unit, and the calculation unit performs optimal smoothing. by each determining an extreme value for each of the node regions of one region partitioned on a grid by the nodes on the spline curved surface on or partial derivative, optimal smoothing spline surface or electrode of the partial derivatives An extreme value detection method using an optimal smoothing spline curved surface, comprising: calculating all of the values and outputting the extreme values to the output means. .

The present invention also includes an input means for inputting pixel data, a storage means for storing a program, an arithmetic means for executing the program and processing the pixel data, and an output means for outputting a processing result of the arithmetic means. An edge detection method for a digital image executed by an image processing apparatus, wherein an arithmetic means generates an optimal smoothed spline curved surface having a plurality of nodes from pixel data including pixel coordinates and luminance values input from an input means. And a step of generating a first curve group comprising a plurality of curves obtained by fixing the optimal smoothing spline curved surface with a plurality of coordinates on the first coordinate axis, and a plurality of components constituting the first curve group by each determining an extreme value for each of the nodes a certain section in a section which is divided by the nodes adjacent on the first derivative of the curve, the first group of curves Determining all of the extreme values of the first derivative, and generating a second curve group comprising a plurality of curves in which the computing means fixes the optimal smoothed spline curved surface at a plurality of coordinates on the second coordinate axis; By obtaining an extreme value for each node section , which is one section divided by adjacent nodes on the first derivative of a plurality of curves constituting two curve groups, 1 of the second curve group is obtained. The step of obtaining all of the extreme values of the second derivative, and the computing means is configured to determine whether the extreme value of the first derivative of the first curve group or the absolute value of the extreme value of the first derivative of the second curve group is predetermined. A digital image edge detection method comprising: determining a portion greater than the threshold value as an edge portion of the digital image and outputting the determination result to an output means.

Further, the present invention provides a processing apparatus comprising input means for inputting data, storage means for storing a program, arithmetic means for executing a program and processing data, and output means for outputting a processing result of the arithmetic means. to, from the discrete data group or function in the a plane input from the input means, calculation means and generating an optimal smoothing spline having a plurality of nodes are computing means optimal smoothing spline curve or on its derivatives The extreme values are obtained for each of the node sections , which are one section divided by the adjacent nodes of each other , to obtain all of the extreme values of the optimal smoothed spline curve or its derivative, and the extreme values are output to the output means. An extreme value detection program using an optimal smoothing spline curve for executing an output step.

Further, the present invention provides a processing apparatus comprising input means for inputting data, storage means for storing a program, arithmetic means for executing a program and processing data, and output means for outputting a processing result of the arithmetic means. to, from the discrete data group or function in inputted space from the input means, step a, calculating means on optimal smoothing spline surface or its partial derivatives to generate optimal smoothing spline curved surface calculating means comprises a plurality of nodes by obtaining the respective extremum for each of the node regions of one region partitioned on a grid by the nodes of the upper, seeking all the optimal smoothing spline surface or extremes of its partial derivatives, extrema An extreme value detection program using an optimal smoothing spline curved surface for executing the step of outputting to the output means.

The present invention also includes an input means for inputting pixel data, a storage means for storing a program, an arithmetic means for executing the program and processing the pixel data, and an output means for outputting a processing result of the arithmetic means. In the image processing apparatus, the calculation means generates an optimal smoothing spline surface having a plurality of nodes from pixel data composed of the pixel coordinates and luminance values input from the input means; Generate a first curve group composed of a plurality of curves fixed at a plurality of coordinates on the first coordinate axis, and classify by adjacent nodes on the first derivative of the plurality of curves constituting the first curve group by each determining an extreme value for each of the nodes section it is has been a section, a step of obtaining all the extrema of the first derivative of the first group of curves, calculation Stage generates a second curve group composed of a plurality of curves and fixed by a plurality of coordinates on the second coordinate axis optimal smoothing spline surface, the first derivative of a plurality of curves that constitute the second group of curves by each determining an extreme value for each of the nodes section which is adjacent a section which is divided by the nodal point of the upper, and determining all the extrema of the first derivative of the second group of curves, arithmetic means Where the absolute value of the first derivative of the first curve group or the absolute value of the first derivative of the second curve group is greater than a predetermined threshold is determined as the edge portion of the digital image. , A digital image edge detection program for executing the step of outputting the determination result to the output means.

  According to the extreme value detection method using the optimum smoothing spline of the present invention, since the designed spline is targeted, the extreme values (minimum value, maximum value) of the polynomial defined for each node may be obtained. As a result, all extreme values are obtained, and the minimum value and the maximum value can be obtained.

(First embodiment)
Hereinafter, an extreme value detection method and an extreme value detection program using an optimal smoothed spline curve according to a first embodiment of the present invention will be described in detail with reference to the drawings.

  As is well known, spline functions are used in various engineering fields, and extensive research has been conducted so far. In particular, the theory of “dynamic splines” based on optimal control theory has been proposed, and such an approach is used for analysis of B-spline functions and design of optimal smoothing curves. In addition, an optimal interpolation and smoothing spline framework is used to design and redesign characters and strings as found in Japanese calligraphy.

  In this embodiment, first, an optimal smoothing spline curve is designed for a given set of discrete data. In designing such a curve, a normalized uniform B-spline function is used as a basis function. First, the basic results for obtaining a simple calculation procedure for the design of the optimal smoothing curve are shown. Next, an algorithm for calculating all the extreme values, that is, the minimum value and the maximum value of the optimum curve will be described. In so doing, the fact that splines are piecewise polynomials and are continuous and differentiable is utilized. Since the spline curve is a piecewise polynomial and can be represented by control points, the algorithm for detecting and calculating its extreme values is simple and easy to use for numerical calculations.

Let B _k (·) be a k-order normalized uniform B-spline function with integer nodes, and generate a spline curve x (t) as the sum of B-spline functions shifted and weighted at regular intervals. To do. That is, the k-th order spline curve x (t) is

It is expressed as

Here, m is an integer and is a design parameter that can be determined in advance. τ _i ∈R is a weighting factor called a control point. α (> 0) is a design parameter that can be determined in advance, and is a constant for adjusting the interval (node interval) of nodes t _i arranged at equal intervals according to t _{i + 1} −t _i = 1 / α. is there.

FIG. 1 shows a state in which the spline curve x (t) is formed by Equation 1. Each bell-shaped waveform represents a function weighted by control points τ _i (indicated by “x” marks) corresponding to appropriately shifted k-th order B-splines B _k (•), and the spline curve x (t) is All of these are added together. By appropriately selecting the control point τ _i ∈R, an arbitrary k-th order spline curve x (t) can be designed in the interval [t ₀ , t _m ].

The k-order B-spline function B _k (t) is given by

Defined in B _k (t) has integer-valued nodes and is specified by basis functions N _{j, k} (t−j) in each node interval [j, j + 1] for j = 0,. In the interval other than k + 1], it is uniformly 0.

Where the basis functions N _{j, k} (t) are

Can be generated sequentially from the initial function N _0,0 (t) = 1.

In practice, when k = 3, that is, when a cubic spline function is selected, the basis function N _{j, 3} (t) is

Thus, four functions (that is, k + 1) are included.

  This means that the value of the kth-order spline function at each t is only affected by k + 1 basis functions (and thus corresponding k + 1 control points) in the interval containing t, and is outside that range. Is not affected by the basis function (and hence the corresponding control point).

(Optimization process)
Next, the set of discrete data D is

And are given by, Also, the control point vector defined Tau∈R ^M a (M = m + k) as follows.

  Then, the basic problem of designing an optimal smoothing spline curve is the evaluation function

A spline curve x (t) or equivalently find control point vector Tau∈R ^M problem to minimize. However, λ> 0 and w _i ε (0,1] ∀i.

The optimal solution τ for this problem is obtained as the solution of the following linear equation.

Where Q∈R ^{M × M} is

And can be calculated in advance regardless of the given discrete data D.

And W∈R ^{N × N} , d∈R ^N and B∈R ^{M × N} are

It is.

B (t) ∈R ^M is in equation 9, 12 However,

Is a vector.

Equation 8 is solvable, so there is always a solution τ. Of course, only when λQ + BWB ^T > 0. By using this solution τ in Equation 1, a k-th order optimal smoothing spline x (t) in the interval [t ₀ , t _m ] is designed.

