Applications to 3DCNN

This section elucidates an interesting model in 3DCNN of the application of the method. The method is elucidated by considering a_0,0,0 = a, a_1,0,0 = a_x, a_0,1,0 = a_y and a_0,0,1 = a_z, which are nonzero; in other cases, a_α,β,γ and b_α,β,γ are zero.

Then, the 3DCNN is of the form as in Eq. (6) du_i,j,k

dt =−u_i,j,k+ w + af (u_i,j,k) + a_xf (u_i+1,j,k) + a_yf (u_i,j+1,k) + a_zf (u_i,j,k+1).

The stationary solution to Eq. (6) satisfies u_i,j,k = w + av_i,j,k+ a_xv_i+1,j,k+ a_yv_i,j+1,k

+ a_zv_i,j,k+1, for (i, j, k)∈ Z³ as in Eq. (7).

Firstly, consider the mosaic solution u = (u_i,j,k) to Eq. (7). If u_i,j,k ≥ 1, i.e. v_i,j,k = 1, then

(a− 1) + w + a_xv_i+1,j,k+ a_yv_i,j+1,k

+ a_zv_i,j,k+1≥ 0. (83) If u_i,j,k≤ −1, i.e. v_i,j,k =−1, then

(a− 1) − w − (a_xv_i+1,j,k+ a_yv_i,j+1,k+ a_zv_i,j,k+1)

≥ 0. (84)

Equation (7) has five parameters w, a, a_x, a_y and a_z. Three procedures are adopted to partition these parameters:

Procedure (I). The parameters ax, a_y and a_z are initially expressed into three-dimensional coor-dinates, to solve Eqs. (83) and (84), as in Fig. 3.

Clearly 2³octants (I)–(VIII) exist in (a_x, a_y, a_z) three-dimensional coordinates.

Procedure (II). In each octant are 3! relations (i) :|a_x| > |a_y| > |a_z|,

(ii) :|ax| > |az| > |ay|, (iii) :|a_y| > |a_x| > |a_z|, (iv) :|ay| > |az| > |ax|, (v) :|az| > |ax| > |ay|, (vi) :|a_z| > |a_y| > |a_x|.

(85)

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Fig. 3. Primary partition of (ax, ay, az).

Procedure (III). Each relation, denoted by |a₁| >

|a2| > |a3|, two situations apply (1)|a₁| > |a₂| + |a₃|

(2)|a1| < |a2| + |a3|. (86)

However, in the (a, w)-planes, two sets of 2³straight lines are important. The first set is

⁺_r : (a− 1) + w + axv_i+1,j,k+ a_yv_i,j+1,k + a_zv_i,j,k+1= 0,

which is related to Eq. (83). The second set is

⁻_r : (a− 1) − w − (axv_i+1,j,k+ a_yv_i,j+1,k + a_zv_i,j,k+1) = 0,

which is related to Eq. (84), where v_i+1,j,k, v_i,j+1,k, v_i,j,k+1 ∈ {−1, 1} and 1 ≤ r ≤ 8. When (a_x, a_y, a_z) lines in the open region (I)–(VIII), (i)–(vi) and (1)–(2) as in Fig. 3, Eqs. (85) and (86) are used to partition the (w, a− 1)-plane, as in Fig. 4.

The symbols [m, n] in Fig. 4 have the follow-ing meanfollow-ings. Consider, for example, (a_x, a_y, a_z) lies in regions (VIII), (i) and (1) as in Fig. 3, Eqs. (85) and (86). This situation is expressed as (VIII)-(i)-(1), and considered a_x < a_y < a_z < 0

Fig. 4. Partition of (w, a − 1)-plane.

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and |a_x| > |a_y| + |a_z|. Denoted by

Table 1. The intersects of
⁺_i and
⁻_j.

