A = JopDct(T, n...[;dims=()])

where A is a type II discrete cosine transfrom (DCT), T is the element type of the domain and range if A, and n is the size of A. If dims is not set, then A is the N-dimensional DCT and where N=length(n). Otherwise, A is a DCT over the dimensions specified by dims.

Examples

1D

A = JopDct(Float64, 128)
m = rand(domain(A))
d = A*m

2D

A = JopDct(Float64, 128, 64)
m = rand(domain(A))
d = A*m

2D transform over 3D array

A = JopDct(Float64,128,64,32;dims=(1,2))
m = rand(domain(A))
d = A*m

Notes

• the adjoint pair is achieved by application sqrt(2/N) and sqrt(2) factors. See:
  • https://en.wikipedia.org/wiki/Discretecosinetransform
  • http://www.fftw.org/fftw3doc/1d-Real002deven-DFTs-0028DCTs0029.html#gt1d-Real002deven-DFTs-0028DCTs0029

A = JopDwt(T, n...[, wt=wavelet(WT.haar, WT.Lifting)])

A is the N-D (N=1,2 or 3) wavelet transform operator, operating on the domain of type T and dimensions n. The optional argument wt is the wavelet type. The supported wavelet types are WT.haar and WT.db (Daubechies).

Notes

• For 2 and 3 dimensional transforms the domain must be square. In other-words, size(dom,1)==size(dom,2)==size(dom,3).
• If your domain is rectangular, please consider using 1D wavelet transforms in combination wih the kronecker product (JopKron).
• The wavelet transform is provide by the Wavelets.jl package: http://github.com/JuliaDSP/Wavelets.jl
• You may try other wavelet classes supported by Wavelets.jl; however, these have yet to be tested for correctness with respect to the

transpose.

Example

A = JopDwt(Float64, 64, 64; wt=wavelet(WT.db5))
d = A*rand(domain(A))

A = JopFft(T, n...[; dims=()])

where A is a Fourier transform, the domain of A is (T,n). If dims is not set, then A is an N-dimensional Fourier transform and where N=ndims(dom). Otherwise, A is a Fourier transform over the dimensions specifed by dims.

For a complex-to-complex transform: range(A)::JetSpace. For a real-to-complex transform: range(A)::JetSpaceSymmetric. Normally, JopFft is responsible for constructing range(A). In the event that you need to manually construct this space, we provide a convenience method:

R = symspace(JopLn{JopFft_df!}, T, symdim, n...)

where T is the precision (generally Float32 or Float64), n... is the dimensions of the domain of A, and symdim is the first dimension begin Fourier transformed.

Examples

1D, real to complex

A = JopFft(Float64,128)
m = rand(domain(A))
d = A*m

1D, complex to complex

A = JopFft(ComplexF64,128)
m = rand(domain(A))
d = A*m

2D, real to complex

A = JopFft(Float64,128,256)
m = rand(domain(A))
d = A*m

3D, real to complex, transforming over only the first dimension

A = JopFft(Float64,128,256; dims=(1,))
m = rand(domain(A))
d = A*m

op = JopSft(dom,freqs,dt)

Polychromatic slow Fourier transforms for a specified list of frequencies. Tranform along the fast dimension of dom::JetSpace{<:Real}.

Notes: - it is expected that the domain is 2D real of size [nt,ntrace] - the range will be 2D complex of size [length(freqs),ntrace] - regarding the factor of (2/n): the factor "1/n" is from the Fourier transform, and the "2" is from Hermittian symmetry.

A = JopSlantStack(dom[; dz=10.0, dh=10.0, h0=-1000.0, ...])

where A is the 2D slant-stack operator mapping for z-h to tau-p. The domain of the operator is nz x nh with precision T, dz is the depth spacing, dh is the half-offset spacing, and h0 is the origin of the half-offset axis. The additional named optional arguments along with their default values are,

• theta=-60:1.0:60 - range of opening angles. The ray parameter is: p=sin(theta/2)/c
• padz=0.0,padh=0.0 - padding in depth and offset to apply before applying the Fourier transfrom
• taperz=(0,0) - beginning and end taper in the z-direction before transforming from z-h to kz-kh
• taperh=(0,0) - beginning and end taper in the h-direction before transforming from z-h to kz-kh
• taperkz=(0,0) - beginning and end taper in the kz-direction before transforming from kz-kh to z-h
• taperkh=(0,0) - beginning and end taper in the kh-direction before transforming from kz-kh to z-h

Notes

• It should be possible to extend this operator to 3D

• If your domain is t-h rather than z-h, you can still use this operator