Object Specific Compressed Sensing by minimizing a weighted L2-norm

1
Object Specific Compressed Sensing by minimizing
a weighted L2-norm

• A. Mahalanobis

2
Background

• Lockheed Martin has been working on the DARPA ISP
  program
• Team includes Duke, JHU, Yale and NAVAIR
• An adaptive sensing scheme has been developed
  that allocates sensor resources (spectral and
  spatial) based on relevant information content
• Algorithms are currently working in a coded
  aperture hyperspectral imager hardware
• Compressed Sensing is a natural extension of this
  ISP concept

3
Motivation

• Can we create an efficient sensing process where
  objects of interest are well resolved, but other
  parts of the scene are heavily compressed?
• Economize on number of data measurements required
  and the computations needed to reconstruct the
  image
• Currently, Compressed Sensing is focused on the
  general reconstruction problem
• We are not interested in the perfect
  reconstruction of the whole scene
• Our approach embeds pattern recognition
  objectives (detection, discrimination) and
  compression in the sensing process, while
  producing visually meaningful images.

4
Approach

• It has been shown that under certain conditions,
  minimizing the L-1 Norm yields the optimum
  solution for perfect reconstruction, but the
  optimization requires iterative (potentially
  cumbersome) techniques
• L-2 norm techniques are well known, analytical
  closed form solutions that are easy to implement
• Computationally attractive for the formation of
  large images
• However, the minimum L2 norm solution does not
  yeild good reconstruction
• Can a weighted L2-norm arrive close to the
  optimum solution when we are interested in
  specific objects ?
• How can we incorporate prior knowledge about the
  objects ?

5
The general solution

• Assume that the image vector y can be represented
  as linear combination of basis vectors (columns
  of the matrix A) such that
• h is the coefficient vector we seek to estimate
  from a small number of measurements, and hence
  re-construct y
• In compressed sensing, we measure a smaller
  vector u, (i.e. the projection of the image y
  through a random mask W)
• The most general family of solution for the
  estimate h that satisfies the above linear
  constraints is
• All solutions (including those which minimize the
  L-0, L-1 or L-2 norm) belong to this family
• The particular solution is the minimum L-2 norm
  solution
• The homogeneous solution can be viewed as a
  correction to the L-2 norm that results in other
  solutions with different properties

A random vector
Particular solution
Homogeneous Solution
6
Weighted L-2 norm

• Minimizing the L-2 norm does not relate to a
  well-defined information metric for
  reconstruction
• It minimizes the variation in the estimate when
  white noise is present in the measurement
• Rather, we seek a weighting that minimizes the
  L-2 norm of the coefficient vector while
  maximizing information about the objects of
  interest
• This results in attenuation of those weights
  which do not bear useful information for
  reconstruction
• Or maximize
• This implies that the best choice for the weights
  is
• We envision that can be calculated
  apriori from a set of representative images of
  the class of objects of interest, or a suitable
  statistical model may be used.

7
Solution using the methods of Lagrange
multipliers

• Problem is stated as
• Minimize the quadratic subject to the linear
  constraints
• D is a diagonal matrix whose diagonal elements
  are calculated apriori from a set of
  representative images or a statistical model
• The well known solution for the estimate of the
  coefficient vector is now
• h is estimate of the coefficients based on the
  measurements u
• A is a matrix that can be used as a basis to
  represent the image
• W is a random matrix on which the image is
  projected to obtain u
• D is a weight vector that maximizes information
  for the objects of interest

8
Reconstruction Equation

• The Reconstructed Image is given by
• where depends on the
  basis functions and the weights
• Without weights, R I, and the solution does not
  depends on the underlying basis set
• Minimum L-2 norm solution is then simply
• We will use i) DCT and ii) KL basis sets to
  demonstrate performance
• For the KL basis set, D is the same as the
  eigen-values

9
Example using ideal weights
WEIGTED L2 norm
K256 mse0.19
Original 32 x 32 image
K192 mse0.25

• Original image is 32 x 32 (1024 elements)
• DCT is used as a basis set
• Any other basis set that allows compact
  representation can be used
• ideal coefficients are used as a place-holder
  for weights
• In practice, these will be estimated
  representative images of the class of objects of
  interest, or statistically modeled.
• Weighted L2 norm produces recognizable results
  using 1/4th the data (256 measurements)
• Conventional L2 norm does not perform well

K64 mse0.5
10
DCT Basis Set and Weights
A
(as a 2D image)

• The DCT of the image shows good compaction
  properties.
• Indicates it should be possible to achieve nearly
  zero mse with only 50 of the coefficients
• Other basis sets should yield much greater
  compactness

11
Example 2 weights estimated for a class
Object
DCT of Object
Average DCT

• The goal is to sense all objects that belong to a
  class
• Exact weights for any one image is not known, but
  an average estimate for the class is used
• The average DCT is estimated using 1600
  representative views and the inverse of the DCT
  coefficients is used weights in the
  reconstruction process

12
Weighted vs. Conventional approach using DCT basis

• Comparison of conventional and weighted minimum
  L2 norm reconstruction using the DCT basis
  functions. Weighting the reconstruction process
  makes a significant difference in the
  reconstruction error

13
Reconstruction based on DCT with and without
weights
(a) Weighted
(b) Unweighted

• Reconstructions using 512 projections and the DCT
  basis set with weighting estimated over the class
  shows better performance than without weighting,
  i.e. the conventional minimum L2 norm solution

14
Using the K-L Basis set

• The weights are the reciprocal of the square-root
  of the eigen-values of the auto-correlation
  matrix estimated using representative images of
  the class of vehicles of interest.
• Only M450 basis functions are necessary for
  accurately representing the images, which reduces
  the size of the matrix R and hence the overall
  computations

15
Weighted vs. Conventional approach using KL basis

• Reconstruction using the KL basis far
  out-performs DCT when weights are used
• Performance of unweighted scheme is comparable to
  the unweighted DCT (not surprising)

16
Other Computational advantages of the KL set
(a) Esimated using 256 measurements and 450
eigen-vectors
(b) Esimated using 256 measurements and All 1024
eigen-vectors

• KL transform offers computational advantages in
  Two ways
• Fewer measurements are necessary (reduces the
  number of rows of R)
• Fewer basis functions as required to represent
  the image (reduces the number of columns of R)
• Image on the left was reconstructed using the
  first 450 eigen-vectors of the KL decomposition,
  whereas all 1024 were used on the right. The two
  images are almost identical, although the image
  in (a) requires considerably less computations.

17
Example of full scene reconstruction (back to DCT)
Original Image

• L2-norm approach easily reconstructs large scene
• Computationally straightforward
• Weighted optimization clearly demonstrates
  ability to heavily compress uninteresting regions
  of the scene, while achieving reasonable
  reconstruction where true objects are present

18
Summary

• Minimizing the L-2 norm is a viable way of
  reconstructing objects of interest in a
  compressed sensing scheme
• Requires prior knowledge of the weights that are
  representative of the class of objects
• Embeds attributes of pattern recognition in the
  sensing process to preserve visual detail for the
  human user, while effectively achieving
  detection, discrimination and compression
• Selection of basis set is important
• Good basis sets require fewer measurements and
  fewer terms in the representation which speeds up
  the computations.
• The selection of basis sets and criterion for
  choosing weights both require further research