Section $2.$ Hadamard transform and beyond

The Hadamard is an example of a generalized class of Fourier transforms. It performs

an orthogonal, symmetric, involutive, linear operation. The Hadamard transform $H_m$ is a $2^m{2}^m$ matrix,

that transforms $2^m$ real numbers into $2^m$ real numbers. The Hadamard transform can be defined recursively:

$H_0=(1),$ and $H_m=\frac{1}{\sqrt[\phantom{2}]{2}}([H_{m-1},H_{m-1}],[H_{m-1},-H_{m-1}])$

The Hadamard transformation $($for both forward and inverse transformation$)$ for blocks of size $22$

is then defined as follows: $H_1=\frac{1}{\sqrt[\phantom{2}]{2}}([1,1],[1,-1])$

Calculate the Hadamard transform coefficients $F(u,v)$ of the $2x2$ pixel block $f(x,y)$; and the inverse

transformation. $f(x,y)=([1,2],[2,1])$

Recall. $F(u,v)=H_1,f(x,y)*H_1^t$

Let $g(x,y)$ denote the following constant $44$ digital image that is zero outside $0x3,0y$

$3,$ with $r$ a constant value.

Give the standard Hadamard Transform of $g(x,y)$ without carrying out any mathematical

manipulations.

$g(x,y)=([r,r,r,r],[r,r,r,r],[r,r,r,r],[r,r,r,r])$

Let $h(x,y),k(x,y)$ and $l(x,y)$ denote the following constant $44$ digital images that are zero

outside $0x3,0y3,$ with $r$ a constant value.

$h(x,y)=([r,r,r,r],[r,r,r,r],[0,0,0,0],[0,0,0,0]),k(x,y)=([r,r,0,0],[r,r,0,0],[r,r,0,0],[r,r,0,0]),l(x,y)=([2r,2r,r,r],[2r,2r,r,r],[r,r,0,0],[r,r,0,0])$

Give the standard Hadamard transform of $h(x,y),k(x,y)$ and $1(x,y).$

Can you easily deduce the Hadamard transform of $k(x,y)$ from the one of $h(x,y)?$ the Hadamard

transform of $1(x,y)$ from the one of $h(x,y)$ and $k(x,y)?$

Assuming that a codec uses Hadamard transform and that the encoder sends to the decoder a limited

number of coefficients $($non$-$transmitted$/$received coefficients are set to zero by the decoder$),$ what

will the decoder display for $1(x,y)$ if the encoder sends:

a$.1$ coefficient only $($out of $16)$: the top left one;

b$.9$ coefficients only $($out of $16)$: the top left ones $(0u2,0v2).$

$($if time permits$)$ Assuming that resulting coefficients are all kept and sent by the encoder to the

decoder but quantized like it is implemented in JPEG using the following quantization table:

$([5,15,25,35],[15,15,25,35],[25,25,25,35],[35,35,35,45])$

Which picture the decoder will display for $1(x,y)$ if $r$ equal to $5?$