和文要旨
画像処理のための複素数離散ウェーブレット変換の設計とその応用に関する研究
近年，カメラ等の撮像デバイスや計算機の高度化により，画像処理技術が広く普及している．
画像処理技術によって，人間が視覚で行う検査，診断，監視等の様々な作業支援や作業の代替が期待されている．これらの技術の実現には，画像に映る物体等の特徴的な情報を抽出する特徴抽出という処理が必要となる．特徴抽出手法の一つにウェーブレット変換(WaveletTransform, WT)がある．これは画像の様々な周波数成分を計算し，多くの特徴を抽出できる．
そのため従来から特徴抽出手法として多くの画像に対し利用されてきた．
近年，画像の方向性エッジを特徴として計算出来る．2D-CDWT(2-Dimensional Complex
Discrete Wavelet Transform, 2D-CDWT)が注目されている．この機能を方向選択性と呼び，物体の形状情報を得る手法として期待されているが，2D-CDWT から得られる方向性エッジは，抽出する特徴の種類が少なく，大まかな形状情報しか得られない．そこで，2D-CDWT 等を基に，詳細な方向性エッジを抽出する新たなウェーブレット変換を提案する．
始めに，方向選択性の原理の基礎検討を行った．その結果，ヒルベルト変換ペアの実数部と虚数部のウェーブレットが，方向性の波形を構成することが確認された．また，ウェーブレッ
トの周波数特性と方向に一意の関係があることを確認した．その関係を基に，方向性エッジの方向・角度範囲の評価手法を提案した．それにより，2D-CDWT 等の方向選択性の抽出機能を定量的に評価可能にした．
次に，2D-CDWT の発展形である 2 次元複素数離散ウェーブレットパケット変換(2-Dimensional Complex Discrete Wavelet Packet Transform, 2D-CWPT)に方向調節フィルタを付与し，2D-CWPT が計算する方向成分を変化させ，任意の方向性エッジを抽出する手法を提案した．またこの手法から詳細な方向性エッジの抽出を可能にした．この提案手法を半導体ウェーハの画像欠陥検査に応用し，K-measn 法と組み合わせた欠陥検査手法を提案した．
その結果，画像欠陥を従来の 2D-CDWT 等の手法よりも高精度に認識できることを確認した．
さらに，直接，方向成分に合わせて周波数特性を設計した方向性フィルタを提案した．そして，方向性フィルタによる新たな方向性ウェーブレット変換を提案した．この手法においても，画像から任意の方向性エッジを抽出可能であることを確認した．またこの手法を医用画像処理に応用した．CT(Computed Tomography)画像内に映る腫瘤の自動検出について検討し，サポートベクターマシン(Support Vector Machine, SVM)と併用すると，腫瘤を高精度に検出可能であった．また，従来手法である 2D-CDWT，ガボールフィルタと比較し，方向性ウェーブレット変換が最も高い検出率を示した．
また，2 次元画像から 3 次元 CT，MRI(Magnetic Resonance Imaging)を解析可能な3D-CDWT を開発している．
これらの結果から，方向性ウェーブレット変換や 2D-CWPT を基にした提案手法が画像の特徴抽出手法として有効であることを確認した．

ABSTRUCTThe Design and Application of Complex Discrete Wavelet
Transform for Image Processing
Takeshi KatoIn recent years, due to the improvement of the imaging devices and computers, theimage processing techniques are widely spread industries, agricultures and medical fields.
The image processing techniques are expected to achieve the inspection, diagnosis andmonitoring which are previously carried out by people. These techniques require theprocess which extracts the characteristic information. This process is called ‘Feature
Extraction’ and one of the feature extraction methods is a Wavelet Transform. TheWavelet Transform extracts the multiple frequency components from the image by using
mother wavelet and filters. Therefore, the Wavelet Transform was used for variable image
processing applications.
The 2-D Complex Discrete Wavelet Transform (2D-CDWT) is especially interested. The
2D-CDWT can extract the singularities different with orientations (directional edges).
Here, singularities mean the edge, border, lines and ridges in the image. This property iscalled directional selection and this property expected as the method to obtain the shapeinformation from each directional edge. However, the number of directional edges is littleand we can’t get detail features about directionalities. For instance when we use thedirectional information for the image recognition and object detection, we calculate morethan 8 directional components. In this paper, we propose the novel 2D-CDWT which canextract the many directional edges and detail features.
At first, the principle of directional selection is considered. We confirmed that theHilbert Transform pair of the real part of wavelet and the imaginary part of wavelet which
has the phase difference makes the specified waveforms. This waveform has theamplitude only the specified direction and we confirmed that this waveform extracts thedirectional edges. Moreover the frequency characteristics of this waveform and its
direction have the unique relationship. Based on this relationship, we construct theevaluation method for the direction and the angular range of directional edges. According
to the evaluation method, we can estimate the extracting ability of the directional
selection quantitatively.
Secondary, we focused on the 2-D Complex Discrete Wavelet Packet Transform(2D-CWPT) which is the advanced Wavelet Transform from the 2D-CDWT. Based on the2D-CWPT, we propose the novel 2D-CWPT that can extract the arbitrary direction andangular range. The novel 2D-CWPT is added the modified filter to adjust the directionaledges. We confirmed that we can get the desirable directional component using thismethod. But we also confirmed that there are error between the result and requested
angular range. We apply the novel 2D-CWPT to the semiconductor wafer image for theautomated defect detection. We used the novel 2D-CWPT as the feature extraction andK-means method as the pattern recognition method for the defect detection. As thedetection result, the novel 2D-CWPT can detect the defect precisely rather than previous
2D-CDWT, 2D-CDWT, 2D-DWT, 2D-STFT (Short Time Fourier Transform).
Thirdly, we propose the novel directional wavelet transform by using designed
directional filter. The directional filter is designed based on the relationship the directionof wavelet and its frequency characteristics. The directional wavelet transform also candetect the desirable directional edges without the error between the desirable angularrange and the result. As the application, we apply the directional wavelet transform tomedical image processing. We consider the tumor detection from CT (ComputedTomography) image by using the directional wavelet transform and support vector
machine. As the result the directional wavelet transform has the most accurate detection
rate rather than previous 2D-CDWT and Gabor filters.
Besides the 2D methods, we developed the 3D-CDWT which can be used for 3D-CT and3D-MRI images. Previously the directional selection for the 3D images has not achieved
yet. We construct the 3D directional selection based on the principle of the directional
selection. The 3D directional selection extracts the 3D directional edges that haveorientation and angular range in 3D space domain. We confirmed that the 3D directionaledges show the features of 3D images. Moreover in the 3D CT image, we can detect andrecognize the tumor using 3D-CDWT and SVM.
Finally in this paper, we confirmed the validation of the novel 2D-CWPT and thedirectional wavelet transform as the image feature extraction method by using wafer
images and medical CT images. Also we can show that the 3D directional selection can get
3D image features.