study streaming multigrid for gradient domain operations on large images

Streaming Multigrid for Gradient-Domain Operations on Large

ImagesMichael Kazhdan, Johns Hopkins University

Hugues Hoppe, Microsoft Research

SIGGRAPH 08

Abstract

• Develop a streaming multigrid solver – 2 passes over out-of-core data

1. Solver of Poisson eq. with unconstrained boundary conditions

2. Construct Relaxation, Restriction and Prolongation operators in multigrid method based on the B-spline basis

3. Build up a framework to pipeline multigrid with a window of rows of images

Outline

1. Introduction2. Related work3. Review of finite-difference multigrid4. Our finite-element multigrid approach5. Streaming multigrid solver6. Efficient convergence of 2nd-order elements7. Implementation8. Application results9. Conclusions & future work

Outline

Gradient-domain problem as Poisson solution

Solve the problem that

• Find U to minimize

• The Poisson equation

Image processing on gradient-domain

• Lighting removal by zeroing small gradient [Horn 74]

• HDR image tone-mapped by adaptively attenuating luminance gradients [Weiss 01]

• Overlapping images stitched seamlessly by merging gradients [P´erez et al.03; Agarwala et al.04;Levin et al.04]

• Shadow removal by zeroing large luminance gradients in regions of constant chromaticity [Finlayson et al. 02]

• Undesirable reflections removed in flash and ambient image pairs [Agrawal et al. 05]

• Photographic tone management is improved using gradient constraints [Bae et al. 06]

Painting on gradient-domain

• Painting with interactive gradient-domain modeling [McCann & Pollard 08]

• Diffusion curves [Orzan et al. 08]

Large images on gradient-domain

• Processing of large images [Kopf et al. 07]

• GB pixel images are too large to fit main memory

• Direct solution (ex: Cholesky factorization) is impractical

• Iteration techniques (ex: conjugate gradients and multigrid method) are in-efficient– Requiring many iterations over out-of-core data

Standard multigrid V-cycle

Contributions

• A general, accurate and efficient solver for Poisson eq. over out-of-core images

– Sufficient accuracy in 2 V-cycles on GB images• 2nd order B-spline finite elements gets more

accuracy than traditional finite element in multigrid method

– Efficiency on temporally locality• Small moving windows of data in memory• Pipelined V-cycle : 3 Gauss-Seidel relaxations,

restriction and prolongation

Contributions

• On a single CPU core– Solve a 3-channel, 16-MB gradient-domain

problem with an rms error on the order of 10−5 in 15 seconds.

• High ratio of local computation to memory bandwidth– Temporal locality in L1 cache

Outline

Solvers of Poisson equation

• Iterative solvers– Gauss-Seidel and conjugate gradient

• Memory-efficient • Require many iterations

– Reduce # of iterations using multi-resolution pre-conditioners [Gortler and Cohen 95; Szeliski 06]

– Reduce # of iterations using multigrid solvers [Brandt 77; Briggs et al. 00]

– GPU + multigrid [Bolz et al. 03; Goodnight et al. 03; G¨oddeke et al. 08]

Out-of-core• Multigrid are difficult to schedule out-of-core [Toledo 99]

• Solve out-of-core problem on a coarser-resolution grid and then upsample the resulting approximation [Kopf et al. 07a]– Can not maintain robust sharp features

• Adaptive partition to solve Poisson eq.For image stitching, – Poisson eq. in the image seams [Agarwala 07] – Poisson eq. in the context of fluid flow simulation [Losasso et al. 04]– Poisson eq. in the surface reconstruction [Kazhdan et al. 06].

• Our method addresses the general case where1. The Poisson eq. is solved accurately everywhere2. The problem size can not be reduced3. No initial solution guess is available

Outline

Gradient-domain image processing

),(),(),(

),(),(

),(,),(

yxGUyxU

Poisson Equation

iUu ii

x0 x1xi

xi+1xi-1 xi

ui-1uN

ui+1ui

ΔU= f

jijijijiji

jijijijijiji

uuuuuh

,1,1,,1,1

),(),(

,1,1,,1,1,

uuuuuu ji

jijijijijiji

u1,1 u1,2 u1,3

u2,1 u2,2 u2,3

u3,1 u3,2 u3,3

• Gauss-Seidel relaxation on u

Lu = f

(L+D+R) u = f where L = (L+D+R)

D u = f – (L+R) u

u (k) = D -1 (f – (L+R) u(k-1)) = D -1 f + RJ u(k-1)

ki LuLfu ,

Error, residual

• Consider Lu = f

Error ek = u – u(k)

Residual rk = f - L u(k) = L ek

• u (k) = D -1 f + RJ u(k-1)

ek = u – u(k)

= (D -1 f + RJ u) – (D -1 f + RJ u(k-1))

= RJ (u - u(k-1) )

= RJ ek-1

= RJk e0

• e0 depends on RJ and eo

Limits of Gauss-Seidel

ek= RJk e0

• Gauss-Seidel relaxation is smooth but converges slowly on low-freq. components

• Why use coarse grids ? [Briggs et al. 00]– Coarse grids can be used to compute an

improved initial guess for the fine-grid relaxation

– Relaxation on the coarse-grid is much cheaper

coarse-grid correction

1. Relax k times on Lhuh = fh on fine-grid, initial u(0) arbitraryu (k) = D -1 f + RJ u(k-1)

2. Compute the residual of the find-grid

r = Lh uh(k) - fh

3. Restrict the residual to the coarse-grid

rH = R rh where H = 2h, R is the restriction

4. Compute the error on the coarse-grid

LH eH = rH, where LH = R Lh P5. Prolongate (interpolate) the error to the fine-grid

eh= P eH , where P is the prolongation6. Correct the fine-grid solution

uh(k+1) = uh

(k) + eh

6Fine grid

coarse grid

Restriction & Prolongation

Restriction operator• Fine grid coarse grid• Typically using local weighted

averaging

Prolongation operator• Coarse grid fine grid• Typically using bilinear

interpolation

Restriction (1D)

prolongation (1D)