  In this way, by obtaining an optimization solution related to the control point vector τ and using Equation 1, the k-th order optimal smoothing spline curve x (t) is generated.

(Extreme value of spline curve)
Next, a method for detecting the extreme value of the optimal smoothing spline curve x (t) will be described. Specifically, find the point t∈ (t ₀ , t _m ) that satisfies x ⁽¹⁾ (t) = 0 and x ⁽²⁾ (t)> 0 or x ⁽²⁾ (t) <0. It ’s fine.

The k-th order optimal smoothing spline x (t) designed in the interval [t ₀ , t _m ] is represented by each interval [t _j , t _{j + 1} ) (k = 0, 1, 2,..., M− Each is a k-th order polynomial in 1). Therefore, by detecting extreme values for each section, all extreme values including the maximum value and the minimum value of x (t) can be obtained.

x (t) is in the interval [t _j , t _{j + 1} )

Next, k + 1 control points _{τ jk, τ jk + 1,} ···, a k-th order polynomial, which is determined by τ _j.

  Variable conversion from t to δ is performed as follows.

At this time, the interval [t _j , t _{j + 1} ) of the variable t is normalized to the unit interval [0, 1) for the variable δ, and the function x (t) is a function of δ.

Is converted to x (δ) is a kth order polynomial for δ,

And

  In addition, regarding the mathematical formulas used below, all the functions x having δ as a variable have “hat (^)”, but they are omitted in the specification text.

Obviously x (t) = x (δ), and x ⁽¹⁾ (t) = αx ⁽¹⁾ (δ), x ⁽²⁾ (t) = α ² x ⁽²⁾ (δ) and α> 0 Thus, the extreme value in the interval [t _j , t _{j + 1} ) of x (t) can be obtained as the extreme value in the unit interval [0, 1) of x (δ). If x (δ) is δ = δ _i (0 ≦ δ _i <1) and has an extreme value x (δ _i ), x (t) will be an extreme value x (t at t = (1 / α) δ _i + t _j ) = X (δ _i ).

The method of obtaining the extreme value in the unit interval [0, 1) of x (δ) is as described in Step 103 to Step 105 in FIG. At that time, the root of the first derivative x ⁽¹⁾ (δ) can be calculated using an existing polynomial root calculation program.

(Cubic spline)
In the case of cubic splines that are often used in practice (ie, k = 3), the above steps can be written as a more detailed algorithm by making the following assumption 1. This is because x ⁽¹⁾ (δ) is a quadratic polynomial and its root can be expressed explicitly.

Here, the following assumptions are introduced for practical purposes.

(Assumption 1) The number of extreme values of x (t) in each node interval [t _j , t _{j + 1} ) is at most one. However, j = 0, 1, 2, ..., m−1

Note that x (t) is an optimal smoothing spline curve, assumption 1 is valid if the node interval [t _j , t _{j + 1} ) is small.

In the case of a cubic spline (ie k = 3), the polynomial x (δ) depends only on the four weighting factors τ _j−3 , τ _j−2 , τ _j−1 , τ _j and the domain of δ [0, 1) can be expressed as follows.

That is, the coefficients p _j , q _j , r _j , s _j are τ _j−3 , τ _j−2 , τ _j−1 among the weighting coefficients (vectors) τ already obtained in the optimization process. , Τ _j is uniquely determined.

As described above, if it is found that x (δ) has an extreme value when δ = δ _e ∈ [0, 1), then x (t) becomes t _e = t _j + δ _e (1 / α ) ∈ [t _j , t _{j + 1} ) and has an extreme value, and the value is obtained by x (t _e ) = x (δ _e ).

In order to obtain [delta] _e, the first derivative of x ([delta]) of Equation 18, i.e.
Plays an important role. If p _j ≠ 0, the two roots of x ⁽¹⁾ (δ) are

It can be expressed as At that ^{time, x (1) (0)} = r j / 2 and x ^{(1) (1)} = Since the r _{j + 1/2,} whether extreme exists for r _j and r _{j + 1} It can be examined by the sign. Specifically, the following three cases are obtained by geometric observation under Assumption 1.

(P1) If r _j · r _{j + 1} <0, the extremum exists at [0, 1).

(P2) If r _j · r _{j + 1} = 0, the extremum exists at [0, 1) under some additional conditions.

(P3) If r _j · r _{j + 1} > 0, there is no extrema in [0, 1).

  The following results are obtained for the cases of (P1) and (P2).

(Proposition 1) Assume that r _j · r _{j + 1} <0. Then, if r _j > 0 and r _{j + 1} <0 (r _j <0 and r _{j + 1} > 0), the function x (δ) is maximal (minimum) with δ = δ _e ∈ (0,1) ) However,

(Proposition 2) When r _j · r _{j + 1} = 0, the following two cases occur.

(i) When r _j = 0: If q _j <0 (q _j > 0), then x (δ) has a maximum (minimum) at δ = 0. If q _j = 0, there is no extreme value.

(ii) If r _j ≠ 0 and r _{j + 1} = 0: If r _j > 0 and q _{j + 1} > 0 (r _j <0 and q _{j + 1} <0) then x (δ ) Has a maximum (minimum) at δ = r _j / p _j ∈ (0, 1). If r _j · q _{j + 1} ≦ 0, there is no extreme value.

  Therefore, the algorithm for detecting and calculating the extreme value of the cubic optimal smoothing spline function x (t) can be summarized as follows.

Weight coefficients τ ₋₃ , τ ₋₂ , τ ₋₁ ,..., Τ _m−1 are given, and r _j and r _{j + 1} are set to j = 0, 1, 2,. If r _j · r _{j + 1} <0 and r _j · r _{j + 1} = 0, then propositions 1 and 2 are used, respectively. [delta] = [delta] if found extreme x ([delta]) at _e, calculates an extreme value x (t _e) at _{t e = t j + (1} / α) δ e.

  FIG. 2 is a block diagram showing a processing device used in the extreme value detection method and extreme value detection program based on the kth-order optimal smoothing spline curve of this embodiment. The processing device 1 used in the extreme value detection method using the optimal smoothing spline of this embodiment has a CPU 5 (calculation means) and a memory device 4 (RAM, ROM, etc .: storage means), and is stored in this memory device 4. 3 and 4, and the operations described in the flowcharts of FIGS. 3 and 4 are executed, as shown in the block diagram of FIG. 2, the CPU 5, the memory device 4, the mouse, keyboard, scanner, data logger, etc. A processing apparatus comprising a device 6 (input means), a bus 3, a display device 7 (output means), a printer device 8 (output means), and a network device 9 (input means / output means) connected to an external network 10. Function as.

  FIG. 3 is a flowchart showing a processing flow of an extreme value detection method using a general kth-order optimal smoothing spline according to this embodiment.

Step 101 (step is abbreviated as S in the figure) is a step of obtaining and inputting the weighting coefficient τ _i for the discrete data D. That is, the CPU 5 sets the optimum weighting factors τ _i , i = −k, −k + 1,... For the input discrete data D (u _i , d _i ), i = 1,. Determine m−1 and design an optimal smoothing spline curve x (t).

In step 102, the CPU 5 normalizes the optimal smoothing spline curve x (t) in the node interval [t _j , t _{j + 1} ), and obtains the k-order polynomial x (in the unit interval [0, 1). δ) is generated. That is, each coefficient a ₀ ,..., A _k of x (δ) is calculated.

x (δ) is defined by Equation 16. Here, the basis functions (polynomials) N _{0, k} (δ), N _{1, k} (δ),..., N _{h, k} (δ) are obtained in advance from a sequential expression of Equation 3 using a computer. Both are k-order polynomials. Now

And then, when Equation 17 the polynomial representation of x ([delta]) to obtain, the coefficients a _0, ···, a _k is determined from the following equation according to Equation 17, Equation 22.

In step 103, the CPU 5 obtains the root of the first derivative x ⁽¹⁾ (δ) of the polynomial x (δ) obtained in step 102.

  In step 104, the CPU 5 determines whether or not the polynomial x (δ) has a real root in the unit interval [0, 1). If it has a real root, the process proceeds to step 105, and if it does not have a real root, the process proceeds to step 106.

