DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF JOENSUU JOENSUU, FINLAND Image Compression Lecture 9 Optimal Scalar Quantization Alexander Kolesnikov

Quantization error for discrete data X C2C2 C1C1 CMCM... Quantization error for the data X with M cells: Cell’s centroids:

Optimal scalar quantization: History J.D.Bruce [1963]: O(MN 2 ) DP X. Wu [1991]: O(MN 2 )  O(MN) DP+Monge X,Y,Z [2001]: O(N M )  O(N M-1 ) Exhaustive search in paper ”A fast algorithm for multilevel thresholding” Some researchers still believe that complexity of the optimal algorithm is exponential, … and some researchers are still re-inventing the optimal DP algorithm of O(MN 2 ) complexity.

Problem formulation Let X={x 1, x 2, …, x N } be a finite ordered set of real numbers ( intensity values ). Let P={p 1, p 2, …, p N } be the correspondent set of a probabilities for the values X (histogram). Let {r 0,r 1,r 2, …,r M+1 } be an ordered set of integers such that defines a partition of the set X into M parts: r 0 = 0 < r 1 <... < r j < r j+1 <... < r M = N. x 1 x 2 xNxN p1p1 p2p2

Sequence partition problem Quantization error for one (part) cell: Cell’s centroid y j : Partition indices: r 0 = 0 < r 1 <... < r j <... < r M =N. We introduced r 0 = 0 for x 0 = . The total quantization error:

Scheme of partition into cells Quantization error for the cell #2 (j=2): Data: x 0 =  < x 1 <... < x j <... < x N Partition indices: r 0 = 0 < r 1 <... < r j <... < r M =N. Cells:... ]( ] ( #1 #2 #3 (x 0 =  ) x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 14 x N (r 0 =0) r 1 =4 r 2 =10 r M =N=15 N=15, the number of cells M=3

Optimization task For a given data X, probabilities P and number of cells M find such a partition {r o,r 1, r 2, …, r M } that the total quantization error is minimal: where and

Cost function D M (0,N] Let us introduce cost function D m (0,n] that is minimum quantization error for quantization of data sub-set X n ={x 1, x 2, …, x n } with m cells: Then D M (0,N] gives us solution of the problem in question.

Dynamic programming approach In other words: Let’s rewrite the cost function:

Reccurent equations Initialization: Recursion:

Calculation of quantization error for a cell Complexity of the quantization error calculation is O(N) Can we calculate it faster?

Calculation of quantization error for a cell

where cumulants S 0 (n), S 1 (n), S 2 (n) are defined as follows: Complexity of quantization error calculation for one cell is O(1).

DP search in the state space  M N1 0 b n m j State space  C(n,m) C(j,m-1) C(N,M) Start state e 2 (j,n ) C(n,m) = min {C(m, m-1) + e 2 (m, n], C(m+1,m-1) + e 2 (m+1,n],... C(j,m-1) + e 2 (j, n]* ),... C(n-1, m-1) + e 2 (n, n]} m-1 * A(n,m)=j opt

Scheme of the DP algorithm // Initialization FOR n = 1 TO N DO C(n,1)= e 2 (j,n] // Minimum search FOR m = 2 TO M DO FOR n = m TO N DO d min   FOR j= m-1 TO n-1 DO c  C(j, m  1) + e 2 (j,n] IF(c < c min ) c min  c; j min  j ENDIF ENDFOR C(n, m)  d min A(n, m)  j min ENDFOR Complexity: O(MN 2 ) C(n,m)=D m (0,n]

Backtrack in the state space  M N1 0 b n m j State space  A(N,M) Start state m-1 S(M+1)= N FOR m = M+1 TO 2 DO S(m  1) = A(S(m), m)) E 2  C(N,M) S(M+1)= N FOR m = M+1 TO 2 DO S(m  1) = A(S(m), m)) E 2  C(N,M) N=22, M=8: S={22,18,14,12,9,6,4,3,1} (x 0,x 3 ], (x 3,x 4 ], (x 4,x 6 ], (x 6,x 9 ], (x 9,x 12 ], (x 12,x 14 ], (x 14,x 18 ], (x 18,x 22 ]

Optimal scalar quantization DP algorithm a) Time complexity: O(MN 2 ) b) Space complexity: O(MN) Error balance property: e 2 (r j-1,r j ]  Const Optimal scalar quantizer as weighted k -link shortest path in directed acyclic graph (DAG) Wu [1991] reduced time complexity of optimal DP algorithm to O(MN) using Monge property of quantization error with L 2 metrics. Space complexity: O(N 2 ).

Can we do it faster? Wu [1991] reduced time complexity of optimal DP algorithm to O(MN) using Monge property (monotonicity property) of quantization error with L 2 metrics. Space complexity: O(N 2 ).

Example 1: M=3 Input image Uniform Optimal

Example 1: M=3 Uniform Optimal

Example 2: M=5 Optimal Uniform

Example: M=5 Uniform Optimal

Example: M=12 Centroid dencity is higher when probability dencity is also higher... Centroid dencity is higher when probability dencity is also higher...

Summary 1) Optimal scalar quantization