[PDF] Covariant Derivatives and Vision - Todor Georgiev PDF CovariantSlides.pdf

1 Covariant Derivatives and Vision Todor Georgiev Adobe Photoshop Presentation at ECCV 2006 The covariant derivative is a rigorous mathematical tool

Chapter 4: Applications of Derivatives EXAMPLE 1 The absolute extrema of the following functions on their domains can be seen in Figure 4 2 Each function

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[PDF] Covariant Derivatives and Vision - Todor Georgiev

1 Covariant Derivatives and Vision Todor Georgiev Adobe Photoshop Presentation at ECCV 2006 The covariant derivative is a rigorous mathematical tool

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Covariant Derivatives and Vision

Todor Georgiev

Adobe Photoshop

Presentation at ECCV 2006

Original

Selection to clone

Poisson cloning from dark area

Selection to clone

Poisson cloning from illuminated area

Poisson

cloning

Covariant

cloning (see next) 8

Poisson cloning can be viewed as an

approximation to covariant cloning.

Outline of our theory:

9 10

Thanks to Jan Koenderink

11 •The image is just a record of pixel values. •We do not see pixel values directly. •What we see is an illusiongenerated from the above record through internal adaptation.

We can not compare pixels.

Retina / Cortex Adaptation

Models of Image Space:

13 •A pair (location, intensity) •Multiple copies of the intensity line. •We can compare intensities. The image is a function that specifies an intensity at each point. (1) Cartesian Product 14 •Two spaces and a mapping (vertical projection)

Total space E

Base space B

•Fiber is the set of points that map to a single point. We will use vector bundles, where fibers are vector spaces. (2) Fiber Bundle 15 •Mapping from base B to total space E

Sections

replace functions •We can not compare intensities. Horizontal projection is not defined. We have forgotten it. •Perceptually correct model of the image

Image = graph of a section

Section in a Fiber Bundle

Derivatives in a Fiber Bundle

Definition:

Derivative is a mapping from one section to

another that satisfies the Leibniz rule relative to multiplication by functions:

In the Cartesian product space this definition is

equivalent to the conventional derivative. 17 If we express a section as a linear combination of some basis sections then the derivative will be:

Derivatives in a Fiber Bundle

If we represent the section in terms of the functions that define it in a given basis (not writing the basis vectors), the last equation can be written as:

The functions are called "color channels" in

Photoshop, and D is called "Covariant Derivative".

It corresponds to the derivative in the Cartesian

product model. 19 The covariant derivativeis a rigorous mathematical tool for perceptual pixel comparison in the fiber bundle model of image space. It replaces the conventional derivative of the Cartesian product model as: 20 21
- Reconstructing images with the covariant Laplace equation based on adaptation vector field, A. - Reconstructing surfaces based on gradient field. Recent work by R. Raskar et. al. Covariant Laplace should produce better results than Poisson.

How can we know A?

It can be extracted based on the idea of covariantly constant section, next: 22
Assume perceived gradient of image g(x, y) is zero.

This means complete adaptation:

Substitute in covariant Laplace:

Covariant

cloning

Poisson equation

Poisson

cloning

Covariant

cloning 24

PoissonCovariant

Detailed Example:

Original Damaged Area

Laplace

Poisson

Laplace

Inpainting

Thanks to Guillermo Sapiro and Kedar Patwardhan

Poisson

Covariant

Inpainting

Thanks to Guillermo Sapiro and Kedar Patwardhan

Structure and Texture Inpainting

Bertalmio - Vese - Sapiro - Osher

Covariant Inpainting

36
Day 37
Night 38

Covariant cloning from day

Poisson cloning from day

40
Thanks to R. Raskar and J. YuCloning from night to day 41
Gradient Domain HDR CompressionChanging the lighting conditions. The visual system is robust. It compensates for the changes in illumination by adaptation vector field A:

Simplest energy invariant to those transforms is:

42
Gradient Domain HDR CompressionEuler-Lagrange equation for the above energy: Exactly reproduces the result of the Fattal-Lischinski-

Werman paper. They assume log; we derive log.

Any good visual system needs to be logarithmic!

43
Conclusion:The covariant (adapted) derivative provides a way to perform perceptual image processing according to how we see images as opposed to - how images are recorded by the camera. Useful for Poisson editing, inpainting or any PDE,

HDR compression, surface reconstruction from

gradients, night/day cloning, graph cuts, bilateral and trilateral filters in terms of jet bundles, and practically any perceptual image editing. 44
Bilateral interpreted in 3D image spaceThe image is a distribution in 3D: or -perceptual? Integrate the following 3D filter expression over z and evaluate it on the original surface. Result: (1)

Appendix:

45
(1)

Same procedure on the logarithmic expression

produces (using "delta function of function" formula): (2) Now, bilateral filter is exactly expression (2) divided by expression (1). Paris-Durand paper derives a similar result (based on intuition) and a speed up algorithm.

Bilateral

quotesdbs_dbs17.pdfusesText_23

[PDF] [PDF] Covariant Derivatives and Vision - Todor Georgiev

Covariant Derivatives and Vision

Todor Georgiev

Adobe Photoshop

Presentation at ECCV 2006

Original

Selection to clone

Poisson cloning from dark area

Selection to clone

Poisson cloning from illuminated area

Poisson

Covariant

Poisson cloning can be viewed as an

Outline of our theory:

Thanks to Jan Koenderink

We can not compare pixels.

Retina / Cortex Adaptation

Models of Image Space:

Total space E

Base space B

Sections

Image = graph of a section

Section in a Fiber Bundle

Derivatives in a Fiber Bundle

Definition:

Derivative is a mapping from one section to

In the Cartesian product space this definition is

Derivatives in a Fiber Bundle

Derivatives in a Fiber Bundle

The functions are called "color channels" in

It corresponds to the derivative in the Cartesian

How can we know A?

This means complete adaptation:

Substitute in covariant Laplace:

Covariant

Poisson equation

Poisson

Covariant

PoissonCovariant

Detailed Example:

Original Damaged Area

Laplace

Poisson

Laplace

Inpainting

Thanks to Guillermo Sapiro and Kedar Patwardhan

Poisson

Covariant

Inpainting

Thanks to Guillermo Sapiro and Kedar Patwardhan

Structure and Texture Inpainting

Bertalmio - Vese - Sapiro - Osher

Covariant Inpainting

Covariant cloning from day

Poisson cloning from day

Simplest energy invariant to those transforms is:

Werman paper. They assume log; we derive log.

Any good visual system needs to be logarithmic!

HDR compression, surface reconstruction from

Appendix:

Same procedure on the logarithmic expression

Bilateral