Diffusion Tensors for PrDiffusion Tensors for Processing Shearedocessing Sheared

and Rotated Rectanglesand Rotated Rectangles

Gabriele Steidl and Tanja TeuberGabriele Steidl and Tanja Teuber

指導教授 張元翔 指導教授 張元翔

學生 陳昱辰學生 陳昱辰

I. INTRODUCTIONI. INTRODUCTION

Image restoration and Image restoration and simplification methods that simplification methods that respect important features such respect important features such as edges play a fundamental role as edges play a fundamental role in digital image processing.in digital image processing.

I. INTRODUCTIONI. INTRODUCTION

we adapt the diffusion tensor for we adapt the diffusion tensor for anisotropic diffusion to avoid anisotropic diffusion to avoid this effects in images containing this effects in images containing rotated and sheared rectanglesrotated and sheared rectangles

I. INTRODUCTIONI. INTRODUCTION

Diffusion tensorDiffusion tensor

Convolved GaussionConvolved Gaussion

FluxFlux

Diffusion equationDiffusion equation

ANISOTROPIC DIFFUSION ANISOTROPIC DIFFUSION MODELMODELStructure Tensor and Diffusion TensorStructure Tensor and Diffusion Tensor EED applies the structure tensor concept to

define useful diffusion tensors.By definition,j is a rank 1 matrix with spectral decomposition

Fig.1Fig.1

ANISOTROPIC DIFFUSION ANISOTROPIC DIFFUSION MODELMODEL

Adaptation to Rotated RectanglesAdaptation to Rotated Rectangles

the new matrix can only distinguish between angles and its smoothing should nicely relate the vertices to the corresponding edges.

Fig.2Fig.2

The local character of the angle smoothing via the structure tensor is obvious.

ANISOTROPIC DIFFUSION ANISOTROPIC DIFFUSION MODELMODEL

• Remark: we can also consider rotated and sheared rectangles, where we know the shear matrices

slightly modifying the diffusion tensor.We just define to be the angles of S

Fig.3Fig.3

we can estimate the shear parameters by the gradients of the two nonhorizontal edges of each sheared rectangle.

ANISOTROPIC DIFFUSION ANISOTROPIC DIFFUSION MODELMODEL

Adaptation to Sheared RectanglesAdaptation to Sheared Rectangles To process such images while preserving sharp vertices, we

want to incorporate an estimation of the shear parameters into the diffusion tensor

ANISOTROPIC ANISOTROPIC REGULARIZATION MODELREGULARIZATION MODEL

• we can use regularization methods to process images consisting of linearly transformed rectangles if we know the transformation matrix at each pixel

• Perona-Malik diffusivity leading to forward-backward diffusion. This PDE becomes well-posed by using the smoothed image in the diffusivity

Fig.6 Fig.7Fig.6 Fig.7

Fig.5Fig.5

DISCRETIZATION ISSUESDISCRETIZATION ISSUES

•1) Anisotropic Diffusion

•To discretize diffusion model we apply an explicit time discretization and discretize partial spatial derivatives by central differences, where we additionally use a smoothing filter

DISCRETIZATION ISSUESDISCRETIZATION ISSUES

•2) Anisotropic Regularization

• To minimize numerically,we compute the minimizer of its discrete counterpart.

• In this paper, we prefer, due to the observed fast

convergence, to minimize the functional by SOCP.

Fig.8Fig.8

Fig.9Fig.9

Fig.10 11 12Fig.10 11 12

SUMMARY AND SUMMARY AND CONCLUSIONSCONCLUSIONS

Preserving vertices is still a problem Preserving vertices is still a problem in image processing.in image processing.

Future work has to be invested for Future work has to be invested for processing images containingprocessing images containing

both rotated and sheared rectangles both rotated and sheared rectangles as well as arbitrary multipleas well as arbitrary multiple

orientations at corners and junctions.orientations at corners and junctions.