In step 105, CPU 5 is the x ([delta]) 2 derivative x ⁽²⁾ of ([delta]), by substituting the real roots [delta] _e of the polynomial determined in step ^{103 x (δ), x (} 2) (δ e )> 0, t _e = t _j + (1 / α) δ _e and a minimum value x (δ _e ), and if x ⁽²⁾ (δ _e ) <0, t Let _e = t _j + (1 / α) δ _e and have a maximum value x (δ _e ). As a result, the maximum value or the minimum value in the predetermined node interval [t _j , t _{j + 1} ) of the optimum smoothing spline function x (t) is obtained.

In step 106, when j is smaller than m−1, the process returns to step 102, and when j is m−1, the process proceeds to step 107, and all the node intervals [t t [t ₀ , t _m ] Steps 102 to 105 are repeated until the extreme value detection procedure for _j , t _{j + 1} ) is completed.

In step 107, the CPU 5 discriminates the magnitude of all the extreme values from all the extreme values existing in the entire interval [t ₀ , t _m ] of the optimum smoothing spline function x (t), thereby performing the optimal smoothing. Find the maximum and minimum values of the spline function x (t). Then, the maximum value, the minimum value, the maximum value, and the minimum value of the optimal smoothing spline function x (t) are output to the display device 7 and the printer device 8.

  FIG. 4 is a flowchart showing a case where a cubic spline (k = 3) is used in the processing flow of the extreme value detection method using the optimal smoothing spline of this embodiment. Since only the steps 102 to 104 are different from the flowchart of FIG. 3, the corresponding portions are described as steps 102a and 103a.

Step 102a corresponds to step 102 in the flowchart of FIG. In step 102a, the CPU 5 calculates coefficients p _j , q _j , r _j , and s _j for x (δ) that is a cubic polynomial. As described above, each coefficient p _j , q _j , r _j , s _j is uniquely determined from four of τ _j−3 , τ _j−2 , τ _j−1 and τ _j .

Step 103a corresponds to steps 103 and 104 in the flowchart of FIG. In step 103a, the CPU 5 obtains the root of the first derivative x ⁽¹⁾ (δ) of the polynomial x (δ) obtained in step 102a. When q _j ² −p _j r _j ≧ 0, the process proceeds to step 105 in FIG. 3, and when q _j ² −p _j r _j <0, the process proceeds to step 106 in FIG.

In the present embodiment, the extreme value is obtained when the discrete data D is given, but the same applies when the function f (t), t ₀ ≦ t ≦ t _m is given as input data. It is possible to obtain extreme values by the method. Also in this case, first, an optimal spline that approximates f (t) is designed as follows.

  When the function f (t) is given, the evaluation function is

Is used. This optimal solution τ is a linear equation

Is obtained as a solution of Here, Q is a matrix defined by Equation 9, and Q ₀ is

Any of these matrices can be calculated in advance regardless of f (t). The right hand side can be calculated using an appropriate numerical integration method, given f (t).

Once the optimal vector τ is determined, the subsequent handling such as detection of extreme values is the same as in the case of discrete data. However, it is effective as follows by using it together with the existing optimization algorithm. In general, when a function f (t) is given, a method of finding a maximum value and a minimum value in a certain interval (which can be considered as [t ₀ , t _m ] without losing generality) is a numerical search method by iteration. It is normal.

  In that case, how to take the initial value of the iterative formula becomes a problem, and in the case of a multimodal function, the extreme values reached are different. There is no guarantee that maximum or minimum values can be obtained. According to this method, an optimal spline that approximates f (t) is obtained, and the maximum value and the minimum value thereof are obtained. Further, if a numerical search method is applied with the corresponding value of t as an initial value, the maximum value and the minimum value of f (t) can be obtained more accurately.

(Numerical experiment example)
FIG. 5 is a diagram showing an optimal smoothing spline curve and its extreme value according to the first embodiment of the present invention. FIG. 5 is a diagram showing the performance of extracting extreme values from the optimal smoothing spline curve x (t) (function indicated by the solid line in the figure). FIG. 5 shows the results when k = 3, α = 1, t ₀ = 1 and t _m = m = 50. However, the triangle and the inverted triangle indicate the detected maximum value and minimum value, respectively. Here, data d _i (marked with * in the figure) is a function f (t) = e ^{a (t−1)} cos (b (t−1)) + 1 (function indicated by a dotted line in the figure) Is generated by sampling. However, a = 1 / (m−1) log1 / 4, b = 2π × 3 / (m−1). The number of data is N = 30, u _i are data points randomly arranged in the interval [t ₀ , t _m ] = (1, 50), and the magnitude of noise added to the data is σ = 0.02. To do. Furthermore, a so-called cross-validation method was employed to estimate the smoothing parameter λ, and the optimum value was obtained as λ ^★ = 0.1585.

  As can be seen from the figure, it can be seen that all the extreme values of the original function f (t) are detected with high accuracy even though the data noise is considerably large. In particular, it can be seen that the value of t giving an extreme value is obtained almost accurately.

In the above embodiment, the extreme value of the optimal smoothing spline function x (t) is obtained, but the extreme value of the i-th derivative x ⁽ⁱ⁾ (t) of x (t) is obtained by the same method. It is possible.

The i-th derivative x ⁽ⁱ⁾ (t) (i = 1, 2,..., K) of the optimal spline x (t) designed in the interval [t ₁ , t _m ] is also applied to each node interval [t _j , T _{j + 1} ) (k = 0, 1,..., M−1). That is, in the node interval [t _j , t _{j + 1} ), the relationship of x ⁽ⁱ⁾ (t) = α ⁱ x ⁽ⁱ⁾ (δ) is established by the variable transformation of Equation 14 and Equation 15. Since α> 0, the extreme value in the node interval [t _j , t _{j + 1} ) of x ⁽ⁱ⁾ (t) is the extreme value in the unit interval [0, 1) of x ⁽ⁱ⁾ (δ). This i th derivative x ⁽ⁱ⁾ (δ) is obtained by using the results of Equations 17 and 23.

Can be calculated by

The extreme value of x ⁽ⁱ⁾ (δ) (where i ≦ k−2) is detected by replacing the polynomial x (δ) in step 102 in FIG. 3 with x ⁽ⁱ⁾ (δ), and accordingly, x ^{(1 )} (δ), x ⁽²⁾ (δ) may be replaced with x ^{(i + i)} (δ) and x ^{(i + 2)} (δ), respectively, and a similar method may be applied after step 103. Here, i ≦ k−2 is limited because an i + second derivative that is a k− (i + 2) order polynomial is required, and therefore k− (i + 2) ≧ 0 is satisfied. This is necessary.

  As described above, according to the extreme value detection method using the optimal smoothing spline of the present embodiment, since the designed spline is targeted, the extreme value (minimum value, maximum value) of the polynomial defined for each node interval is used. Value). As a result, all extreme values are obtained, and the minimum value and the maximum value can be obtained.

  In addition, since the B-spline function is used as the basis function, the expression of the designed spline is simple and the derivative thereof can be easily obtained.

  The target of extreme value calculation may be discrete data or a function. In any case, an optimal smoothing spline can be designed, and at that time, it can be approximated with sufficient accuracy by adjusting the node spacing and the order of the B-spline, and it is a numerically very stable method that is not easily affected by noise. It can be said. Of course, this extreme value detection method is not limited to the optimal smoothed spline curve, but can be applied to any spline curve expressed by Equation 1.

  In particular, when a cubic B-spline function is used, a cubic polynomial is obtained for each section, and the extreme value can be obtained analytically.

(Second Embodiment)
Hereinafter, an extreme value detection method and an extreme value detection program using an optimal smoothing spline curved surface according to a second embodiment of the present invention will be described in detail with reference to the drawings.

(Optimal smoothing spline surface)
In the first embodiment of the present invention, an optimal smoothing spline curve is generated for discrete data in a plane and extreme values are detected. However, in this embodiment, an optimal smoothing spline curved surface that approximates discrete data in space is generated. The extreme value is detected. In addition, the present embodiment provides a method in which the method for detecting the extreme value of the optimal smoothing spline curve according to the first embodiment of the present invention is straightforwardly extended to the optimal smoothing spline curved surface.

  FIG. 6 is a diagram schematically showing the optimal smoothing spline curved surface of the present embodiment. FIG. 6A is a diagram showing a distribution of optimal control points (weighting factors) obtained from discrete data in space, and FIG. 6B is an optimal smoothed spline curved surface x (s) generated based on the discrete data in space. , T).