(v_i+1,j,k, v_i,j+1,k, v_i,j,k+1) −(axv_i+1,j,k+ayv_i,j+1,k+azv_i,j,k+1) c⁺₁ =c⁻₈ (−1, −1, −1) ax+ay+az

c⁺₂ =c⁻₇ (−1, −1, 1) ax+ay− az

c⁺₃ =c⁻₆ (−1, 1, −1) a_x− a_y+a_z

c⁺₄ =c⁻₅ (−1, 1, 1) ax− ay− az

c⁺₅ =c⁻₄ (1, −1, −1) −ax+ay+az

c⁺₆ =c⁻₃ (1, −1, 1) −a_x+a_y− a_z

c⁺₇ =c⁻₂ (1, 1, −1) −ax− ay+az

c⁺₈ =c⁻₁ (1, 1, 1) −ax− ay− az

Then, c⁺₈ > c⁺₇ > c⁺₆ > c⁺₅ > 0 > c⁺₄ > c⁺₃ > c⁺₂ >

c⁺₁ > are the intersects of ⁺_i and ⁻_j on the w-axis displayed in Fig. 4.

With reference to the local patterns on cube-cells, +1 is represented by the symbol + and −1 is represented by the symbol−. The 2⁴ local patterns can be listed and ordered, as in Fig. 5.

Now, when (w, a − 1) lies in region [m, n] in Fig. 4, the only admissible patterns are exactly

, 21 , . . . , m and 1^, 2^, . . . , n^. For instance, in region (VIII)-(i)-(1) and (a − 1, w) ∈ [4, 8] only

, 21 , 3, 4 and 1^, 2^, 3^, 4^5^, 6^7^, 8^ can

be produced. This fact is equivalent to the hold-ing of inequalities in Eqs. (83) and (84) if and only if v_i,j,k, v_i+1,j,k, v_i,j+1,k and v_i,j,k+1 are of the form

, 21 , 3, 4 and 1^, 2^, 3^, 4^5^, 6^7^, 8^. Next, the transition matrix of local patterns in region (VIII)-(i)-(1)-[4,8] can be derived as

A_x;2×2×2= G⊗ E ⊗ E ⊗ E.

Then, according to Proposition 3.9, the admissible local patterns in Σ_2×m2×m3 and its corresponding transition matrices are

A_x;2×m2×2 =⊗(G ⊗ E)^m2−1⊗ (⊗E²),

Fig. 5. Ordering of local patterns in partition (VIII)-(i)-(1).

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A_ˆx;2×m2×2 = (⊗G^m2−1)⊗ (⊗E^m2+1),

A_{ˆx;2×m2×m3} =⊗((⊗G^m2−1)⊗ E)^m3−1⊗ (⊗E^m2), as in Eqs. (43), (44) and (45).

Finally, the connecting operator is adopted to examine the complexity of the set of mosaic pat-terns in 3DCNN. That is, the lower bound of spa-tial entropy in the region (VIII)-(i)-(1)-[4,8] can be estimated. the spatial entropy can be exactly computed as

h(A_x;2×2×2) = log g as in Proposition 3.9.

Proof. According to Eq. (44),

Aˆx;2×m2×2= (⊗G^m2−1)⊗ (⊗E^m2+1) is obtained. Evidently,

A_{ˆx;2×m2×2;1}=⊗E^m2 and

(A^(r)_ˆx;2×m2×2)^(c)_;1 = (⊗G^m2−1)⊗ E.

By Remark 4.3, the connecting operator C_z;m1;m22;11= A_{ˆx;2×m2×2;1}◦ (A^(r)_ˆx;2×m2×2)^(c)_;1

= (⊗G^m2−1)⊗ E.

Therefore, based on Remark 4.13, the lower bound of spatial entropy is estimated as

h(Ax;2×2×2)≥ lim method described above can be applied, as stated in Remark 4.14. The details are omitted here for brevity.

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在文檔中 Three-dimensional Cellular Neural Networks and pattern generation problems (頁 24-29)