Efficient Local Statistical Analysis via Integral Histograms with Discrete Wavelet Transform

Teng-Yok Lee & Han-Wei Shen

IEEE SciVis ’13 Uncertainty & Multivariate Analysis

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Local distributions (region histograms) are widely used…

Example 1: Transfer Function Design

Example 2:Local vector field analysis

Example 3:Time-varying data overview

… but computing region histograms for arbitrary sizes is not always efficient

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(x0, y0)+

–

–(x1, y0)

(x0, y1) (x1, y1)

Integral Histograms: A Solution in Image Processing

I(x, y): Integral histogram of (x, y)The histogram of the region bounded

by (0, 0) and (x, y)

(x, y)

(0, 0)

Histogram of the region bounded by (x0, y0) and (x1, y1) :

I(x1, y1) – I(x1, y0) – I(x0, y1) + I(x0, y0)

+

(x0, y0)

F. Porikli, Integral histogram: a fast way to extract histograms in Cartesian spaces. In CVPR ‘05.

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Integral Histograms: Properties

• Widely used in image processing – Easy to implement – Performance: A region histogram can be computed by combining 4

integral histograms

• Storage-challenging – Each grid point is associated with one histogram, implying that the

data is magnified by the number of histogram bins– Especially true for large images, videos, and 3D volumes

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Our Solution: WaveletSAT

• WaveletSAT: Efficient integral histogram compression with Discrete Wavelet Transform (DWT)

• Contributions Efficient region histogram query with limited storage overhead A single shot algorithm: No need to build the integral histograms and

then apply DWT Efficient compression: limited memory footprints & easy to parallelize

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From Integral Histograms to Bin SATs

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The bin values of integral histogram at x is the sum from the left to x in the binary function

A 1D Example

Integral histograms

Bin SAT: The SAT formed by the values of histogram bin b of all integral histograms

Hist

ogra

m B

inHi

stog

ram

Bin

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Bin SAT: Monotonically Increasing and Smooth

The image Mandrill.Lots of high frequency details.

The bin SATs of its integral histograms of 32 bins. The bin SATs are smooth.

BinSATs can be transformed to sparse coefficients via FFT/DCT/DWT But … Need to process all bin SATs

Require all data points

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Bin SATs & Step Functions• A bin SAT = sum of step functions • Only part of grid points contribute to a bin SAT

• For DWT, DCT, or FFT– No need to wait for all points– The transform of bin SAT = Sum of the transform of these step functions

1 1

1 1 1 1 1 11

1 1

1 1 1

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Input function

The bins SAT for bin 2 = Sum of step functions sx where x’s value is in bin 2

0 0 0 0 1 1 1 1 2 2

1 1

1

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Efficient Wavelet Transform for Step Functions• With DWT, each step function can be efficiently transformed

• DWT: Computing the local difference with wavelet functions at different scales

Only the wavelet that covers the edge has a non-0 coefficient

Wavelet function: A windowed function to compute local difference

(Wavelet Coefficient)

+ + – –

0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1

+ + – – + + – – + + – –

Before the edge: Wavelet Coef = 0

After the edgie: Wavelet Coef = 0

Cover the edge: Wavelet Coef ≠ 0

+ + – –

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WaveletSAT: Algorithm

• As DWT has O(log N) scales, give a step function– Each scale has only 1 non-zero wavelet coefficient– This step function is transformed to O(log N) non-zero coefficients

• WaveletSAT algorithm– Input: An 1D array of N points– Output: Wavelet coefficients for all bin SATs

For each pointFind the corresponding bin BUpdate the O(log N) non-zero wavelet coefficients for bin B

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WaveletSAT: Benefits• As each point only contributes to a limited number of bins, the time complexity

for N points is O(N log N)

• The complexity is independent to the number of bins

• Each step function can be transformed separately & out-of-order– Easy to parallelize– No need to pre-compute the integral histograms

For each pointFind the corresponding bin BUpdate the O(log N) non-0 wavelet coefficients for bin B

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Query of Integral Histograms

+ + + + + + + +

+ + + + – – – –

+ + – –

+ + – –

+ –

+ –

+ –

+ –

× w0

× w1

× w2

× w3

× w4

× w5

× w6

× w7

Bin SAT

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=

Wavelet Functions Wavelet Coef.Reconstruction of bin SATs via Inverse DWT.

Inverse DWT: Linear combination of wavelet functions with their wavelet coefficients

When query a single integral histogram at x, only its bin value at x is needed

The wavelet functions that do not cover x can be discarded

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Optimization for Region Histogram Query

+ + + + + + + +

+ + + + – – – –

+ + – –

+ + – –

+ –

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× w0

× w1

× w2

× w3

× w4

× w5

× w6

× w7

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Wavelet Functions Wavelet Coef for bins

w0

w1

w2

w3

w4

w5

w6

w7

…

…

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……

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…

Performance issue: The reconstruction is needed for all bins.

Recall: A region histogram is the combination of multiple integral histograms.

These integral histograms can share the same wavelet functions and coefficients.

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WaveletSAT for High Dimensional Data

• Now a Bin SAT is the sum of multiple D-dimensional step function• For each step function, sequentially apply DWT to all dimensions • A D-dimensional step function has O(log N D) non-zero wavelet coefficients

A 2D Array DWT along the row DWT along the column

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Result: Encoding Time & Compression Rates• Comparison: ZIP compression for bin SATs• Encoding time (lower is better)

– WaveletSAT is more efficient when #bins increases – GPUs bring 4 – 6 time speed up

• Compression rate (CR, higher is better)– WaveletSAT achieves higher CR when #bins increases– CR of WaveletSAT can be further boosted by ZIP

Blue curves: Integral histograms with different zip levels. Red Curves: WaveletSAT (◊: CPU; □: GPU)

A 2D slice of dataset Ocean

Blue curves: Integral histograms with different zip levels. Red Curves: WaveletSAT (□: w/o ZIP; ◊: w/ ZIP)

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Result: Query Time

• If integral histograms can be fully loaded into the memory – Faster than WaveletSAT, but getting slower when #bins is increasing– Not doable for larger datasets

• The optimization of region histogram query reduces the performance gap

A 2D slice of dataset MJOBlue curves: Integral histograms in core. Red Curves: WaveletSAT

Black Curves: With the optimization for region histograms

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Summary

• WaveletSAT: An efficient algorithm for region histogram query Efficient in terms of encoding, storage, & reconstruction

• Future works– Utilize WaveletSAT to decide the scale of salient features– Parallelize WaveletSAT for distributed environments

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Acknowledgements

• Dataset sources– Ocean: M. Maltrud at Los Alamos National Laboratory – MJO: S. Hagos & R. L. Leung at Pacific Northwest National Laboratory– 3D volumes: The repo maintained by C. Scheidegger et al.– Mandrill: ?

• Source code: https://code.google.com/p/wavelet-satQuestions?

This work was supported in part by NSF grant IIS-1017635, NSF grant IIS-1065025, US Department of Energy OESC0005036, Battelle Contract No. 137365, and Department of Energy SciDAC grant DE-FC02-06ER25779, program manager Lucy Nowell.