(Curved surface generation)
The optimal smoothing spline curved surface x (s, t) of the present embodiment can be expressed by the following formula 28, which is a natural extension of formula 1, using a k-th order B spline function B _k (•). .

Here, α and β (> 0) are constants, m ₁ and m ₂ (> 2) are integers, and s _i and t _j represent nodes arranged at equal intervals as follows.

At this time, by appropriately selecting the control points τ _{i, j} , an arbitrary k-th order spline curved surface can be represented in the region (s, t).

Can be designed above. These control points are collectively expressed as a matrix τ∈RM ₁ ^{× M 2} (where M ₁ = m ₁ + k, M ₂ = m ₂ + k).

  Now, discrete data in space

Suppose that At this time, the optimal smoothing spline curved surface is designed so that the following evaluation function J (τ) is minimized.

  The solution to this problem, ie the optimal control point, is obtained as the solution of the following linear equation.

The matrixes and vectors that appear in Equation 34 are shown below. First, Q is an M ₁ M ₂ × M ₁ M ₂ matrix given by the following equation.

Where Q _l ^(ij) ∈ R ^{M1 × M2} is

Defined by However, the vector ^{_{b 1 (t) ∈R M1,}} b 2 (t) ∈R M2 is composed of B-spline function.

  Each matrix shown in Equation 36, and thus Q, can be calculated in advance from the B-spline independently of the given data.

The matrix W∈RN ^{× N} is a diagonal matrix composed of the weights in the evaluation function shown in Equation 33.

Matrix B∈R ^{M1M2 × N,} and vector D∈R ^N are defined as follows by the given data.

  As described above, each matrix and vector appearing in Equation 34 is determined. By solving this equation, the optimum τ, and hence the optimum control point matrix τ, is obtained, and the optimum smoothed curved surface x (s, t) can be constructed by Equation 28.

(Extreme value of curved surface)
Next, a method for obtaining the extreme value of the optimum smoothed curved surface x (s, t) obtained by the above method will be described.

(Target curved surface)
The extreme value of the k-th order spline curved surface of Expression 28 is obtained. Here, it is assumed that the parameters k, α, β, m ₁ and m ₂ have been set, and the control points τ _{i, j} have already been given by the above-described method or the like. Note that the domain of (s, t) is S = [s ₀ , sm ₁ ] × [t ₀ , t _m2 ] ⊂R ² .

(Extreme value)
The extreme value of the surface x (s, t) is

Is obtained as a point (s, t) ∈S that satisfies

The minimum value is given when the Hessian of the surface x (s, t) is positive definite, and the maximum is given when the Hessian of the surface x (s, t) is negative definite. The problem of finding the extrema in the region S of x (s, t) is the extrema in the subregion S _{κ, μ} = [s _κ , s _{κ + 1} ) × [t _κ , t _{κ + 1} ) All regions of the problem of obtaining _{κ = 0,1, ···, m 1} -1; μ = 0,1, ···, may be repeated over m ₂ -1.

(Normalization of partial area)
_{Focusing on the} region S _{κ, μ} and performing variable transformation such that u = α (s−s _κ ), v = β (t−t _μ ), both of the domains of u and v are [0, 1) and Normalized. That is, the region S _{κ, μ} of (s, t) becomes the unit region ε = [0, 1) × [0, 1) of (u, v) regardless of κ, μ. At this time, the function x (s, t) in S _{κ, μ} is

It can be expressed as. In addition, regarding the mathematical expressions used below, all the functions x having u and v as variables have “hat (^)”, but they are omitted in the specification text.

  Furthermore, the relationship between the partial derivative of x (s, t) and the partial derivative of x (u, v) is

It becomes. Since α and β> 0, the extreme value on the region S _{κ, μ} of x (s, t) coincides with the extreme value on the region ε of x (u, v).

(Extreme value in partial area)
The gradient vector of the curved surface x (s, t) shown in Equation 40 is 0 and the Hessian of the curved surface x (s, t) is a positive definite value, or the Hessian of the curved surface x (s, t) is A point (u ^* , v ^* ) ∈ε that becomes a negative definite value may be detected. At this time, x (s, t) is an extreme value x (s ^* , t ^* ) at the point (s ^* , t ^* ) = ((1 / α) u ^* + _sκ , (1 / β) v ^* + _tμ ). t ^* ) = x (u ^* , v ^* ).

(How to find the extreme value of x (u, v))
(Expression of function x (u, v))
x (u, v) is a k-order bivariate polynomial for u and v, respectively, and can be simply expressed as follows.

  The (k + 1) × (k + 1) matrix A is

Defined by S, _{Tκ, and μ} are (k + 1) × (k + 1) matrices obtained as follows.

The matrix S is a matrix in which each row is composed of coefficients of B-spline basis functions N _{i, k} (t), i = 0, 1,..., K. That is,

Is a matrix S such that N _k (t) = Sh _{k + 1} (t). For example, when k = 3, S in Equation 4 is as follows.

The matrix T _{κ, μ} is composed of (k + 1) ^two control points appearing in the definition formula of the curved surface x (u, v) as follows.

That is, it is obtained as a block of (k + 1) × (k + 1) in the upper left from the elements T _{κ and μ} in the entire control point matrix τ (formula 31) constituting the curved surface x (s, t). It is a submatrix of τ.

(Gradient vector and Hessian)
The gradient vector and Hessian of x (u, v) are expressed as follows:

C _k is a (k + 1) × k matrix defined as follows.

(Candidate points for extreme values)
A point (u, v) ∈ ε at which the gradient vector of the curved surface x (u, v) becomes 0 is obtained. Each element of the gradient vector of the curved surface x (u, v) is given as follows.

  However,

Is defined by the following equation.

In order for x _u (u, v) = 0 and x _v (u, v) = 0 to hold simultaneously, the final expression when these are regarded as polynomials concerning u needs to be zero. That is,

  However, R (v) is the following (2k−1) × (2k−1) matrix.

| R (v) | is a polynomial related to v. The root of this polynomial is obtained, and the root satisfying 0 ≦ v <1 is defined as v = v ^* . When there is no such root, x (u, v) has no extreme value in the unit region ε. The v u corresponding to ^* is obtained as the solution of simultaneous equations relating ^{_{u x u (u, v *}} ) = 0, x v (u, v *) = 0. Specifically, two polynomials obtained as q _i (bar) = q _i (v ^* ) and r _i (bar) = r _i (v ^* )

Find the common factor f ₀ (u). At that time, for example, the Euclidean mutual assistance method can be applied. Note that f ₁ (u) and f ₂ (u) are guaranteed to have a common factor, and of course, f ₀ (u) is a polynomial of degree 1 or higher with respect to u. When the root u of f ₀ (u) satisfies 0 ≦ u <1, u = u ^* . When not observed, x (u, v) has no extreme value in the region ε.

(Determination of extreme values)
The obtained point (u ^* , v ^* ) ∈ε is a candidate for the extreme value of the function x (u, v), and the Hessian matrix for this point is a positive definite (local minimum) or negative definite (local maximum) pole It may be determined whether or not the value condition is satisfied. When calculating the Hessian matrix, a polynomial defined as follows:

It is preferable to use the following equation obtained by introducing q and then using q (v) and r (v).

  FIG. 7 is a block diagram showing a processing device used in the extreme value detection method and the extreme value detection program using the optimal smoothing spline of this embodiment. The extreme value detection processing apparatus 11 of the present embodiment includes a CPU 15 (calculation means) and a memory device 14 (RAM, ROM, etc .: storage means), executes a program stored in the memory device 14, As shown in the block diagram of FIG. 7, by executing the operation described in the flowchart of FIG. Processing provided with device 16 (input means), bus 13, display device 17 (output means), printer device 18 (output means), and network device 19 (input means / output means) connected to external network 20 Functions as a device.

  FIG. 8 is a flowchart showing an outline of the processing of the extreme value detection method and the extreme value detection program by the optimum smoothing spline of this embodiment. 9 is a flowchart showing details of the process of step 203 in the flowchart of FIG. 8, FIG. 10 is a flowchart showing details of the process of step 206 in the flowchart of FIG. 8, and FIG. 11 is a flowchart of FIG. It is a flowchart which shows the detail of a process of step 211 of.

