Is it possible to mount a gzip compressed dd image on the fly?

It depends on whether the disk image is a full disk image, or just a partition.

Washing the partition(s)

If the disk is in good working condition, you will get better compression if you wash the empty space on the disk with zeros. If the disk is failing, skip this step.

If you're imaging an entire disk then you will want to wash each of the partitions on the disk.

CAUTION: Be careful, you want to set the of to a file in the mounted partition, NOT THE PARTITION ITSELF!

mkdir image_source
sudo mount /dev/sda1 image_source
dd if=/dev/zero of=image_source/wash.tmp bs=4M
rm image_source/wash.tmp
sudo umount image_source

Making a Partition Image

mkdir image
sudo dd if=/dev/sda1 of=image/sda1_backup.img bs=4M

Where sda is the name of the device, and 1 is the partition number. Adjust accordingly for your system if you want to image a different device or partition.

Making a Whole Disk Image

mkdir image
sudo dd if=/dev/sda of=image/sda_backup.img bs=4M

Where sda is the name of the device. Adjust accordingly for your system if you want to image a different device.

Compression

Make a "squashfs" image that contains the full uncompressed image.

sudo apt-get install squashfs-tools
mksquashfs image squash.img

Streaming Compression

To avoid making a separate temporary file the full size of the disk, you can stream into a squashfs image.

mkdir empty-dir
mksquashfs empty-dir squash.img -p 'sda_backup.img f 444 root root dd if=/dev/sda bs=4M'

Mounting a compressed partition image

First mount the squashfs image, then mount the partition image stored in the mounted squashfs image.

mkdir squash_mount
sudo mount squash.img squash_mount

Now you have the compressed image mounted, mount the image itself (that is inside the squashfs image)

mkdir compressed_image
sudo mount squash_mount/sda1_backup.img compressed_image

Now your image is mounted under compressed_image.

EDIT: If you wanted to simply restore the disk image onto a partition at this point (instead of mounting it to browse/read the contents), just dd the image at squash_mount/sda1_backup.img onto the destination instead of doing mount.

Mounting a compressed full disk image

This requires you to use a package called kpartx. kpartx allows you to mount individual partitions in a full disk image.

sudo apt-get install kpartx

First, mount your squashed partition that contains the full disk image

mkdir compressed_image
sudo mount squash.img compressed_image

Now you need to create devices for each of the partitions in the full disk image:

sudo kpartx -a compressed_image/sda_backup.img

This will create devices for the partitions in the full disk image at /dev/mapper/loopNpP where N is the number assigned for the loopback device, and P is the partition number. For example: /dev/mapper/loop0p1.

Now you have a way to mount the individual partitions in the full disk image:

mkdir fulldisk_part1
sudo mount /dev/mapper/loop0p1 fulldisk_part1

Try archivemount

root@srv1:/backup# archivemount windows-2003-S.gz /target/
Unrecognized archive format
root@srv1:/backup# archivemount -o formatraw windows-2003-S.gz /target/
Calculating uncompressed file size. Please wait.
root@srv1:/backup# ls /target/
data
root@srv1:/backup# file /target/data
/target/data: DOS/MBR boot sector; partition 1 : ID=0x7, start-CHS (0x0,1,1), end-CHS (0x3ff,254,63), startsector 63, 58717512 sectors, extended partition table (last)

archivemount is a FUSE-based file system for Unix variants, including Linux. Its purpose is to mount archives (i.e. tar, tar.gz, etc.) to a mount point where it can be read from or written to as with any other file system. This makes accessing the contents of the archive, which may be compressed, transparent to other programs, without decompressing them.

http://linuxaria.com/howto/how-to-mounts-an-archive-for-access-as-a-file-system

After mounting archive you can use it contents like regular file. Maybe get partition table, or convert, mount image with qemu tools.

squashfs useful for booting from image, but much complex for backuping.

If the image is read-only you can also use nbdkit (man page) and its xz plugin (xz should provide better compression and random access times than gzip).

Create the compressed partition image

dd if=/dev/sda1 bs=16M | xz -9 --block-size=16MiB - > sda1.img.xz

A --block-size option of 16 MiB should provide good random access performance.

Note: you may use alternative xz compression programs such as pixz which provides parallel compression, just make sure it splits the output in multiple small blocks, otherwise nbdkit has to decompress a lot of data. For example as of September 2015, pxz does not support this.

Serve it with nbdkit

nbdkit --no-fork --user nobody --group nobody -i 127.0.0.1 \
       xz file=sda1.img.xz

Connect to the NBD server

nbd-client 127.0.0.1 10809 /dev/nbd0 -nofork

Mount it read-only

mount -o ro /dev/nbd0 sda1

When done

umount /dev/nbd0
nbd-client -d /dev/nbd0

Stop nbdkit by pressing Ctrl+C (or with kill).