Mutual Information-based Stereo Matching Combined with SIFT Descriptor in Log-chromaticity Color Space

Yong Seok Heo, Kyoung Mu Lee, and Sang Uk LeeDepartment of EECS, ASRI, Seoul National University, 151-742, Seoul, Korea

IEEE Conference on Computer Vision and Pattern Recognition, 2009.

Outline• Introduction

• System Overview

• Mutual Information as a Stereo Correspondence Measure

• Proposed Algorithm

• Experiments

• Conclusion

Introduction

Introduction• Radiometric variations (between two input images)

• Degrade the performance of stereo matching algorithms.

• Mutual Information :• Powerful measure which can find any global

relationship of intensities• Erroneous as regards local radiometric variations

Introduction• Different camera exposures (global)

• Different light configurations (local) Conventional MI

Conventional MI

Objective• To present a new method:• Based on mutual information combined with

SIFT descriptor• Superior to the state-of-the art algorithms

(conventional mutual information-based)

Mutual Information

global radiometric variations (camera exposure)

SIFT descriptor

local radiometric variations (light configuration)

+

System Overview

Proposed Alogorithm

Mutual Information (as a Stereo Correspondence Measure)

Energy Function• Energy Function:

• In MAP-MRF framework

• f: disparity map

Mutual Information• Used as a data cost:

•Measures the mutual dependence of the two random variables

Disparity Map

Left / RightImage

Mutual Information

• Entropy:

• Joint Entropy:

• P(i): marginal probability of intensity i

• P(iL,iR): joint probability of intensity iL and iR

Entropy Entropy Joint Entropy

1

0

1

0

1

0

Mutual InformationSuppose you are reporting the result of rolling a

fair eight-sided die. What is the entropy?

→The probability distribution is f (x) = 1/8, for x =1··8 , Therefore entropy is:

= 8(1/8)log 8 = 3 bits

Mutual Information

H( IL, IR )

H( IL ) H( IR )

H( IL | IR ) H( IR | IL )MI( IL; IR )

Entropy Entropy Joint Entropy

Mutual Information

• Entropy:

• Joint Entropy:

• P(i): marginal probability of intensity i

• P(iL,iR): joint probability of intensity iL and iR

Entropy Entropy Joint Entropy

1

0

1

0

1

0

i1 i2

Pixel-wise Mutual Information• Previous: Mutual Information of whole images• Difficult to use it as a data cost in an energy

minimization framework → Pixel-wise

Pixel-wise Mutual Information

=

P(.) / P(., .) : marginal / joint probability

G(.) / G(., .) : 1D / 2D Gaussian function

Left Image Intensity

Right Image Intensity

Conventional MI• Cannot handle the local radiometric variations

caused by light configuration change

• Collect correspondences in the joint probability assuming that there is a global transformation

• The shape of the corresponding joint probability is very sparse.

• Do not encode spatial information

Conventional MI• Different camera exposures (global)

• Different light configurations (local)Conventional MI

Conventional MI

Proposed Algorithm

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Log-chromaticity Color Space• Transform the input color images to log-

chromaticity color space [5]

• To deal with local as well as global radiometric variations

• Used to establish a linear relationship between color values of input images

[5] Y. S. Heo, K.M. Lee, and S. U. Lee. Illumination and camera invariant stereo matching. In Proc. of CVPR, 2008

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SIFT Descriptor• Robust and accurately depicts local gradient

information

• Computed for every pixel in the log-chromaticity color space

Energy Function• Data Cost:

• Mutual Information:

• SIFT descriptor distance:

( )constant

Log-chromaticity intensity

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Joint Probability Using SIFT DescriptorA joint probability is computed at each channel

by use of the estimated disparity map from the previous iteration.

Wrong disparity can induce an incorrect joint probability.

Incorporate the spatial information in the joint probability computation step

Adopt the SIFT descriptor

Joint Probability Using SIFT Descriptor

• K-channel SIFT-weighted joint probability:

• 　 : Euclidean distance• VL,K(P) / VR,K(P) : SIFT descriptors for the pixel P• l : SIFT descriptor size

T = 1 If TrueT = 0 If false

i1 i2

Joint Probability Using SIFT Descriptor• A joint probability is computed at each channel • Use estimated disparity map from the previous

iteration.

• is governed by the constraint that corresponding pixels should have similar gradient structures.

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Energy Function• Data Cost:

• Smooth Cost:

MI SIFT

Energy Minimization• The total energy is minimized by the Graph-cuts

expansion algorithm[3].

[3] Y. Boykov, O. Veksler, and R. Zabih. Fast approximate energy minimization via graph cuts. IEEE Trans. PAMI, 23(11):1222–1239, 2001.

Energy Minimization

Experiments

Experimental Results• The std. dev σ of the Gaussian function is 10,

τ = 30, l = 4*4*8 = 128

• The window size of the SIFT descriptor : 9X9

• λ = 0.1, μ = 1.1, VMAX=5

• The total running time of our method for most images does not exceed 8 minutes.

• Aloe image (size : 427 X 370 / disparity range : 0-70) is about 6 minutes on a PC with PENTIUM-4 2.4GHz CPU.

MI vs. SIFT

MI SIFT MI + SIFT

17.6% 11.97 % 9.27 %

26.45% 17.87 % 11.83 %

L: illum(1)-exp(1) / R: illum(3)-exp(1)

MI SIFT MI + SIFT

Error Rate

Error Rate

L: illum(1)-exp(1) / R: illum(3)-exp(1)

Different ExposureLeft Image Right Image Ground Truth Proposed

Rank/BT NCC ANCC MI

Different Light Source ConfigurationsLeft Image Right Image Ground Truth Proposed

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Rank/BT NCC ANCC MI

Different ExposureLeft Image Right Image Ground Truth Proposed

111

Rank/BT NCC ANCC MI

Different Light Source ConfigurationsLeft Image Right Image Ground Truth Proposed

111

Rank/BT NCC ANCC MI

Different ExposureLeft Image Right Image Ground Truth Proposed

111

Rank/BT NCC ANCC MI

Different Light Source ConfigurationsLeft Image Right Image Ground Truth Proposed

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Rank/BT NCC ANCC MI

Exposure Exposure

Light Configuration Light Configuration

Exposure Exposure

Light Configuration Light Configuration

Conclusion

Conclusion• Propose a new stereo matching algorithm

based on :• mutual information (MI) combined with• SIFT descriptor

• Quite robust and accurate to local as well as global radiometric variations