In step 201, the parameters κ, α, β, m ₁ , m ₂ and the obtained optimum control points τ _{i, j} , i = −k, −k used for the optimum design of the curved surface in the extreme value detection processing device 11. _{+1, ···, m 1 -1,} j = -k, -k + 1, ···, inputs a m ₂ -1.

In step 202, the CPU 15 prepares a (k + 1) × (k + 1) matrix S, a (k + 1) × k matrix C _k and a k × (k−1) matrix C _k−1 .

Hereinafter, for all combinations of (κ, μ), κ = 1, 2,..., M ₁ −1, μ = 1, ₂ ,. Detection and calculation of the extreme value of x (s, t) in the region S _{κ, μ} .

The CPU 15 calculates a (k + 1) × (k + 1) matrix A (step 203a), a polynomial q _i (v), i = 1, 2,..., K, a polynomial r _i (v), i = 1 , 2,..., K + 1 (step 203b) and a termination expression | R (v) | are obtained (step 203c), and the root of the termination expression which is a polynomial of v is obtained (step 203). Formula calculation can be used for the calculation of the termination formula | R (v) |. If there is no root v satisfying 0 ≦ v <1, the process proceeds to the next set of (κ, μ) (step 204).

The CPU 15 sets the obtained v (0 ≦ v <1) as v ^* (step 205), and further, q _i (bar) = q _i (v ^* ), i = 1, 2,. , And r _i (bar) = r _i (v ^* ), i = 1, 2,..., K + 1. By defining the polynomials f ₁ (u) and f ₂ (u) of u with coefficients q _i (bar) and r _i (bar) respectively (step 206a), for example, by applying the Euclidean mutual assistant method, f A common factor f ₀ (u) between ₁ (u) and f ₂ (u) is obtained (step 206b). When the root u of f ₀ (u) (step 206c) satisfies 0 ≦ u <1 (step 207), u = u ^* is set (step 208). If there is no such root u, the process proceeds to the next set of (κ, μ).

The CPU 15 calculates the Hessian matrix at the point (u ^* , v ^* ) (step 209), and if the Hessian matrix at the point (u ^* , v ^* ) becomes positive definite, the minimum (u ^* , v ^* ) When the Hessian matrix at is negative, it is determined that it is a maximum (step 210), and the corresponding extreme value x (u ^* , v ^* ) is calculated (step 211a). In cases other than these two cases, the process proceeds to the next set of (κ, μ).

The CPU 15 determines the point (s ^* , t ^* ) having the extreme value of x (s, t) as (s ^* , t ^* ) = ((1 / α) u ^* + _sκ , (1 / β) v. ^* + T _μ ) (step 211b) and output together with the value x (s ^* , t ^* ) = x (u ^* , v ^* ) (step 211c).

(Design of optimal surface for bivariate function)
In this embodiment, the extreme value is obtained when the discrete data D is given. However, the extreme value is obtained by the same method when the function f (s, t) is given as input data. It is possible. The design method of the kth-order optimal spline curved surface x (s, t) when the function f (s, t) is given is as follows. The evaluation function uses the following equation instead of Equation 33.

  Τ (= vecτ) that minimizes the evaluation function is a linear equation instead of the equation 34.

Can be solved. The vector on the right side can be obtained using numerical integration. The matrices Q, Q ₁ ⁽⁰⁰⁾ and Q ₂ ⁽⁰⁰⁾ are defined by Equation 35, and the vectors b ₁ (s) and b ₂ (t) are defined by Equation 37.

(Numerical experiment example)
A numerical example of extreme value detection is shown. The optimal smoothed spline curved surface x (s, t) that is the target of extreme value detection was designed for the data obtained by sampling the following equation.

  The parameter values here are shown in the following table.

The parameters used in designing the curved surface are k = 3, α = β = 1, s ₀ = t ₀ = 0, and m ₁ = m ₂ = 10. Therefore, a cubic spline curved surface was designed with a region S = [0, 10] × [0, 10]. The data used for the design was generated by sampling the above d (s, t) at grid points with 0.5 intervals in both the s and t directions. FIG. 12 shows the result obtained by applying the above-described extreme value detection method to the designed curved surface x (s, t). In the figure, the detected maximum value (Δ mark) and minimum value (（mark) are shown superimposed on the three-dimensional display of the curved surface x (s, t). In addition, the contour display of the curved surface and the extreme points are shown on the st plane. It can be confirmed from all the figures that all extreme values have been detected.

  As described above, according to the extreme value detection method and the extreme value detection program using the optimal smoothing spline curved surface of the present embodiment, all extreme values (local minimum values) of a given k-th order spline curved surface x (s, t) are obtained. , The maximum value). Therefore, the minimum value and the maximum value are also obtained. The design of the curved surface x (s, t) may be the above-described method based on the given discrete space data D (Equation 32), or the design of the curved surface approximating the given function f (s, t). Also good. In the latter case, the maximum and minimum values of the designed spline curved surface x (s, t) are obtained, and the numerical search method such as the hill-climbing method is further added to the function f (s, t) using the given points as initial values. By applying, the maximum and minimum values of f (s, t) can be obtained in a more accurate and systematic manner.

  In the above embodiment, the extreme value of the optimal smoothing spline function x (s, t) is obtained. However, the extreme value of an arbitrary partial derivative of x (s, t) can be obtained by a similar method. Is possible.

(Extreme value of partial derivative)
For simplification of symbols, the partial derivatives of x (s, t) and x (u, v) are set as follows.

In addition, i, j ≦ k−2 is set because the problem has a meaning. At this time, the relationship of Equation 42 is x _{i, j} (s, t) = α ^{i βj} x _{i, j} (u, v), and the extreme value of the partial derivative x _{i, j} (s, t) is , X _{i, j} (u, v). That is, if x _{i, j} (u, v) takes an extreme value at the point (u ^* , v ^* ) ∈ε, x _{i, j} (s, t) is the point (s ^* , t ^* ) = (( 1 / α) u ^* + s _κ , (1 / β) v ^* + t _μ ), and extrema x _{i, j} (s, t) = α ⁱ β ^j x _{i, j} (u ^* , v ^* ) It can be seen that

(Expression of function x _{i, j} (u, v))
x _{i, j} (u, v) is obtained from Equation 43.

It is expressed. here,

It is. Since this x _{i, j} (u, v) has the same format as x (u, v) in Equation 43, its extreme value is the same as in the case of detection and calculation of the extreme value of x (u, v). Can be determined according to simple procedures. Of course, all extreme values (minimum value, maximum value) of the partial derivatives x _{i, j} (s, t) can be obtained, and therefore the minimum value and maximum value thereof have also been obtained.

(Third embodiment)
Hereinafter, an extreme value detection method and an extreme value detection program using an optimal smoothing spline according to a third embodiment of the present invention will be described in detail with reference to the drawings.

  In the second embodiment of the present invention, an optimum smoothing spline curved surface that approximates discrete data in space is generated and extreme values are detected, but in this embodiment, luminance value data of each pixel of a digital image is used as discrete data. Detect extreme values for. Further, here, edge detection of the digital image is performed by detecting extreme values of the first-order partial derivatives in the s direction and t direction of the optimal smoothed spline curved surface x (s, t).

(Curved surface generation)
The above extreme value detection method was applied to edge detection of digital images. The target is a 256 × 256 (pixel) monochrome image, and the optimal smoothed spline surface x (s, t) is first designed using the luminance value of each pixel as data, and the edge of the image is the luminance value. Based on the idea that the rate of change of sigma exists in the part where the rate of change changes abruptly (that is, the part where the extreme value of the first-order partial derivative in the s direction and t direction of the optimal smoothing spline curved surface x (s, t) exists) To detect.

  In the present embodiment, the optimal smoothed spline curved surface x (s, t) is processed for each line (for example, s is fixed to s = p). In that case, the edge may be obtained as the extreme value of the first derivative of the spline curve x (p, t) with respect to the variable t.

  FIG. 6 is a diagram schematically showing the optimal smoothing spline curved surface of the present embodiment. FIG. 6A is a diagram showing the distribution of optimum control points (weighting factors) obtained from the position (coordinates) of each pixel and its luminance distribution, and FIG. 6B is based on discrete data in the space of each pixel. It is a figure which shows the produced | generated optimal smoothing spline curved surface x (s, t) and the function x (s, q), x (p, t) produced | generated for every line.

