應用在圖形辨識的自發性比例式記憶體細胞非線性網路的穩定度分析

Abstract

人腦可以處理對於一些用現今的數位影像方法仍然無法有令人滿意的影像問題，例如影像分割與辨識。 一些研究人員相信大腦新皮質以自體聯結的方式回憶出形態。 他們指出人腦可快速處理影像問題的原因乃由於人腦直接從記憶中取出答案。

為了處理影像辨識，我們提出自發性比例式記憶體細胞非線性網路(以下簡稱比例式記憶體網路) 來製作自體聯結的記憶體。 比例式記憶體網路是一個非線性系統，它的每個細胞與附近的鄰居細胞彼此連接， 這些彼此連接的結構對於在每個細胞上的類比輸入即時訊號一起同時處理。人類視網膜亦具有這兩種特質: 每個細胞與其附近的鄰居細胞彼此連接以及類比輸入即時訊號一起同時處理。這兩種特質使得比例式記憶體網路 適合以類比式超大型積體電路來製作。

對於影像辨識，暸解穩態值及暫態吸引區對於穩定度分析是很重要的事。一開始的影像點將會依其所在的 暫態吸引區來收斂到他所對應的穩態值。藉由李雅普諾夫定理，本論文提出一個輸入影像的保守的暫態吸引區。 並且藉由李雅普諾夫定理，本論文提出一個圖形法來建構此暫態吸引區。

從電腦模擬及離散電路執行的觀點來看，找到數值積分法的最大時間步而不引起數值不穩定 是極其重要的事。除了數值不穩定外，另外一個議題就是錯誤的記憶點，一些並非黑或白的記憶點 可能存在。這些錯誤的記憶點會降低圖形的辨識率。本論文使用擴散模式及特性空間分解法來探討 在三種情況下的一維度比例式記憶體網路的歐拉數值積分的最大時間步。 這三種情況分別是所有細胞處在線性區，一端細胞進入飽和區，兩端細胞進入飽和區。每一種情況有其對應的 擴散模式的邊界條件。從這三種情況的特性空間分解法，我們推導出神經元增益的公式，來降低錯誤記憶點的數目。

此外，由此特性空間分解法可推論出在一個充份條件下， 使用歐拉數值積分的比例式記憶體網路的穩態輸出等於類比連續式的比例式記憶體網路的穩態輸出。 雖然本論文並未推導二維度的比例式記憶體網路的特性空間分解，我們建議類似的設計原則存在， 本論文以一維度及二維度的比例式記憶體網路的例子來支持本論文的設計方程式。

Humans brains can resolve many complex image tasks such as pattern recognition and segmentation which are still difficult for digital image processing algorithms to have a satisfying result. Some researchers believe that the neocortex recalls patterns auto-associatively. They pointed out the reason for the brain to efficiently resolve these image tasks is that the brain retrieves the answer from memory.

We propose the ARMCNN (Autonomous Ratio-Memory Cellular Nonlinear Network) structure to implement the associative memory for pattern recognition. The ARMCNN is a nonlinear system with each cell locally coupled to its neighbors. This locally connected structure processes the input real-time analog signals at each cell simultaneously. Human retinas also have these two characterictics : the locally connected structure and the ability to process the input real-time analog signal . These two characterictics make ARMCNNs suitable to implement in analog VLSI.

For pattern recognition, stability analysis is essential to understand the steady state values and the domain of attraction. The initial image point will converge to its corresponing steady state value depending on which corresponding domain of attraction the initial point belong. This thesis aims to provide a conservative domain of attraction for the input image by Lyapunov stability analysis. From this Lyapunov stability analysis, a graphical method is proposed to construct the domain of attraction.

From the view point of computer simulation and discrete-time circuit implementation, it is crucial to obtain the maximum time step in the numerical integration algorithm without causing numerical instability. Besides the numerical instability, another issue is the spurious memory points. Some non-binary equilibrium points may exist. These spurious memory points lower the recognition rate. This thesis uses the diffusion model and the eigenspace decomposition method to examine the maximum forward Euler time step for one-dimensional ARMCNN in three cases: all neurons in linear regions, one end neuron entering the saturation region, and two end neurons entering the saturation region. Each case has a distinct boundary condition for its corresponding diffusion model. From the eigenspace decomposition in these three cases, an analytic neuron gain is derived to lower the amount of spurious memory points.

In addition, the eigenspace decomposition implied a sufficient condition to guarantee that the forward Euler ARMCNN has the same steady state output as the continuous time ARMCNN. Although the eigenspace decomposition for 2-D ARMCNN is not derived, this thesis suggests a similar design principle exists for 2-D ARMCNN. 1-D and 2-D examples are given in this thesis to support the ARMCNN design equations.