R = PT

2D stencils of the multigrid

2D Restriction 2D Prolongation 2D Relaxation (Laplacian)

1/41/21/4

1/211/2

1/41/24/1

1/161/81/16

1/81/41/8

1/161/816/1

Standard multigrid V-cycle

fl-1=Rll-rl ul=Pl

l-1uPl-1+uR

Llminulmin=flmin

Base solution

Outline

Representing the Poisson equation with 1D B-spline basis

• Solving Poisson eq. reduces to solving Lu = f– L as the N x N matrix with Li,j= < ΔBi(x), Bj(x)> = <>

– f as the vector with fj = < F(x), Bj(x)>

Li,j= <▽Bi(x), －▽ Bj(x)>

Fitting forward-difference gradient constrains

• Gradient t of unknown image v

• Difference of B-spline

2nd order B-spline1D case

2nd order B-spline2D case

2D stencils of the multigrid operators using B-splines

• Prolongation

• Restriction

• Laplacian

Outline

How to pipeline ?• phases of each row

– Al R Al-1 R Al-2 P Al-1 P Al

– Does neighborhood in Laplacian stencil exist ?

Gauss-Seidel Relaxation (A), Restriction (R), Prolongation (P)

Rn-1 Al R Al-1 R Al-2 P Al-1 P Al

Rn Al R Al-1 R Al-2 P Al-1 P

Rn+1 Al R Al-1 R Al-2 P Al-1

Rn-1 Al Al Al

Rn Al Al Al

Rn+1 Al Al Al

Perform 3 times relaxations (A) as a single streaming operations

Temporally blocked relaxation • Maintain a window of rows [i-1, i+2k+1] to perform a skipping, counter-

current relaxation sweep, updating pixels in rows {i+2k-1, i+2k-3, …, i+1}

• Row 4 {11}

– Relaxed twice: Row 3 {02,06}, row 2{03,08}

– Relaxed once: Row 5 {7}, row {10}

Row 2Row 3

Row 4Row 5

The pixel row is memory-resident

i The ith relaxation globally

Perform k=3 times relaxations (A) as a single streaming operations

Full data pipeline for gradient-domain processing

Memory analysis

• Implement the active windows on ulR and fl

as circular memory buffer of images rows

• Window size (w, h) – w: the image width at the coarser level – h: 2k+3 (restriction), 2k+5 (prolongation)

• Memory usage is O(Nx) for an Nx x Ny

Outline

Parameters (n,k,v)

• n-order of the finite element (n-order B-spline)• K times Gauss-Seidel relaxations• v passes of V-cycles

2-order B-spline basis (2, 3, 2)

(n, v)

Plot of the rms and max errors vs. # of multigrid V-cycles

2nd-order element give the fastest convergence !

(n, k=2, v=1)

Parameter selection

• Sufficiently accurate solution with a minimum number of V-cycles– 8-bit channel image processing– Max error < 1/256

(2,5,1): 2nd order B-spline basis, 5 Gauss-Seidel updates with a single-V cycle in all our applications

Outline

Implementation

• Maximizing disk throughput– Larger block (4MB)

transfer to minimize disk latency

– 2X speed improvement• Storing the intermediate

u,f floating-point value on dist at half precision

• Relaxation optimization– leverage the vertical and

horizontal symmetries of the stencil

– Make use of CPU SSE2- 4-vector instructors

• Non-power-of-two image– Padding the input image

to 2lmax-lmin times the coarest level

• Multi-channel image– Stitching requires full-

color gradient– Interleave the per-

channel solutions to reduce the total # of passes

Outline

Environment

• NB– 2.2 GHz Core 2 Duo processor– 4 GB RAM

• Timing– I/O to read the gradient field from disk– Write JPEG-compressed ouput to disk

• Initial solution– u=0 in all experiments

• (2,5,1) 2nd order B-spline basis, 5 Gauss-Seidel updates with a single-V cycle

Image stitching

19,588 x 4,457 (87 MB) panorama form 9 photos

Copy the image

gradients and solving the Poisson eq.

Image stitching

SM: streaming multigrid solverQT: quadtree (AT) solver of Agarwala [07]

Tone-mapping (HDR normal tone)

before after

Tone-mapping (HDR normal tone)

Stream multigraid is the first one to solve Poisson eq in time that is linear on the # of pixels

GB stitching and tone-mapping

May not capture the true scene contrast

GB stitching and tone-mapping

Outline

Conclusion

• Streaming multigrid– Out-of-core tech. for solving large global

linear system• Local access• A few passes of sequential I/O

– 2nd order B-spline finite element formulation is compatible with traditional multigrid

• Efficient accurate solution in a single V-cycle

Future work• Dirichlet boundary condition

– Modify 2D stencils– Construct f

• Soft constraint to match some original image u0

• Weighted minimization

where w(x,y) is a spatially varying 2x2 diagonal matrix that weight difference in x and y independently

• Bilaplacian

Future work

• Reduce disk bandwidth by using compression/decompression of the streamed temporary data

• Parallelization– Many-core CPUs or GPUs for instance by

partitioning image rows

study streaming multigrid for gradient domain operations on large images

r j u u

u i u n h u i

r j u d

x n u i

difference multigrid

streaming multigrid

multigrid bolz

f error e

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