  FIG. 13 is a block diagram showing an edge detection processing apparatus for a digital image using the extreme value detection method and the extreme value detection program using the optimum smoothing spline according to this embodiment.

  The image edge detection processing device 21 of the present embodiment includes a CPU 25 (calculation means) and a memory device 24 (RAM, ROM, etc .: storage means), executes a program stored in the memory device 24, and FIG. As shown in the block diagram of FIG. 13, by executing the operations described in the flowcharts ˜16, the CPU 25 (arithmetic means), the memory device 24 (storage means), a mouse, a keyboard, a scanner, a data logger, etc. An input device 26 (input means), a bus 23, a display device 27 (output means), a printer device 28 (output means), and a network device 29 (input means / output means) connected to an external network 30 are provided. Functions as a processing device.

  FIG. 14 is a flowchart showing an outline of processing of the edge detection method of a digital image by the optimal smoothing spline of this embodiment.

In step 301, pixel data (i, j, fi _{, j} ), i = 1, 2,..., I, j = 1, 2,. It is the process of inputting to. Here, f _{i, j} represents the luminance value of the pixel at the coordinates (i, j) of the pixel. The pixel data can be acquired as data by an optical reading unit (not shown), for example. Further, the pixel data may be obtained from an external device (not shown) via the network device 29. Then, the threshold value ρ is stored in the memory device 24. The threshold value ρ is a parameter for determining whether an extreme value is an edge portion. x ⁽¹⁾ A point where the absolute value of the extreme value (maximum, minimum) of (s, t) is larger than a certain level is regarded as an edge. Since the degree varies depending on the state of the image and edge detection conditions, a threshold value ρ is set in edge detection. If the absolute value of the extreme value is larger than the threshold value ρ, the portion is regarded as an edge portion. In other cases, the extreme value is not regarded as an edge portion.

In step 302, the CPU 25 generates an optimal smoothed spline curved surface based on the input pixel data (i, j, fi _{, j} ). Specifically, the CPU 25 determines the optimum weighting factor τ _ij (i = −k, −k + 1,..., I−1, j = −k, −k + 1,. ) And an optimal smoothed spline curved surface represented by Equation 28 is generated. The method for generating the optimal smoothed spline curved surface is as described in the second embodiment of the present invention. The parameters at that time are α = β = 1, s ₀ = t ₀ = 1, m ₁ = I−1, and m ₂ = J−1.

  After generating the optimal smoothed spline curved surface, the curved surface is processed for each line (for example, s is fixed to s = q). In order to perform the process for the s-axis direction in FIG. 6, the process proceeds to step 303, and to perform the process in the t-axis direction in FIG. 6, the process proceeds to step 304.

In step 303, the CPU 25 determines a plurality of curves x with the t coordinate fixed at q (q = 1, 2,..., J) for the optimal smoothed spline curved surface x (s, t) generated in step 302. (s, q) (see FIG. 6B) is generated, and an extreme value is detected for the first derivative x ⁽¹⁾ (s, q) of the curve x (s, q) (step 303a). Then, the CPU 25 performs edge determination processing based on the extreme values (step 303b).

In step 304, the CPU 25 determines a plurality of curves x with the s coordinate fixed at p (p = 1, 2,..., I) for the optimum smoothed spline curved surface x (s, t) generated in step 302. (p, t) (see FIG. 6B) is generated, and extreme values are detected for the first derivative x ⁽¹⁾ (p, t) of the curve x (p, t) (step 304a). Then, the CPU 25 performs edge determination processing based on the extreme values (step 304b).

  In step 305, the CPU 25 summarizes the edge portions detected by the determinations obtained in steps 303 and 304 as a sum. That is, at least one of the s direction and the t direction is determined as an edge.

  In step 306, the CPU 25 converts the edge detection result for the input image into an image based on the result of step 305, and outputs the image to the display device 27 and the printer device 28.

  FIG. 15 is a flowchart showing details of the processing in step 303. Steps 401 to 403 correspond to step 303a in FIG. 14, and steps 404 to 407 correspond to step 303b in FIG.

In step 401, the CPU 25 calculates a weight coefficient τ _{i, j} (i = −k, −k + 1,..., I) for the pixel data (i, j, fi _{, j} ) obtained in step 301 of FIG. −1, j = −k, −k + 1,..., J−1).

  In addition, regarding the mathematical formulas used below, all the functions x having δ as a variable have “hat (^)”, but they are omitted in the specification text.

In step 402, the CPU 25 generates the optimum smoothed spline curved surface x (s, t) generated in step 302 with the t coordinate fixed at q (q = 1, 2,..., J). Normalization is performed in the nodal interval [s _j , s _{j + 1} ) for a plurality of curves x (s, q). Then, a normalized curve x (δ _s , q) in the unit interval [0, 1) is generated to obtain a first derivative x ⁽¹⁾ (δ _s , q), and a polynomial x ⁽¹⁾ For (δ _s , q), the coefficients a ₀ ,..., a _k−1 are calculated.

In step 403, the CPU 25 detects extreme points δ _s1 , δ _s2 ,..., Δ _sm (m ≦ k−1) of the polynomial x ⁽¹⁾ (δ _s , q). The extreme points δ _s1 , δ _s2 , ..., δ _sm of the polynomial x ⁽¹⁾ (δ _s , q) are unit intervals [0, 1 among the roots of x ⁽²⁾ (δ _s , q). ) Can be found and detected.

In steps 404 and 405, the CPU 25 determines x ⁽¹⁾ (δ _sl , q) · x ⁽³⁾ (δ _sl , q) for all the extreme points δ _s1 , δ _{s2 ,.} It is determined whether <0 and | x ⁽¹⁾ (δ _sl , q) |> ρ (0 ≦ l ≦ m). If x ⁽¹⁾ (δ _sl , q) · x ⁽³⁾ (δ _sl , q) <0 and | x ⁽¹⁾ (δ _sl , q) |> ρ (0 ≦ l ≦ m) Advances to step 406. Otherwise, return to step 404.

In step 406, the CPU 25 determines that x ⁽¹⁾ (δ _sl , q) · x ⁽³⁾ (δ _sl , q) <0 and | x ⁽¹⁾ (δ _sl , q) |> ρ (0 ≦ When l ≦ m), it is determined that there is an edge at the point (p + (1 / α) δ _sl , q). This is because the location of the extreme value of x ⁽¹⁾ (t, q) does not necessarily correspond to the position of the edge, and x ⁽¹⁾ (δ _sl , q) ・ x ⁽³⁾ (δ _sl , q) This is because a location satisfying <0 corresponds to an actual edge.

  In step 407, the CPU 25 repeats steps 402 to 406 if p <I−1, and in step 408 repeats steps 402 to 407 if q <J. , A plurality of curves x (s, t, generated by fixing the t-coordinate of the optimal smoothed spline curved surface x (s, t) generated in step 302 with q (q = 1, 2,..., J). Judge edges for all of q).

  FIG. 16 is a flowchart showing details of the process in step 304.

In step 501, a weight coefficient τ _{i, j} (i = −k, −k + 1,..., I) is obtained for the pixel data (i, j, f _{i, j} ) obtained in step 301 of FIG. −1, j = −k, −k + 1,..., J−1).

In step 502, the CPU 25 generates the optimal smoothed spline curved surface x (s, t) generated in step 302 with the s coordinate fixed at p (p = 1, 2,..., I). Normalization is performed in the interval [t _j , t _{j + 1} ) for a plurality of curves x (p, t). Then, a normalized curve x (p, δ _t ) in the unit interval [0, 1) is generated to obtain a first derivative x ⁽¹⁾ (p, δ _t ), and a polynomial x ⁽¹⁾ For (p, δ _t ), the coefficients b ₀ ,..., b _k−1 are calculated.

In step 503, the CPU 25 detects extreme points δ _t1 , δ _t2 ,..., Δ _tn (n ≦ k−1) of the polynomial x ⁽¹⁾ (p, δ _t ). The extreme points δ _t1 , δ _t2 , ..., δ _tn of the polynomial x ⁽¹⁾ (p, δ _t ) are the unit intervals [0, 1 among the roots of x ⁽²⁾ (p, δ _t ) ) Can be found and detected.

