1. Efficient Local Statistical Analysis via Integral Histograms with Discrete Wavelet Transform Teng-Yok Lee & Han-Wei Shen IEEE SciVis ’13Uncertainty & Multivariate Analysis

2. 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 2

3. (x 0, y 0 ) + – – (x 1, y 0 ) (x 0, y 1 )(x 1, y 1 ) 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 (x 0, y 0 ) and (x 1, y 1 ) : I(x 1, y 1 ) – I(x 1, y 0 ) – I(x 0, y 1 ) + I(x 0, y 0 ) + (x 0, y 0 ) F. Porikli, Integral histogram: a fast way to extract histograms in Cartesian spaces. In CVPR ‘05. 3

4. 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 4

5. 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 5

6. 1 11 111 11 11 From Integral Histograms to Bin SATs 6 1 11 111 11 11 0 1 0 0 0 1 1 0 0 0 1 2 0 0 0 1 2 0 0 1 1 2 0 1 1 1 2 1 1 1 1 2 2 1 1 1 2 3 1 1 1 2 3 2 1 1 2 3 2 2 11 0000111122 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 Histogram Bin

7. Bin SAT: Monotonically Increasing and Smooth 7 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

8. 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 8 11 111111 1 11 111 11 1 Input function The bins SAT for bin 2 = Sum of step functions s x where x’s value is in bin 2 0000111122 11 1

9. 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 9 Wavelet function: A windowed function to compute local difference (Wavelet Coefficient) ++–– 0000011111111111 ++––++––++–– Before the edge: Wavelet Coef = 0 After the edgie: Wavelet Coef = 0 Cover the edge: Wavelet Coef ≠ 0 ++––

10. 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 10 For each point Find the corresponding bin B Update the O(log N) non-zero wavelet coefficients for bin B

11. 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 11 For each point Find the corresponding bin B Update the O(log N) non-0 wavelet coefficients for bin B

12. Query of Integral Histograms 12 ++++++++ ++++–––– ++–– ++–– +– +– +– +– × w 0 × w 1 × w 2 × w 3 × w 4 × w 5 × w 6 × w 7 Bin SAT + + + + + + + = Wavelet FunctionsWavelet 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

13. Optimization for Region Histogram Query 13 ++++++++ ++++–––– ++–– ++–– +– +– +– +– × w 0 × w 1 × w 2 × w 3 × w 4 × w 5 × w 6 × w 7 + + + + + + + Wavelet FunctionsWavelet Coef for bins w0w0 w1w1 w2 w2 w3w3 w4w4 w5w5 w6w6 w7w7 … … … … … … … … 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.

14. 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 14 A 2D ArrayDWT along the rowDWT along the column

15. 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 15 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)

16. Result: Query Time 16 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 MJO Blue curves: Integral histograms in core. Red Curves: WaveletSAT Black Curves: With the optimization for region histograms

17. 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 17

18. 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-sat Questions? 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. 18