In step 504 and 505, CPU 25, all the extreme point of the [delta] _t1, [delta] _t2, · · ·, for ^{_{δ tn, x (1) (}} p, δ tl) · x (3) (p, δ tl) It is determined whether <0 and | x ⁽¹⁾ (p, δ _tl ) |> ρ (0 ≦ l ≦ n). When x ⁽¹⁾ (p, δ _tl ) · x ⁽³⁾ (p, δ _tl ) <0 and | x ⁽¹⁾ (p, δ _tl ) |> ρ (0 ≦ l ≦ n) Advances to step 506. Otherwise, the process returns to step 504.

In step 506, the CPU 25 determines that x ⁽¹⁾ (p, δ _tl ) · x ⁽³⁾ (p, δ _tl ) <0 and | x ⁽¹⁾ (p, δ _tl ) |> ρ (0 ≦ In the case of l ≦ n), it is determined that there is an edge at the point (p, q + (1 / α) δ _tl ).

  In step 507, when q <J−1, the CPU 25 repeats the processes in steps 502 to 506, and in step 508, repeats the processes in steps 502 to 507 when p <I. , The optimal smoothed spline curved surface x (s, t) generated in step 302, and a plurality of curves x (p generated by fixing the s coordinates at p (p = 1, 2,..., I). , T) detect edges for all.

(Example of edge detection for digital images)
FIGS. 17 and 18 are diagrams illustrating an example in which edge detection is actually performed using the edge detection method of the digital image used in the extreme value detection method using the optimum smoothing spline according to the third embodiment of the present invention. . FIG. 17 illustrates an example in which noise is not included in the original image (a), and FIG. 18 illustrates an example in which noise is included in the original image (a).

  FIGS. 17B and 18B are images generated by applying the edge detection method of the present embodiment. Further, as a comparative example, images detected by using a canny filter and a sobel filter are obtained as shown in FIGS. 17 (c) and 18 (c) (canny filter) and FIGS. 17 (d) and 18 (d) (sobel filter). )Pointing out toungue.

  As can be seen from the figure, it can be seen that the edge can be detected with higher accuracy than other filters, particularly when the original image (a) contains noise. Therefore, the effectiveness of the edge detection method of the digital image used in the extreme value detection method by the optimum smoothing spline of the present invention was proved.

  As described above, according to the extreme value detection method using the optimal smoothing spline of the present embodiment, since the design of the spline curved surface is a curved surface approximation based on the information of the entire image information, the edge detection of the digital image is performed globally. Method. Although this method is completely different from the conventional local method (based on partial information of the image), the extreme value detection method using the optimal smoothing spline according to the present embodiment results in the influence of image noise. It is hard to receive.

  In each of the above-described embodiments, the case of a spline curve (one-dimensional) and the case of a spline curved surface (two-dimensional) have been described. However, the present invention is not limited to this, and is extended and applied to cases of higher dimensions. Is possible.

It is a figure which shows the optimal smoothing spline curve roughly. It is a block diagram which shows the processing apparatus used for the extreme value detection method by the optimal smoothing spline of 1st Embodiment of this invention. It is a flowchart which shows the flow of a process of the extreme value detection method by the optimal smoothing spline of 1st Embodiment of this invention. It is a flowchart which shows the case where an optimal smoothing spline is cubic in the processing flow of the extreme value detection method by the optimal smoothing spline of 1st Embodiment of this invention. It is a figure which shows the optimal smoothing spline curve of 1st Embodiment of this invention, and its extreme value. It is a figure which shows distribution of the optimal control point (weighting coefficient) calculated | required from the discrete data in space. It is a figure which shows schematically the optimal smoothing spline curved surface. It is a block diagram which shows the edge detection processing apparatus of the digital image used for the extreme value detection method by the optimal smoothing spline of 2nd Embodiment of this invention. It is a flowchart which shows the outline | summary of the process of the extreme value detection method by the optimal smoothing spline of 2nd Embodiment of this invention, and an extreme value detection program. It is a flowchart which shows the detail of the process in step 203 of the flowchart of FIG. It is a flowchart which shows the detail of the process in step 206 of the flowchart of FIG. It is a flowchart which shows the detail of the process in step 211 of the flowchart of FIG. It is a figure which shows the example which actually performed the extreme value detection using the extreme value detection method and the extreme value detection program by the optimal smoothing spline of 2nd Embodiment of this invention. It is a block diagram which shows the edge detection processing apparatus of the digital image using the extreme value detection method by the optimal smoothing spline of 3rd Embodiment of this invention, and an extreme value detection program. It is a flowchart which shows the outline | summary of a process of the edge detection method of the digital image by the optimal smoothing spline of 3rd Embodiment of this invention. It is a flowchart which shows the detail of the process in step 303 of the flowchart of FIG. It is a flowchart which shows the detail of the process in step 304 of the flowchart of FIG. It is a figure which shows the example which actually performed edge detection using the edge detection method of the digital image using the extreme value detection method by the optimal smoothing spline of 3rd Embodiment of this invention. It is a figure which shows the example which actually performed edge detection using the edge detection method of the digital image using the extreme value detection method by the optimal smoothing spline of 3rd Embodiment of this invention.

Explanation of symbols

1: Extreme value detection processing device with optimum smoothing spline 3: Bus 4: Memory device 5: CPU
6: input device 7: display device 8: printer device 9: network device

Claims (20)

Executed by a processing device comprising: input means for inputting data; storage means for storing a program; arithmetic means for executing the program and processing the data; and output means for outputting a processing result of the arithmetic means. An extreme value detection method,
A step of generating an optimal smoothing spline curve having a plurality of nodes by the computing means from a discrete data group or function in a plane inputted from the input means;
The calculating means, by each determining an extreme value for each of the optimal smoothing spline curve or on each node interval is the nodal one section which is partitioned by neighboring on its derivatives, the optimum smoothing spline curve Or determining all of the extreme values of the derivative and outputting the extreme values to the output means;
An extreme value detection method using an optimal smoothing spline curve. Obtaining all of the extreme values of the optimal smoothed spline curve or its derivative and outputting the extreme values to the output means,
The arithmetic means normalizing each nodal section of the optimal smoothed spline curve or its derivative to generate a normalized polynomial function or its derivative in a unit section;
The computing means obtains all extreme values of the optimal smoothed spline curve or derivative thereof by obtaining extreme values for the normalized polynomial function or derivative thereof in the unit interval;
The method of detecting an extreme value using an optimal smoothed spline curve according to claim 1, wherein: The step of generating the optimum smoothed spline curve by the computing means includes:
The computing means obtaining an optimal weighting factor for each node from a discrete data group or function in the plane;
A step of generating an optimal smoothed spline curve by configuring the arithmetic means to add a normalized uniform B-spline function for each node weighted by an optimal weighting factor for each node; ,
The method of detecting an extremum using an optimal smoothing spline curve according to claim 1 or 2, wherein:   4. The method of detecting extreme values using an optimal smoothed spline curve according to claim 3, wherein the spline function is a cubic spline function. Executed by a processing device comprising: input means for inputting data; storage means for storing a program; arithmetic means for executing the program and processing the data; and output means for outputting a processing result of the arithmetic means. An extreme value detection method,
Generating an optimal smoothed spline curved surface having a plurality of nodes by the computing means from a discrete data group or function in the space inputted from the input means;
By obtaining the respective extremum for each of the nodes a region where the calculating means is in a region partitioned on a grid by the nodes on the optimal smoothing spline surface on or partial derivatives, the optimum smoothing spline Obtaining all of the extreme values of the curved surface or its partial derivative, and outputting the extreme values to the output means;
An extreme value detection method using an optimally smoothed spline curved surface. Obtaining all of the extreme values of the optimal smoothed spline curved surface or its partial derivative, and outputting the extreme values to the output means,
The computing means normalizing the optimal smoothing spline curved surface or each nodal region of the partial derivative thereof to generate a normalized polynomial function or a partial derivative thereof in a unit region;
Obtaining the extreme values of the normalized polynomial function or its partial derivative in the unit region by the computing means to obtain all the extreme values of the optimal smoothed spline curved surface or the partial derivative;
The extreme value detection method by the optimal smoothing spline curved surface of Claim 5 characterized by the above-mentioned. The step of generating the optimum smoothed spline curved surface by the computing means includes:
The computing means obtaining an optimal weighting factor for each node from a discrete data group or function in the space;
A step of generating an optimal smoothed spline curved surface by the arithmetic means comprising a normalized uniform B-spline function for each node, weighted by an optimal weighting factor for each node, and added together; ,
The method of detecting an extremum using the optimal smoothed spline curved surface according to claim 5 or 6, wherein: An image processing apparatus comprising: input means for inputting pixel data; storage means for storing a program; arithmetic means for executing the program and processing the pixel data; and output means for outputting a processing result of the arithmetic means An edge detection method of a digital image executed in
A step of generating an optimal smoothing spline curved surface having a plurality of nodes by the computing means from pixel data comprising pixel coordinates and luminance values inputted from the input means;
The computing means generates a first curve group composed of a plurality of curves in which the optimal smoothed spline curved surface is fixed at a plurality of coordinates on a first coordinate axis, and a plurality of curves constituting the first curve group are generated. Obtain all extrema of the first derivative of the first curve group by obtaining extrema for each of the nodal sections , which are one section segmented by adjacent nodes on the first derivative. Steps,
The computing means generates a second curve group composed of a plurality of curves in which the optimum smoothing spline curved surface is fixed at a plurality of coordinates on a second coordinate axis, and a plurality of curves constituting the second curve group are generated. by each obtaining an extreme value for each of the nodes a certain section in the first derivative on the adjacent said nodes a section which is divided by, finding all extrema of the first derivative of the second group of curves Steps,
A point where the computing means has an absolute value of the first derivative of the first curve group or an absolute value of the first derivative of the second curve group greater than a predetermined threshold value. Determining an edge portion and outputting the determination result to the output means;
A method for detecting an edge of a digital image, comprising: Determining all of the extreme values of the first derivative of the first group of curves,
The arithmetic means normalizes each nodal section of the first derivative of the plurality of curves constituting the first curve group, and the plurality of curves constituting the first curve group normalized in the unit section Determining all of the extreme values of the first derivative of the first curve group by determining the extreme value of the first derivative,
Determining all of the extreme values of the first derivative of the second group of curves,
The computing means normalizes each node section of the first derivative of the plurality of curves constituting the second curve group, and the plurality of curves constituting the second curve group normalized in the unit section Determining all of the extreme values of the first derivative of the second group of curves by determining the extreme value of the first derivative;
The edge detection method for a digital image according to claim 8, comprising: The step of generating the optimum smoothed spline curved surface by the computing means includes:
A step of obtaining an optimum weighting factor for each node from pixel data comprising the coordinates and luminance values of the pixels;
A step of generating an optimal smoothed spline curved surface by the arithmetic means comprising a normalized uniform B-spline function for each node, weighted by an optimal weighting factor for each node, and added together; ,
10. The method for detecting an edge of a digital image according to claim 8 or 9, characterized by comprising: A processing apparatus comprising: input means for inputting data; storage means for storing a program; arithmetic means for executing the program and processing the data; and output means for outputting a processing result of the arithmetic means;
A step of generating an optimal smoothing spline curve having a plurality of nodes by the computing means from a discrete data group or function in a plane inputted from the input means;
By obtaining the respective extremum for each of the nodes interval the arithmetic means is in the optimum smoothing spline curve or on a section which is divided by the nodes adjacent on its derivatives, the optimum smoothing spline curve or Obtaining all of the extreme values of the derivative and outputting the extreme values to the output means;
Program for detecting extreme values using an optimal smoothing spline curve. Finding all of the extreme values of the optimal smoothed spline curve or its derivative, and outputting the extreme values to the output means,
The computing means normalizing each nodal interval of the optimal smoothing spline curve or its derivative to generate a normalized polynomial function or its derivative in a unit interval;
Obtaining the extreme values of the normalized polynomial function or derivative thereof in the unit interval by the computing means to obtain all the extreme values of the optimal smoothed spline curve or derivative thereof;
The extreme value detection program by the optimal smoothing spline curve of Claim 11 characterized by the above-mentioned. The step of generating the optimum smoothed spline curve by the computing means includes:
The computing means obtaining an optimal weighting factor for each node from a discrete data group or function in the plane;
A step of generating an optimal smoothed spline curve by configuring the arithmetic means to add a normalized uniform B-spline function for each node weighted by an optimal weighting factor for each node; ,
The extreme value detection program by the optimal smoothing spline curve according to claim 11 or 12, characterized by comprising:   14. The extreme value detection program using an optimal smoothed spline curve according to claim 13, wherein the spline function is a cubic spline. A processing apparatus comprising: input means for inputting data; storage means for storing a program; arithmetic means for executing the program and processing the data; and output means for outputting a processing result of the arithmetic means;
A step of generating an optimal smoothing spline curved surface having a plurality of nodes by the computing means from a discrete data group or function in the space inputted from the input means;
By obtaining the respective extremum for each of the nodes a region where the calculating means is in a region partitioned on a grid by the nodes on the optimal smoothing spline surface on or partial derivatives, the optimum smoothing spline Obtaining all of the extreme values of the curved surface or its partial derivative, and outputting the extreme values to the output means;
Program for detecting extreme values with optimal smoothing spline curved surface. Obtaining all of the extreme values of the optimal smoothed spline curved surface or its partial derivative, and outputting the extreme values to the output means,
The arithmetic device normalizing each optimally-spline curved surface or each nodal region of its partial derivative to generate a normalized polynomial function or its derivative in a unit region;
Obtaining the extreme values of the normalized polynomial function or the partial derivative thereof in the unit region by the arithmetic unit to obtain all the extreme values of the optimal smoothed spline curved surface or the partial derivative thereof;
The extreme value detection program by the optimal smoothing spline curved surface according to claim 15, characterized by comprising: The step of generating the optimum smoothed spline curved surface by the computing means includes:
A step of calculating an optimal weighting factor for each node from a discrete data group or function in space;
A step of generating an optimal smoothed spline curved surface by the arithmetic means comprising a normalized uniform B-spline function for each node, weighted by an optimal weighting factor for each node, and added together; ,
The extreme-value detection program by the optimal smoothing spline curved surface of Claim 15 or 16 characterized by the above-mentioned. An image processing apparatus comprising: input means for inputting pixel data; storage means for storing a program; arithmetic means for executing the program and processing the pixel data; and output means for outputting a processing result of the arithmetic means In addition,
Generating an optimal smoothing spline curved surface having a plurality of nodes by the computing means from pixel data consisting of pixel coordinates and luminance values inputted from the input means;
The computing means generates a first curve group composed of a plurality of curves in which the optimal smoothed spline curved surface is fixed at a plurality of coordinates on a first coordinate axis, and a plurality of curves constituting the first curve group are generated. Obtain all extrema of the first derivative of the first curve group by obtaining extrema for each of the nodal sections , which are one section segmented by adjacent nodes on the first derivative. Steps,
The computing means generates a second curve group composed of a plurality of curves in which the optimum smoothing spline curved surface is fixed at a plurality of coordinates on a second coordinate axis, and a plurality of curves constituting the second curve group are generated. by each obtaining an extreme value for each of the nodes a certain section in the first derivative on the adjacent said nodes a section which is divided by, finding all extrema of the first derivative of the second group of curves Steps,
A point where the computing means has an absolute value of the first derivative of the first curve group or an absolute value of the first derivative of the second curve group greater than a predetermined threshold value. Determining an edge portion and outputting the determination result to the output means;
Digital image edge detection program to execute. Determining all of the extreme values of the first derivative of the first group of curves,
The computing means normalizes each nodal section of the first derivative of the plurality of curves constituting the first curve group, and the plurality of curves constituting the normalized first curve group in the unit section Determining all the extreme values of the first derivative of the first curve group by determining the extreme values of the first derivative of
Determining all of the extreme values of the first derivative of the second group of curves,
A plurality of curves constituting the second curve group normalized in the unit interval by the arithmetic means normalizing each node interval of the first derivative of the plurality of curves constituting the second curve group. Obtaining all extrema of the first derivative of the second curve group by obtaining extrema of the first derivative of
19. The digital image edge detection program according to claim 18, further comprising: The step of generating the optimum smoothed spline curved surface by the computing means includes:
A step of obtaining an optimum weighting factor for each node from pixel data comprising the coordinates and luminance values of the pixels;
A step of generating an optimal smoothed spline curved surface by the arithmetic means comprising a normalized uniform B-spline function for each node, weighted by an optimal weighting factor for each node, and added together; ,
21. The digital image edge detection program according to claim 18 or 19, characterized by comprising: